these two variables

Correlation in Psychology: Meaning, Types, Examples & coefficient

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

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Correlation means association – more precisely, it measures the extent to which two variables are related. There are three possible results of a correlational study: a positive correlation, a negative correlation, and no correlation.

• A positive correlation is a relationship between two variables in which both variables move in the same direction. Therefore, one variable increases as the other variable increases, or one variable decreases while the other decreases. An example of a positive correlation would be height and weight. Taller people tend to be heavier.

• A negative correlation is a relationship between two variables in which an increase in one variable is associated with a decrease in the other. An example of a negative correlation would be the height above sea level and temperature. As you climb the mountain (increase in height), it gets colder (decrease in temperature).

• A zero correlation exists when there is no relationship between two variables. For example, there is no relationship between the amount of tea drunk and the level of intelligence.

Scatter Plots

A correlation can be expressed visually. This is done by drawing a scatter plot (also known as a scattergram, scatter graph, scatter chart, or scatter diagram).

A scatter plot is a graphical display that shows the relationships or associations between two numerical variables (or co-variables), which are represented as points (or dots) for each pair of scores.

A scatter plot indicates the strength and direction of the correlation between the co-variables.

When you draw a scatter plot, it doesn’t matter which variable goes on the x-axis and which goes on the y-axis.

Remember, in correlations, we always deal with paired scores, so the values of the two variables taken together will be used to make the diagram.

Decide which variable goes on each axis and then simply put a cross at the point where the two values coincide.

Uses of Correlations

• If there is a relationship between two variables, we can make predictions about one from another.
• Concurrent validity (correlation between a new measure and an established measure).

Reliability

• Test-retest reliability (are measures consistent?).
• Inter-rater reliability (are observers consistent?).

Theory verification

• Predictive validity.

Correlation Coefficients

Instead of drawing a scatter plot, a correlation can be expressed numerically as a coefficient, ranging from -1 to +1. When working with continuous variables, the correlation coefficient to use is Pearson’s r.

The correlation coefficient ( r ) indicates the extent to which the pairs of numbers for these two variables lie on a straight line. Values over zero indicate a positive correlation, while values under zero indicate a negative correlation.

A correlation of –1 indicates a perfect negative correlation, meaning that as one variable goes up, the other goes down. A correlation of +1 indicates a perfect positive correlation, meaning that as one variable goes up, the other goes up.

There is no rule for determining what correlation size is considered strong, moderate, or weak. The interpretation of the coefficient depends on the topic of study.

When studying things that are difficult to measure, we should expect the correlation coefficients to be lower (e.g., above 0.4 to be relatively strong). When we are studying things that are easier to measure, such as socioeconomic status, we expect higher correlations (e.g., above 0.75 to be relatively strong).)

In these kinds of studies, we rarely see correlations above 0.6. For this kind of data, we generally consider correlations above 0.4 to be relatively strong; correlations between 0.2 and 0.4 are moderate, and those below 0.2 are considered weak.

When we are studying things that are more easily countable, we expect higher correlations. For example, with demographic data, we generally consider correlations above 0.75 to be relatively strong; correlations between 0.45 and 0.75 are moderate, and those below 0.45 are considered weak.

Correlation vs. Causation

Causation means that one variable (often called the predictor variable or independent variable) causes the other (often called the outcome variable or dependent variable).

Experiments can be conducted to establish causation. An experiment isolates and manipulates the independent variable to observe its effect on the dependent variable and controls the environment in order that extraneous variables may be eliminated.

A correlation between variables, however, does not automatically mean that the change in one variable is the cause of the change in the values of the other variable. A correlation only shows if there is a relationship between variables.

While variables are sometimes correlated because one does cause the other, it could also be that some other factor, a confounding variable , is actually causing the systematic movement in our variables of interest.

Correlation does not always prove causation, as a third variable may be involved. For example, being a patient in a hospital is correlated with dying, but this does not mean that one event causes the other, as another third variable might be involved (such as diet and level of exercise).

“Correlation is not causation” means that just because two variables are related it does not necessarily mean that one causes the other.

A correlation identifies variables and looks for a relationship between them. An experiment tests the effect that an independent variable has upon a dependent variable but a correlation looks for a relationship between two variables.

This means that the experiment can predict cause and effect (causation) but a correlation can only predict a relationship, as another extraneous variable may be involved that it not known about.

1. Correlation allows the researcher to investigate naturally occurring variables that may be unethical or impractical to test experimentally. For example, it would be unethical to conduct an experiment on whether smoking causes lung cancer.

2 . Correlation allows the researcher to clearly and easily see if there is a relationship between variables. This can then be displayed in a graphical form.

Limitations

1 . Correlation is not and cannot be taken to imply causation. Even if there is a very strong association between two variables, we cannot assume that one causes the other.

For example, suppose we found a positive correlation between watching violence on T.V. and violent behavior in adolescence.

It could be that the cause of both these is a third (extraneous) variable – for example, growing up in a violent home – and that both the watching of T.V. and the violent behavior is the outcome of this.

2 . Correlation does not allow us to go beyond the given data. For example, suppose it was found that there was an association between time spent on homework (1/2 hour to 3 hours) and the number of G.C.S.E. passes (1 to 6).

It would not be legitimate to infer from this that spending 6 hours on homework would likely generate 12 G.C.S.E. passes.

How do you know if a study is correlational?

A study is considered correlational if it examines the relationship between two or more variables without manipulating them. In other words, the study does not involve the manipulation of an independent variable to see how it affects a dependent variable.

One way to identify a correlational study is to look for language that suggests a relationship between variables rather than cause and effect.

For example, the study may use phrases like “associated with,” “related to,” or “predicts” when describing the variables being studied.

Another way to identify a correlational study is to look for information about how the variables were measured. Correlational studies typically involve measuring variables using self-report surveys, questionnaires, or other measures of naturally occurring behavior.

Finally, a correlational study may include statistical analyses such as correlation coefficients or regression analyses to examine the strength and direction of the relationship between variables.

Why is a correlational study used?

Correlational studies are particularly useful when it is not possible or ethical to manipulate one of the variables.

For example, it would not be ethical to manipulate someone’s age or gender. However, researchers may still want to understand how these variables relate to outcomes such as health or behavior.

Additionally, correlational studies can be used to generate hypotheses and guide further research.

If a correlational study finds a significant relationship between two variables, this can suggest a possible causal relationship that can be further explored in future research.

What is the goal of correlational research?

The ultimate goal of correlational research is to increase our understanding of how different variables are related and to identify patterns in those relationships.

This information can then be used to generate hypotheses and guide further research aimed at establishing causality.

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What Is a Correlation?

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

 James Lacy, MLS, is a fact-checker and researcher.

What Is a Correlation Coefficient?

Scatter plots and correlation, strong vs. weak correlations, correlation does not equal causation, illusory correlations, frequently asked questions.

A correlation means that there is a relationship between two or more variables. This does not imply, however, that there is necessarily a cause or effect relationship between them. Instead, it simply means that there is some type of relationship, meaning they change together at a constant rate.

A correlation coefficient is a number that expresses the strength of the relationship between the two variables.

At a Glance

Correlation can help researchers understand if there is an association between two variables of interest. Such relationships can be positive, meaning they move in the same direction together, or negative, meaning that as one goes up, the other goes down. Correlations can be visualized using scatter plots to show how measurements of a variable change along an x- and y-axis.

It is important to remember that while correlations can help show a relationship, correlation does not indicate causation.

A correlation coefficient, often expressed as r , indicates a measure of the direction and strength of a relationship between two variables. When the r value is closer to +1 or -1, it indicates that there is a stronger linear relationship between the two variables.

Correlational studies are quite common in psychology, particularly because some things are impossible to recreate or research in a lab setting .

Instead of performing an experiment , researchers may collect data to look at possible relationships between variables. From the data they collect and its analysis, researchers then make inferences and predictions about the nature of the relationships between variables.

Helpful Hint

A correlation is a statistical measurement of the relationship between two variables. Remember this handy rule: The closer the correlation is to 0, the weaker it is. The closer it is to +/-1, the stronger it is.

Types of Correlation

Correlation strength ranges from -1 to +1.

Positive Correlation

A correlation of +1 indicates a perfect positive correlation, meaning that both variables move in the same direction together. In other words, +1 is the strong positive correlation you can find.

Negative Correlation

A correlation of –1 indicates a perfect negative correlation, meaning that as one variable goes up, the other goes down.

Zero Correlation

A zero correlation suggests that the correlation statistic does not indicate a relationship between the two variables. This does not mean that there is no relationship at all; it simply means that there is not a linear relationship. A zero correlation is often indicated using the abbreviation r = 0.

Scatter plots (also called scatter charts, scattergrams, and scatter diagrams) are used to plot variables on a chart to observe the associations or relationships between them. The horizontal axis represents one variable, and the vertical axis represents the other.

Investopedia

Each point on the plot is a different measurement. From those measurements, a trend line can be calculated. The correlation coefficient is the slope of that line. When the correlation is weak ( r is close to zero), the line is hard to distinguish. When the correlation is strong ( r is close to 1), the line will be more apparent.

Correlations can be confusing, and many people equate positive with strong and negative with weak. A relationship between two variables can be negative, but that doesn't mean that the relationship isn't strong.

• A weak positive correlation indicates that, although both variables tend to go up in response to one another, the relationship is not very strong.
• A strong negative correlation , on the other hand, indicates a strong connection between the two variables, but that one goes up whenever the other one goes down.

For example, a correlation of -0.97 is a strong negative correlation, whereas a correlation of 0.10 indicates a weak positive correlation. A correlation of +0.10 is weaker than -0.74, and a correlation of -0.98 is stronger than +0.79.

Correlation does not equal causation. Just because two variables have a relationship does not mean that changes in one variable cause changes in the other.

Correlations tell us that there is a relationship between variables, but this does not necessarily mean that one variable causes the other to change.

An oft-cited example is the correlation between ice cream consumption and homicide rates. Studies have found a correlation between increased ice cream sales and spikes in homicides. However, eating ice cream does not cause you to commit murder. Instead, there is a third variable: heat. Both variables increase during summertime .

An illusory correlation is the perception of a relationship between two variables when only a minor relationship—or none at all—actually exists. An illusory correlation does not always mean inferring causation; it can also mean inferring a relationship between two variables when one does not exist.

For example, people sometimes assume that, because two events occurred together at one point in the past, one event must be the cause of the other. These illusory correlations can occur both in scientific investigations and in real-world situations.

Stereotypes are a good example of illusory correlations. Research has shown that people tend to assume that certain groups and traits occur together and frequently overestimate the strength of the association between the two variables.

For example, suppose someone holds the mistaken belief that all people from small towns are extremely kind. When they meet a very kind person, their immediate assumption might be that the person is from a small town, despite the fact that kindness is not related to city population.

What This Means For You

Psychology research frequently uses correlations, but it's essential to understand that correlation is not the same as causation. Confusing correlation with causation assumes a cause-effect relationship that might not exist. While correlation can help you see that there is a relationship (and tell you how strong that relationship is), only experimental research can reveal a causal connection.

You can calculate the correlation coefficient in a few different ways, with the same result. The general formula is r XY =COV XY /(S X S Y ) , which is the covariance between the two variables, divided by the product of their standard deviations:

In the cell in which you want the correlation coefficient to appear, enter =CORREL(A2:A7,B2:B7), where A2:A7 and B2:B7 are the variable lists to compare. Press Enter .

Finding the linear correlation coefficient requires a long, difficult calculation, so most people use a calculator or software such as Excel or a statistics program.

Correlations range from -1.00 to +1.00. The correlation coefficient (expressed as r ) shows the direction and strength of a relationship between two variables. The closer the r value is to +1 or -1, the stronger the linear relationship between the two variables is.

Correlations indicate a relationship between two variables, but one doesn't necessarily cause the other to change.

Mukaka M. A guide to appropriate use of correlation coefficient in medical research .  Malawi Med J . 2012;24(3):69-71.

Heath W.  Psychology Research Methods: Connecting Research to Students’ Lives . Cambridge University Press.

Chen DT. When correlation does not imply causation: Why your gut microbes may not (yet) be a silver bullet to all your problems . Harvard University.

Association for Psychological Science.  Research states that prejudice comes from a basic human need and way of thinking .

Correlation and regression . In: Swinscow TDV. Statistics at Square One . The BMJ.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Statistics Made Easy

Pearson Correlation Coefficient

The Pearson correlation coefficient (also known as the “product-moment correlation coefficient”) is a measure of the linear association between two variables  X  and  Y.  It has a value between -1 and 1 where:

• -1 indicates a perfectly negative linear correlation between two variables
• 0 indicates no linear correlation between two variables
• 1 indicates a perfectly positive linear correlation between two variables

The Formula to Find the Pearson Correlation Coefficient

The formula to find the Pearson correlation coefficient, denoted as  r , for a sample of data is ( via Wikipedia ):

You will likely never have to compute this formula by hand since you can use software to do this for you, but it’s helpful to have an understanding of what exactly this formula is doing by walking through an example.

Suppose we have the following dataset:

If we plotted these (X, Y) pairs on a scatterplot, it would look like this:

Just from looking at this scatterplot we can tell that there is a positive association between variables X and Y: when X increases, Y tends to increase as well.  But to quantify exactly how positively associated these two variables are, we need to find the Pearson correlation coefficient.

Let’s focus on just the numerator of the formula:

For each (X, Y) pair in our dataset, we need to find the difference between the x value and the mean x value, the difference between the y value and the mean y value, then multiply these two numbers together.

For example, our first (X, Y) pair is (2, 2). The mean x value in this dataset is 5 and the mean y value in this dataset is 7. So, the difference between the x value in this pair and the mean x value is 2 – 5 = -3. The difference between the y value in this pair and the mean y value is 2 – 7 = -5. Then, when we multiply these two numbers together we get -3 * -5 = 15.

Here’s a visual look at what we just did: 

Next, we just need to do this for every single pair:

The last step to get the numerator of the formula is to simply add up all of these values:

15 + 3 +3 + 15 = 36

Next, the denominator of the formula tells us to find the sum of all the squared differences for both x and y, then multiply these two numbers together, then take the square root:

So, first we’ll find the sum of the squared differences for both x and y:

Then we’ll multiply these two numbers together: 20 * 68 = 1,360.

Lastly, we’ll take the square root: √ 1,360 = 36.88

So, we found the numerator of the formula to be 36 and the denominator to be 36.88. This means that our Pearson correlation coefficient is r = 36 / 36.88 = 0.976

This number is close to 1, which indicates that there is a strong positive linear relationship between our variables  X  and  Y . This confirms the relationship that we saw in the scatterplot.

Visualizing Correlations

Recall that a Pearson correlation coefficient tells us the type of linear relationship (positive, negative, none) between two variables as well as the strength of that relationship (weak, moderate, strong).

When we make a scatterplot of two variables, we can  see  the actual relationship between two variables. Here are the many different types of linear relationships we might see:

Pearson correlation coefficient: 0.94

Weak, positive relationship:  As the variable on the x-axis increases, the variable on the y-axis increases as well. The dots are fairly spread out, which indicates a weak relationship.

Pearson correlation coefficient: 0.44

No relationship:  There is no clear relationship (positive or negative) between the variables. 

Pearson correlation coefficient: 0.03

Strong, negative relationship:  As the variable on the x-axis increases, the variable on the y-axis decreases. The dots are packed tightly together, which indicates a strong relationship.

Pearson correlation coefficient: -0.87

Weak, negative relationship:  As the variable on the x-axis increases, the variable on the y-axis decreases. The dots are fairly spread out, which indicates a weak relationship.

Pearson correlation coefficient: – 0.46

Testing for Significance of a Pearson Correlation Coefficient

When we find the Pearson correlation coefficient for a set of data, we’re often working with a  sample  of data that comes from a larger  population . This means that it’s possible to find a non-zero correlation for two variables even if they’re actually not correlated in the overall population.

For example, suppose we make a scatterplot for variables  X  and  Y  for every data point in the entire population and it looks like this:

Clearly these two variables are not correlated. However, it’s possible that when we take a sample of 10 points from the population, we choose the following points:

We may find that the Pearson correlation coefficient for this sample of points is 0.93, which indicates a strong positive correlation despite the population correlation being zero.

In order to test for whether or not a correlation between two variables is statistically significant, we can find the following test statistic:

Test statistic T = r * √ (n-2) / (1-r 2 )

where n  is the number of pairs in our sample,  r is the Pearson correlation coefficient, and test statistic T follows a t distribution with n-2 degrees of freedom.

Let’s walk through an example of how to test for the significance of a Pearson correlation coefficient.

The following dataset shows the height and weight of 12 individuals:

The scatterplot below shows the value of these two variables:

The Pearson correlation coefficient for these two variables is r = 0.836.

The test statistic T = .836 * √ (12 -2) / (1-.836 2 ) = 4.804.

According to our t distribution calculator , a t score of 4.804 with 10 degrees of freedom has a p-value of .0007. Since .0007 < .05, we can conclude that the correlation between weight and height in this example is statistically significant at alpha = .05.

While a Pearson correlation coefficient can be useful in telling us whether or not two variables have a linear association, we must keep three things in mind when interpreting a Pearson correlation coefficient:

1. Correlation does not imply causation. Just because two variables are correlated does not mean that one is necessarily  causing  the other to occur more or less often. A classic example of this is the positive correlation between ice cream sales and shark attacks. When ice cream sales increase during certain times of the year, shark attacks also tend to increase.

Does this mean ice cream consumption is causing  shark attacks? Of course not! It just means that during the summer, both ice cream consumption and shark attacks tend to increase since ice cream is more popular during the summer and more people go in the ocean during the summer.

2. Correlations are sensitive to outliers.  One extreme outlier can dramatically change a Pearson correlation coefficient. Consider the example below:

Variables  X  and  Y  have a Pearson correlation coefficient of 0.00 . But imagine that we have one outlier in the dataset:

Now the Pearson correlation coefficient for these two variables is 0.878 . This one outlier changes everything. This is why, when you calculate the correlation for two variables, it’s a good idea to visualize the variables using a scatterplot to check for outliers. 

3. A Pearson correlation coefficient does not capture nonlinear relationships between two variables.  Imagine that we have two variables with the following relationship:

The Pearson correlation coefficient for these two variables is 0.00 because they have no linear relationship. However, these two variables do have a nonlinear relationship: The y values are simply the x values squared.

When using the Pearson correlation coefficient, keep in mind that you’re merely testing to see if two variables are linearly related. Even if a Pearson correlation coefficient tells us that two variables are uncorrelated, they could still have some type of nonlinear relationship. This is another reason that it’s helpful to create a scatterplot when analyzing the relationship between two variables – it may help you detect a nonlinear relationship.

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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Thank you for a good practical explanation.

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Keyboard Shortcuts

Relationships between two variables, introduction.

Let's get started! Here is what you will learn in this lesson.

Learning objectives for this lesson

Upon completion of this lesson, you should be able to do the following:

• Understand the relationship between the slope of the regression line and correlation,
• Comprehend the meaning of the Coefficient of Determination, R 2,
• Know how to determine which variable is a response and which is an explanatory in a regression equation,
• Understand that correlation measures the strength of a linear relationship between two variables,
• Realize how outliers can influence a regression equation, and
• Determine if variables are categorical or quantitative.

Examining Relationships Between Two Variables

Previously we considered the distribution of a single quantitative variable. Now we will study the relationship between two variables where both variables are qualitative, i.e. categorical, or quantitative. When we consider the relationship between two variables, there are three possibilities:

• Both variables are categorical. We analyze an association through a comparison of conditional probabilities and graphically represent the data using contingency tables. Examples of categorical variables are gender and class standing.
• Both variables are quantitative. To analyze this situation we consider how one variable, called a response variable, changes in relation to changes in the other variable called an explanatory variable. Graphically we use scatterplots to display two quantitative variables. Examples are age, height, weight (i.e. things that are measured).
• One variable is categorical and the other is quantitative, for instance height and gender. These are best compared by using side-by-side boxplots to display any differences or similarities in the center and variability of the quantitative variable (e.g. height) across the categories (e.g. Male and Female).

Correlation Coefficient Calculator

What is the correlation coefficient, how to use this correlation calculator with steps, pearson correlation coefficient formula, spearman correlation coefficient, kendall rank correlation (tau), matthews correlation (pearson phi).

Welcome to Omni's correlation coefficient calculator! Here you can learn all there is about this important statistical concept. Apart from discussing the general definition of correlation and the intuition behind it, we will also cover in detail the formulas for the four most popular correlation coefficients :

• Pearson correlation;
• Spearman correlation;
• Kendall tau correlation (including the variants); and
• Matthews correlation (MCC, a.k.a. Pearson phi).

As a bonus, we will also explain how Pearson correlation is linked to simple linear regression . We will start, however, by explaining what the correlation coefficient is all about. Let's go!

Correlation coefficients are measures of the strength and direction of relation between two random variables. The type of relationship that is being measured varies depending on the coefficient. In general, however, they all describe the co-changeability between the variables in question – how increasing (or decreasing) the value of one variable affects the value of the other variable – does it tend to increase or decrease?

Importantly, correlation coefficients are all normalized , i.e., they assume values between -1 and +1. Values of ±1 indicate the strongest possible relationship between variables, and a value of 0 means there's no relationship at all.

And that's it when it comes to the general definition of correlation! If you wonder how to calculate correlation , the best answer is to... use Omni's correlation coefficient calculator 😊! It allows you to easily compute all of the different coefficients in no time. In the next section, we explain how to use this tool in the most effective way.

If you wonder how to calculate correlation by hand, you will find all the necessary formulas and definitions for several correlation coefficients in the following sections.

To use our correlation coefficient calculator:

• Kendall rank correlation; or
• Matthews correlation.
• Input your data into the rows. When at least three points (both an x and y coordinate) are in place, it will give you your result.
• Be aware that this is a correlation calculator with steps ! If you turn on the option Show calculation details? , our tool will show you the intermediate stages of calculations. This is very useful when you need to verify the correctness of your calculations.
• 0.8 ≤ |corr| ≤ 1.0 very strong ;
• 0.6 ≤ |corr| < 0.8 strong ;
• 0.4 ≤ |corr| < 0.6 moderate ;
• 0.2 ≤ |corr| < 0.4 weak ; and
• 0.0 ≤ |corr| < 0.2 very weak .

The Pearson correlation between two variables X and Y is defined as the covariance between these variables divided by the product of their respective standard deviations :

This translates into the following explicit formula:

where x ‾ \overline{x} x and y ‾ \overline{y} y ​ stand for the average of the sample, x 1 , . . . , x n x_1, ..., x_n x 1 ​ , ... , x n ​ and y 1 , . . . , y n y_1, ..., y_n y 1 ​ , ... , y n ​ , respectively.

Remember that the Pearson correlation detects only a linear relationship – a low value of Pearson correlation doesn't mean that there is no relationship at all! The two variables may be strongly related, yet their relationship may not be linear but of some other type.

In least squares regression Y = a X + b Y = aX + b Y = a X + b , the square of the Pearson correlation between X X X and Y Y Y is equal to the coefficient of determination, R² , which expresses the fraction of the variance in Y Y Y that is explained by X X X :

If you want to discover more about the Pearson correlation, visit our dedicated Pearson correlation calculator website .

The Spearman coefficient is closely related to the Pearson coefficient. Namely, the Spearman rank correlation between X X X and Y Y Y is defined as the Pearson correlation between the rank variables r ( X ) r(X) r ( X ) and r ( Y ) r(Y) r ( Y ) . That is, the formula for Spearman's rank correlation r h o rho r h o reads:

To obtain the rank variables, you just need to order the observations (in each sample separately) from lowest to highest. The smallest observation then gets rank 1 , the second-smallest rank 2 , and so on – the highest observation will have rank n . You only need to be careful when the same value appears in the data set more than once (we say there are ties ). If this happens, assign to all these identical observations the rank equal to the arithmetic mean of the ranks you would assign to these observations where they all had different values.

The Spearman correlation is sensitive to the monotonic relationship between the variables, so it is more general than the Pearson correlation – it can capture, e.g., quadratic or exponential relationships.

There is also a simpler and more explicit formula for Spearman correlation , but it holds only if there are no ties in either of our samples . More details await you in the Spearman's rank correlation calculator .

We most often denote Kendall's rank correlation by the Greek letter τ (tau), and that's why it's often referred to as Kendall tau.

Consider two samples, x and y , each of size n : x 1 , ..., x n and y 1 , ..., y n . Clearly, there are n(n+1)/2 possible pairs of x and y .

We have to go through all these pairs one by one and count the number of concordant and discordant pairs . Namely, for two pairs (x i , y i ) and (x j , y j ) we have the following rules:

• If x i < x j and y i < y j then this pair is concordant.
• If x i > x j and y i > y j then this pair is concordant.
• If x i < x j and y i > y j then this pair is discordant.
• If x i > x j and y i < y j then this pair is discordant.

The Kendall rank correlation coefficient formula reads:

• C C C – Number of concordant pairs; and
• D D D – Number of discordant pairs.

That is, τ \tau τ is the difference between the number of concordant and discordant pairs divided by the total number of all pairs.

Easy, don't you think? However, it is so only if there are no ties . That is, there are no repeating values in both sample x and sample y . If there are ties, there are two additional variants of Kendall tau. (Fortunately, our correlation coefficient calculator can calculate them all!) To define them, we need to distinguish different kinds of ties:

• If x i = x j and y i ≠ y j then we have a tie in x .
• If x i ≠ x j and y i = y j then we have a tie in y .
• If x i = x j and y i = y j then we have a double tie .

The Kendall rank tau-b correlation coefficient formula reads:

• T x T_x T x ​ – Number of ties in x x x ; and
• T y T_y T y ​ – Number of ties in y y y .

Use tau-b if the two variables have the same number of possible values (before ranking). In other words, if you can summarize the data in a square contingency table . An example of such a situation is when both variables use a 5-point Likert scale: strongly disagree, disagree, neither agree nor disagree, agree, or strongly agree .

If your data is assembled in a rectangular non-square contingency table , or, in other words, if the two variables have a different number of possible values , then use tau-c (sometimes called Stuart-Kendall tau-c ):

• m m m – m i n ( r , c ) {\rm min}(r,c) min ( r , c ) ;
• r r r – The number of rows in the contingency table; and
• c c c – The number of columns in the contingency table.

But where is tau-a , you may think? Fortunately, tau-a is defined in the same simple way as before (when we had no ties):

Kendall tau correlation coefficient is sensitive monotonic relationship between the variables.

The Matthews correlation (abbreviated as MCC, also known as Pearson phi) measures the quality of binary classifications . Most often, we can encounter it in machine learning and biology/medicine-related data.

To write down the formula for the Matthews correlation coefficient we need to assemble our data in a 2x2 contingency table, which in this context is also called the confusion matrix :

where we use the following quite standard abbreviations::

• TP – True positive;
• FP – False positive;
• TN – True negative; and
• FN – False negative.

Matthews correlation is given by the following formula:

The interpretation of this coefficient is a bit different now:

• +1 means we have a perfect prediction;
• 0 means we don't have any valid information; and
• -1 means we have a complete inconsistency between prediction and the actual outcome.

If you're interested, don't hesitate to visit our Matthews correlation coefficient calculator .

What does a positive correlation mean?

If the value of correlation is positive, then the two variables under consideration tend to change in the same direction : when the first one increases, the other tends to increase, and when the first one decreases, then the other one tends to decrease as well.

What does a negative correlation mean?

If the value of correlation is positive, then the two variables under consideration tend to change in the opposite directions : when the first one increases, the other tends to decrease, and when the first one decreases, then the other one tends to increase.

How to read a correlation matrix?

A correlation matrix is a table that shows the values of a correlation coefficient between all possible pairs of several variables . It always has ones at the main diagonal (this is the correlation of a variable with itself) and is symmetric (because the correlation between X and Y is the same as between Y and X). For these reasons, the redundant cells sometimes get trimmed . If there is some color-coding , make sure to check what it means: it may either illustrate the strength and direction of correlation or its statistical significance.

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7.1 Systems of Linear Equations: Two Variables

Learning objectives.

In this section, you will:

• Solve systems of equations by graphing.
• Solve systems of equations by substitution.
• Solve systems of equations by addition.
• Identify inconsistent systems of equations containing two variables.
• Express the solution of a system of dependent equations containing two variables.

A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.

Introduction to Systems of Equations

In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.

In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.

The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.

In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y -intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.

Another type of system of linear equations is an inconsistent system , which is one in which the equations represent two parallel lines. The lines have the same slope and different y- intercepts. There are no points common to both lines; hence, there is no solution to the system.

Types of Linear Systems

There are three types of systems of linear equations in two variables, and three types of solutions.

• An independent system has exactly one solution pair ( x , y ) . ( x , y ) . The point where the two lines intersect is the only solution.
• An inconsistent system has no solution. Notice that the two lines are parallel and will never intersect.
• A dependent system has infinitely many solutions. The lines are coincident. They are the same line, so every coordinate pair on the line is a solution to both equations.

Figure 2 compares graphical representations of each type of system.

Given a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.

• Substitute the ordered pair into each equation in the system.
• Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.

Determining Whether an Ordered Pair Is a Solution to a System of Equations

Determine whether the ordered pair ( 5 , 1 ) ( 5 , 1 ) is a solution to the given system of equations.

Substitute the ordered pair ( 5 , 1 ) ( 5 , 1 ) into both equations.

The ordered pair ( 5 , 1 ) ( 5 , 1 ) satisfies both equations, so it is the solution to the system.

We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines. See Figure 3 .

Determine whether the ordered pair ( 8 , 5 ) ( 8 , 5 ) is a solution to the following system.

Solving Systems of Equations by Graphing

There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.

Solving a System of Equations in Two Variables by Graphing

Solve the following system of equations by graphing. Identify the type of system.

Solve the first equation for y . y .

Solve the second equation for y . y .

Graph both equations on the same set of axes as in Figure 4 .

The lines appear to intersect at the point ( −3, −2 ) . ( −3, −2 ) . We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.

The solution to the system is the ordered pair ( −3, −2 ) , ( −3, −2 ) , so the system is independent.

Solve the following system of equations by graphing.

Can graphing be used if the system is inconsistent or dependent?

Yes, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.

Solving Systems of Equations by Substitution

Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method , in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.

Given a system of two equations in two variables, solve using the substitution method.

• Solve one of the two equations for one of the variables in terms of the other.
• Substitute the expression for this variable into the second equation, then solve for the remaining variable.
• Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.
• Check the solution in both equations.

Solving a System of Equations in Two Variables by Substitution

Solve the following system of equations by substitution.

First, we will solve the first equation for y . y .

Now we can substitute the expression x −5 x −5 for y y in the second equation.

Now, we substitute x = 8 x = 8 into the first equation and solve for y . y .

Our solution is ( 8 , 3 ) . ( 8 , 3 ) .

Check the solution by substituting ( 8 , 3 ) ( 8 , 3 ) into both equations.

Can the substitution method be used to solve any linear system in two variables?

Yes, but the method works best if one of the equations contains a coefficient of 1 or –1 so that we do not have to deal with fractions.

Solving Systems of Equations in Two Variables by the Addition Method

A third method of solving systems of linear equations is the addition method . In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.

Given a system of equations, solve using the addition method.

• Write both equations with x - and y -variables on the left side of the equal sign and constants on the right.
• Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.
• Solve the resulting equation for the remaining variable.
• Substitute that value into one of the original equations and solve for the second variable.
• Check the solution by substituting the values into the other equation.

Solving a System by the Addition Method

Solve the given system of equations by addition.

Both equations are already set equal to a constant. Notice that the coefficient of x x in the second equation, –1, is the opposite of the coefficient of x x in the first equation, 1. We can add the two equations to eliminate x x without needing to multiply by a constant.

Now that we have eliminated x , x , we can solve the resulting equation for y . y .

Then, we substitute this value for y y into one of the original equations and solve for x . x .

The solution to this system is ( − 7 3 , 2 3 ) . ( − 7 3 , 2 3 ) .

Check the solution in the first equation.

We gain an important perspective on systems of equations by looking at the graphical representation. See Figure 5 to find that the equations intersect at the solution. We do not need to ask whether there may be a second solution because observing the graph confirms that the system has exactly one solution.

Using the Addition Method When Multiplication of One Equation Is Required

Solve the given system of equations by the addition method .

Adding these equations as presented will not eliminate a variable. However, we see that the first equation has 3 x 3 x in it and the second equation has x . x . So if we multiply the second equation by −3 , −3 , the x -terms will add to zero.

Now, let’s add them.

For the last step, we substitute y = −4 y = −4 into one of the original equations and solve for x . x .

Our solution is the ordered pair ( 3 , −4 ) . ( 3 , −4 ) . See Figure 6 . Check the solution in the original second equation.

Solve the system of equations by addition.

Using the Addition Method When Multiplication of Both Equations Is Required

Solve the given system of equations in two variables by addition.

One equation has 2 x 2 x and the other has 5 x . 5 x . The least common multiple is 10 x 10 x so we will have to multiply both equations by a constant in order to eliminate one variable. Let’s eliminate x x by multiplying the first equation by −5 −5 and the second equation by 2. 2.

Then, we add the two equations together.

Substitute y = −4 y = −4 into the original first equation.

The solution is ( −2 , −4 ) . ( −2 , −4 ) . Check it in the other equation.

See Figure 7 .

Using the Addition Method in Systems of Equations Containing Fractions

First clear each equation of fractions by multiplying both sides of the equation by the least common denominator.

Now multiply the second equation by −1 −1 so that we can eliminate the x -variable.

Add the two equations to eliminate the x -variable and solve the resulting equation.

Substitute y = 7 y = 7 into the first equation.

The solution is ( 11 2 , 7 ) . ( 11 2 , 7 ) . Check it in the other equation.

Identifying Inconsistent Systems of Equations Containing Two Variables

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different y y -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as 12 = 0. 12 = 0.

Solving an Inconsistent System of Equations

Solve the following system of equations.

We can approach this problem in two ways. Because one equation is already solved for x , x , the most obvious step is to use substitution.

Clearly, this statement is a contradiction because 9 ≠ 13. 9 ≠ 13. Therefore, the system has no solution.

The second approach would be to first manipulate the equations so that they are both in slope-intercept form. We manipulate the first equation as follows.

We then convert the second equation expressed to slope-intercept form.

Comparing the equations, we see that they have the same slope but different y -intercepts. Therefore, the lines are parallel and do not intersect.

Writing the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect eventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The graphs of the equations in this example are shown in Figure 8 .

Solve the following system of equations in two variables.

Expressing the Solution of a System of Dependent Equations Containing Two Variables

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as 0 = 0. 0 = 0.

Finding a Solution to a Dependent System of Linear Equations

Find a solution to the system of equations using the addition method .

With the addition method, we want to eliminate one of the variables by adding the equations. In this case, let’s focus on eliminating x . x . If we multiply both sides of the first equation by −3 , −3 , then we will be able to eliminate the x x -variable.

Now add the equations.

We can see that there will be an infinite number of solutions that satisfy both equations.

If we rewrote both equations in the slope-intercept form, we might know what the solution would look like before adding. Let’s look at what happens when we convert the system to slope-intercept form.

See Figure 9 . Notice the results are the same. The general solution to the system is ( x , − 1 3 x + 2 3 ) . ( x , − 1 3 x + 2 3 ) .

Using Systems of Equations to Investigate Profits

Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation R = x p , R = x p , where x = x = quantity and p = p = price. The revenue function is shown in orange in Figure 10 .

The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in Figure 10 . The x x -axis represents quantity in hundreds of units. The y -axis represents either cost or revenue in hundreds of dollars.

The point at which the two lines intersect is called the break-even point . We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.

The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function is the revenue function minus the cost function, written as P ( x ) = R ( x ) − C ( x ) . P ( x ) = R ( x ) − C ( x ) . Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

Finding the Break-Even Point and the Profit Function Using Substitution

Given the cost function C ( x ) = 0.85 x + 35,000 C ( x ) = 0.85 x + 35,000 and the revenue function R ( x ) = 1.55 x , R ( x ) = 1.55 x , find the break-even point and the profit function.

Write the system of equations using y y to replace function notation.

Substitute the expression 0.85 x + 35,000 0.85 x + 35,000 from the first equation into the second equation and solve for x . x .

Then, we substitute x = 50,000 x = 50,000 into either the cost function or the revenue function.

The break-even point is ( 50,000 , 77,500 ) . ( 50,000 , 77,500 ) .

The profit function is found using the formula P ( x ) = R ( x ) − C ( x ) . P ( x ) = R ( x ) − C ( x ) .

The profit function is P ( x ) = 0.7 x −35,000. P ( x ) = 0.7 x −35,000.

The cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a profit, the business must produce and sell more than 50,000 units. See Figure 11 .

We see from the graph in Figure 12 that the profit function has a negative value until x = 50,000 , x = 50,000 , when the graph crosses the x -axis. Then, the graph emerges into positive y -values and continues on this path as the profit function is a straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the left of the break-even point represents operating at a loss.

Writing and Solving a System of Equations in Two Variables

The cost of a ticket to the circus is $ 25.00 $ 25.00 for children and $ 50.00 $ 50.00 for adults. On a certain day, attendance at the circus is 2,000 2,000 and the total gate revenue is $ 70,000. $ 70,000. How many children and how many adults bought tickets?

Let c = the number of children and a = the number of adults in attendance.

The total number of people is 2,000. 2,000. We can use this to write an equation for the number of people at the circus that day.

The revenue from all children can be found by multiplying $ 25.00 $ 25.00 by the number of children, 25 c . 25 c . The revenue from all adults can be found by multiplying $ 50.00 $ 50.00 by the number of adults, 50 a . 50 a . The total revenue is $ 70,000. $ 70,000. We can use this to write an equation for the revenue.

We now have a system of linear equations in two variables.

In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either c c or a . a . We will solve for a . a .

Substitute the expression 2,000 − c 2,000 − c in the second equation for a a and solve for c . c .

Substitute c = 1,200 c = 1,200 into the first equation to solve for a . a .

We find that 1,200 1,200 children and 800 800 adults bought tickets to the circus that day.

Meal tickets at the circus cost $ 4.00 $ 4.00 for children and $ 12.00 $ 12.00 for adults. If 1,650 1,650 meal tickets were bought for a total of $ 14,200 , $ 14,200 , how many children and how many adults bought meal tickets?

Access these online resources for additional instruction and practice with systems of linear equations.

• Solving Systems of Equations Using Substitution
• Solving Systems of Equations Using Elimination
• Applications of Systems of Equations

7.1 Section Exercises

Can a system of linear equations have exactly two solutions? Explain why or why not.

If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company’s profit margins.

If you are solving a break-even analysis and get a negative break-even point, explain what this signifies for the company?

If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?

Given a system of equations, explain at least two different methods of solving that system.

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.

5 x − y = 4 x + 6 y = 2 5 x − y = 4 x + 6 y = 2 and ( 4 , 0 ) ( 4 , 0 )

−3 x − 5 y = 13 − x + 4 y = 10 −3 x − 5 y = 13 − x + 4 y = 10 and ( −6 , 1 ) ( −6 , 1 )

3 x + 7 y = 1 2 x + 4 y = 0 3 x + 7 y = 1 2 x + 4 y = 0 and ( 2 , 3 ) ( 2 , 3 )

−2 x + 5 y = 7 2 x + 9 y = 7 −2 x + 5 y = 7 2 x + 9 y = 7 and ( −1 , 1 ) ( −1 , 1 )

x + 8 y = 43 3 x −2 y = −1 x + 8 y = 43 3 x −2 y = −1 and ( 3 , 5 ) ( 3 , 5 )

For the following exercises, solve each system by substitution.

x + 3 y = 5 2 x + 3 y = 4 x + 3 y = 5 2 x + 3 y = 4

3 x −2 y = 18 5 x + 10 y = −10 3 x −2 y = 18 5 x + 10 y = −10

4 x + 2 y = −10 3 x + 9 y = 0 4 x + 2 y = −10 3 x + 9 y = 0

2 x + 4 y = −3.8 9 x −5 y = 1.3 2 x + 4 y = −3.8 9 x −5 y = 1.3

− 2 x + 3 y = 1.2 − 3 x − 6 y = 1.8 − 2 x + 3 y = 1.2 − 3 x − 6 y = 1.8

x −0.2 y = 1 −10 x + 2 y = 5 x −0.2 y = 1 −10 x + 2 y = 5

3 x + 5 y = 9 30 x + 50 y = −90 3 x + 5 y = 9 30 x + 50 y = −90

−3 x + y = 2 12 x −4 y = −8 −3 x + y = 2 12 x −4 y = −8

1 2 x + 1 3 y = 16 1 6 x + 1 4 y = 9 1 2 x + 1 3 y = 16 1 6 x + 1 4 y = 9

− 1 4 x + 3 2 y = 11 − 1 8 x + 1 3 y = 3 − 1 4 x + 3 2 y = 11 − 1 8 x + 1 3 y = 3

For the following exercises, solve each system by addition.

−2 x + 5 y = −42 7 x + 2 y = 30 −2 x + 5 y = −42 7 x + 2 y = 30

6 x −5 y = −34 2 x + 6 y = 4 6 x −5 y = −34 2 x + 6 y = 4

5 x − y = −2.6 −4 x −6 y = 1.4 5 x − y = −2.6 −4 x −6 y = 1.4

7 x −2 y = 3 4 x + 5 y = 3.25 7 x −2 y = 3 4 x + 5 y = 3.25

−x + 2 y = −1 5 x −10 y = 6 −x + 2 y = −1 5 x −10 y = 6

7 x + 6 y = 2 −28 x −24 y = −8 7 x + 6 y = 2 −28 x −24 y = −8

5 6 x + 1 4 y = 0 1 8 x − 1 2 y = − 43 120 5 6 x + 1 4 y = 0 1 8 x − 1 2 y = − 43 120

1 3 x + 1 9 y = 2 9 − 1 2 x + 4 5 y = − 1 3 1 3 x + 1 9 y = 2 9 − 1 2 x + 4 5 y = − 1 3

−0.2 x + 0.4 y = 0.6 x −2 y = −3 −0.2 x + 0.4 y = 0.6 x −2 y = −3

−0.1 x + 0.2 y = 0.6 5 x −10 y = 1 −0.1 x + 0.2 y = 0.6 5 x −10 y = 1

For the following exercises, solve each system by any method.

5 x + 9 y = 16 x + 2 y = 4 5 x + 9 y = 16 x + 2 y = 4

6 x −8 y = −0.6 3 x + 2 y = 0.9 6 x −8 y = −0.6 3 x + 2 y = 0.9

5 x −2 y = 2.25 7 x −4 y = 3 5 x −2 y = 2.25 7 x −4 y = 3

x − 5 12 y = − 55 12 −6 x + 5 2 y = 55 2 x − 5 12 y = − 55 12 −6 x + 5 2 y = 55 2

7 x −4 y = 7 6 2 x + 4 y = 1 3 7 x −4 y = 7 6 2 x + 4 y = 1 3

3 x + 6 y = 11 2 x + 4 y = 9 3 x + 6 y = 11 2 x + 4 y = 9

7 3 x − 1 6 y = 2 − 21 6 x + 3 12 y = −3 7 3 x − 1 6 y = 2 − 21 6 x + 3 12 y = −3

1 2 x + 1 3 y = 1 3 3 2 x + 1 4 y = − 1 8 1 2 x + 1 3 y = 1 3 3 2 x + 1 4 y = − 1 8

2.2 x + 1.3 y = −0.1 4.2 x + 4.2 y = 2.1 2.2 x + 1.3 y = −0.1 4.2 x + 4.2 y = 2.1

0.1 x + 0.2 y = 2 0.35 x −0.3 y = 0 0.1 x + 0.2 y = 2 0.35 x −0.3 y = 0

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

3 x − y = 0.6 x −2 y = 1.3 3 x − y = 0.6 x −2 y = 1.3

− x + 2 y = 4 2 x −4 y = 1 − x + 2 y = 4 2 x −4 y = 1

x + 2 y = 7 2 x + 6 y = 12 x + 2 y = 7 2 x + 6 y = 12

3 x −5 y = 7 x −2 y = 3 3 x −5 y = 7 x −2 y = 3

3 x −2 y = 5 −9 x + 6 y = −15 3 x −2 y = 5 −9 x + 6 y = −15

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

0.1 x + 0.2 y = 0.3 −0.3 x + 0.5 y = 1 0.1 x + 0.2 y = 0.3 −0.3 x + 0.5 y = 1

−0.01 x + 0.12 y = 0.62 0.15 x + 0.20 y = 0.52 −0.01 x + 0.12 y = 0.62 0.15 x + 0.20 y = 0.52

0.5 x + 0.3 y = 4 0.25 x −0.9 y = 0.46 0.5 x + 0.3 y = 4 0.25 x −0.9 y = 0.46

0.15 x + 0.27 y = 0.39 −0.34 x + 0.56 y = 1.8 0.15 x + 0.27 y = 0.39 −0.34 x + 0.56 y = 1.8

−0.71 x + 0.92 y = 0.13 0.83 x + 0.05 y = 2.1 −0.71 x + 0.92 y = 0.13 0.83 x + 0.05 y = 2.1

For the following exercises, solve each system in terms of A , B , C , D , E , A , B , C , D , E , and F F where A – F A – F are nonzero numbers. Note that A ≠ B A ≠ B and A E ≠ B D . A E ≠ B D .

x + y = A x − y = B x + y = A x − y = B

x + A y = 1 x + B y = 1 x + A y = 1 x + B y = 1

A x + y = 0 B x + y = 1 A x + y = 0 B x + y = 1

A x + B y = C x + y = 1 A x + B y = C x + y = 1

A x + B y = C D x + E y = F A x + B y = C D x + E y = F

Real-World Applications

For the following exercises, solve for the desired quantity.

A stuffed animal business has a total cost of production C = 12 x + 30 C = 12 x + 30 and a revenue function R = 20 x . R = 20 x . Find the break-even point.

An Ethiopian restaurant has a cost of production C ( x ) = 11 x + 120 C ( x ) = 11 x + 120 and a revenue function R ( x ) = 5 x . R ( x ) = 5 x . When does the company start to turn a profit?

A cell phone factory has a cost of production C ( x ) = 150 x + 10 , 000 C ( x ) = 150 x + 10 , 000 and a revenue function R ( x ) = 200 x . R ( x ) = 200 x . What is the break-even point?

A musician charges C ( x ) = 64 x + 20,000 C ( x ) = 64 x + 20,000 where x x is the total number of attendees at the concert. The venue charges $80 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

A guitar factory has a cost of production C ( x ) = 75 x + 50,000. C ( x ) = 75 x + 50,000. If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

For the following exercises, use a system of linear equations with two variables and two equations to solve.

Find two numbers whose sum is 28 and difference is 13.

A number is 9 more than another number. Twice the sum of the two numbers is 10. Find the two numbers.

The startup cost for a restaurant is $120,000, and each meal costs $10 for the restaurant to make. If each meal is then sold for $15, after how many meals does the restaurant break even?

A moving company charges a flat rate of $150, and an additional $5 for each box. If a taxi service would charge $20 for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

A total of 1,595 first- and second-year college students gathered at a pep rally. The number of first-years exceeded the number of second-years by 15. How many students from each year group were in attendance?

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If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40% saline solution, how many gallons of 10% and 60% solutions were mixed?

An investor earned triple the profits of what they earned last year. If they made $500,000.48 total for both years, how much did the investor earn in profits each year?

An investor invested 1.1 million dollars into two land investments. On the first investment, Swan Peak, her return was a 110% increase on the money she invested. On the second investment, Riverside Community, she earned 50% over what she invested. If she earned $1 million in profits, how much did she invest in each of the land deals?

If an investor invests a total of $25,000 into two bonds, one that pays 3% simple interest, and the other that pays 2 7 8 % 2 7 8 % interest, and the investor earns $737.50 annual interest, how much was invested in each account?

If an investor invests $23,000 into two bonds, one that pays 4% in simple interest, and the other paying 2% simple interest, and the investor earns $710.00 annual interest, how much was invested in each account?

Blu-rays cost $5.96 more than regular DVDs at All Bets Are Off Electronics. How much would 6 Blu-rays and 2 DVDs cost if 5 Blu-rays and 2 DVDs cost $127.73?

A store clerk sold 60 pairs of sneakers. The high-tops sold for $98.99 and the low-tops sold for $129.99. If the receipts for the two types of sales totaled $6,404.40, how many of each type of sneaker were sold?

A concert manager counted 350 ticket receipts the day after a concert. The price for a student ticket was $12.50, and the price for an adult ticket was $16.00. The register confirms that $5,075 was taken in. How many student tickets and adult tickets were sold?

Admission into an amusement park for 4 children and 2 adults is $116.90. For 6 children and 3 adults, the admission is $175.35. Assuming a different price for children and adults, what is the price of the child’s ticket and the price of the adult ticket?

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Home » Correlation Analysis – Types, Methods and Examples

Correlation Analysis – Types, Methods and Examples

Table of Contents

Correlation Analysis

Correlation analysis is a statistical method used to evaluate the strength and direction of the relationship between two or more variables . The correlation coefficient ranges from -1 to 1.

• A correlation coefficient of 1 indicates a perfect positive correlation. This means that as one variable increases, the other variable also increases.
• A correlation coefficient of -1 indicates a perfect negative correlation. This means that as one variable increases, the other variable decreases.
• A correlation coefficient of 0 means that there’s no linear relationship between the two variables.

Correlation Analysis Methodology

Conducting a correlation analysis involves a series of steps, as described below:

• Define the Problem : Identify the variables that you think might be related. The variables must be measurable on an interval or ratio scale. For example, if you’re interested in studying the relationship between the amount of time spent studying and exam scores, these would be your two variables.
• Data Collection : Collect data on the variables of interest. The data could be collected through various means such as surveys , observations , or experiments. It’s crucial to ensure that the data collected is accurate and reliable.
• Data Inspection : Check the data for any errors or anomalies such as outliers or missing values. Outliers can greatly affect the correlation coefficient, so it’s crucial to handle them appropriately.
• Choose the Appropriate Correlation Method : Select the correlation method that’s most appropriate for your data. If your data meets the assumptions for Pearson’s correlation (interval or ratio level, linear relationship, variables are normally distributed), use that. If your data is ordinal or doesn’t meet the assumptions for Pearson’s correlation, consider using Spearman’s rank correlation or Kendall’s Tau.
• Compute the Correlation Coefficient : Once you’ve selected the appropriate method, compute the correlation coefficient. This can be done using statistical software such as R, Python, or SPSS, or manually using the formulas.
• Interpret the Results : Interpret the correlation coefficient you obtained. If the correlation is close to 1 or -1, the variables are strongly correlated. If the correlation is close to 0, the variables have little to no linear relationship. Also consider the sign of the correlation coefficient: a positive sign indicates a positive relationship (as one variable increases, so does the other), while a negative sign indicates a negative relationship (as one variable increases, the other decreases).
• Check the Significance : It’s also important to test the statistical significance of the correlation. This typically involves performing a t-test. A small p-value (commonly less than 0.05) suggests that the observed correlation is statistically significant and not due to random chance.
• Report the Results : The final step is to report your findings. This should include the correlation coefficient, the significance level, and a discussion of what these findings mean in the context of your research question.

Types of Correlation Analysis

Types of Correlation Analysis are as follows:

Pearson Correlation

This is the most common type of correlation analysis. Pearson correlation measures the linear relationship between two continuous variables. It assumes that the variables are normally distributed and have equal variances. The correlation coefficient (r) ranges from -1 to +1, with -1 indicating a perfect negative linear relationship, +1 indicating a perfect positive linear relationship, and 0 indicating no linear relationship.

Spearman Rank Correlation

Spearman’s rank correlation is a non-parametric measure that assesses how well the relationship between two variables can be described using a monotonic function. In other words, it evaluates the degree to which, as one variable increases, the other variable tends to increase, without requiring that increase to be consistent.

Kendall’s Tau

Kendall’s Tau is another non-parametric correlation measure used to detect the strength of dependence between two variables. Kendall’s Tau is often used for variables measured on an ordinal scale (i.e., where values can be ranked).

Point-Biserial Correlation

This is used when you have one dichotomous and one continuous variable, and you want to test for correlations. It’s a special case of the Pearson correlation.

Phi Coefficient

This is used when both variables are dichotomous or binary (having two categories). It’s a measure of association for two binary variables.

Canonical Correlation

This measures the correlation between two multi-dimensional variables. Each variable is a combination of data sets, and the method finds the linear combination that maximizes the correlation between them.

Partial and Semi-Partial (Part) Correlations

These are used when the researcher wants to understand the relationship between two variables while controlling for the effect of one or more additional variables.

Cross-Correlation

Used mostly in time series data to measure the similarity of two series as a function of the displacement of one relative to the other.

Autocorrelation

This is the correlation of a signal with a delayed copy of itself as a function of delay. This is often used in time series analysis to help understand the trend in the data over time.

Correlation Analysis Formulas

There are several formulas for correlation analysis, each corresponding to a different type of correlation. Here are some of the most commonly used ones:

Pearson’s Correlation Coefficient (r)

Pearson’s correlation coefficient measures the linear relationship between two variables. The formula is:

   r = Σ[(xi – Xmean)(yi – Ymean)] / sqrt[(Σ(xi – Xmean)²)(Σ(yi – Ymean)²)]

• xi and yi are the values of X and Y variables.
• Xmean and Ymean are the mean values of X and Y.
• Σ denotes the sum of the values.

Spearman’s Rank Correlation Coefficient (rs)

Spearman’s correlation coefficient measures the monotonic relationship between two variables. The formula is:

   rs = 1 – (6Σd² / n(n² – 1))

• d is the difference between the ranks of corresponding variables.
• n is the number of observations.

Kendall’s Tau (τ)

Kendall’s Tau is a measure of rank correlation. The formula is:

   τ = (nc – nd) / 0.5n(n-1)

• nc is the number of concordant pairs.
• nd is the number of discordant pairs.

This correlation is a special case of Pearson’s correlation, and so, it uses the same formula as Pearson’s correlation.

Phi coefficient is a measure of association for two binary variables. It’s equivalent to Pearson’s correlation in this specific case.

Partial Correlation

The formula for partial correlation is more complex and depends on the Pearson’s correlation coefficients between the variables.

For partial correlation between X and Y given Z:

  rp(xy.z) = (rxy – rxz * ryz) / sqrt[(1 – rxz^2)(1 – ryz^2)]

• rxy, rxz, ryz are the Pearson’s correlation coefficients.

Correlation Analysis Examples

Here are a few examples of how correlation analysis could be applied in different contexts:

• Education : A researcher might want to determine if there’s a relationship between the amount of time students spend studying each week and their exam scores. The two variables would be “study time” and “exam scores”. If a positive correlation is found, it means that students who study more tend to score higher on exams.
• Healthcare : A healthcare researcher might be interested in understanding the relationship between age and cholesterol levels. If a positive correlation is found, it could mean that as people age, their cholesterol levels tend to increase.
• Economics : An economist may want to investigate if there’s a correlation between the unemployment rate and the rate of crime in a given city. If a positive correlation is found, it could suggest that as the unemployment rate increases, the crime rate also tends to increase.
• Marketing : A marketing analyst might want to analyze the correlation between advertising expenditure and sales revenue. A positive correlation would suggest that higher advertising spending is associated with higher sales revenue.
• Environmental Science : A scientist might be interested in whether there’s a relationship between the amount of CO2 emissions and average temperature increase. A positive correlation would indicate that higher CO2 emissions are associated with higher average temperatures.

Importance of Correlation Analysis

Correlation analysis plays a crucial role in many fields of study for several reasons:

• Understanding Relationships : Correlation analysis provides a statistical measure of the relationship between two or more variables. It helps in understanding how one variable may change in relation to another.
• Predicting Trends : When variables are correlated, changes in one can predict changes in another. This is particularly useful in fields like finance, weather forecasting, and technology, where forecasting trends is vital.
• Data Reduction : If two variables are highly correlated, they are conveying similar information, and you may decide to use only one of them in your analysis, reducing the dimensionality of your data.
• Testing Hypotheses : Correlation analysis can be used to test hypotheses about relationships between variables. For example, a researcher might want to test whether there’s a significant positive correlation between physical exercise and mental health.
• Determining Factors : It can help identify factors that are associated with certain behaviors or outcomes. For example, public health researchers might analyze correlations to identify risk factors for diseases.
• Model Building : Correlation is a fundamental concept in building multivariate statistical models, including regression models and structural equation models. These models often require an understanding of the inter-relationships (correlations) among multiple variables.
• Validity and Reliability Analysis : In psychometrics, correlation analysis is used to assess the validity and reliability of measurement instruments such as tests or surveys.

Applications of Correlation Analysis

Correlation analysis is used in many fields to understand and quantify the relationship between variables. Here are some of its key applications:

• Finance : In finance, correlation analysis is used to understand the relationship between different investment types or the risk and return of a portfolio. For example, if two stocks are positively correlated, they tend to move together; if they’re negatively correlated, they move in opposite directions.
• Economics : Economists use correlation analysis to understand the relationship between various economic indicators, such as GDP and unemployment rate, inflation rate and interest rates, or income and consumption patterns.
• Marketing : Correlation analysis can help marketers understand the relationship between advertising spend and sales, or the relationship between price changes and demand.
• Psychology : In psychology, correlation analysis can be used to understand the relationship between different psychological variables, such as the correlation between stress levels and sleep quality, or between self-esteem and academic performance.
• Medicine : In healthcare, correlation analysis can be used to understand the relationships between various health outcomes and potential predictors. For example, researchers might investigate the correlation between physical activity levels and heart disease, or between smoking and lung cancer.
• Environmental Science : Correlation analysis can be used to investigate the relationships between different environmental factors, such as the correlation between CO2 levels and average global temperature, or between pesticide use and biodiversity.
• Social Sciences : In fields like sociology and political science, correlation analysis can be used to investigate relationships between different social and political phenomena, such as the correlation between education levels and political participation, or between income inequality and social unrest.

Advantages and Disadvantages of Correlation Analysis

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Study Guides > Boundless Algebra

Systems of equations in two variables, introduction to systems of equations, learning objectives, key takeaways.

• A system of linear equations consists of two or more linear equations made up of two or more variables, such that all equations in the system are considered simultaneously.
• To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time.
• In order for a linear system to have a unique solution, there must be at least as many equations as there are variables.
• The solution to a system of linear equations in two variables is any ordered pair [latex](x, y)[/latex] that satisfies each equation independently. Graphically, solutions are points at which the lines intersect.
• system of linear equations : A set of two or more equations made up of two or more variables that are considered simultaneously.
• dependent system : A system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system.
• inconsistent system : A system of linear equations with no common solution because they represent parallel lines, which have no point or line in common.
• independent system : A system of linear equations with exactly one solution pair [latex](x, y)[/latex].

Types of Linear Systems and Their Solutions

• The system has a single unique solution.
• The system has no solution .
• The system has infinitely many solutions .

An independent system has exactly one solution pair [latex](x, y)[/latex]. The point where the two lines intersect is the only solution.

Solving Systems Graphically

• To solve a system of equations graphically, graph the equations and identify the points of intersection as the solutions. There can be more than one solution to a system of equations.
• A system of linear equations will have one point of intersection, or one solution.
• To graph a system of equations that are written in standard form, you must rewrite the equations in slope -intercept form.
• system of equations : A set of equations with multiple variables which can be solved using a specific set of values.
• The graphical method : A way of visually finding a set of values that solves a system of equations.
• The graphical method
• The substitution method
• The elimination method

A system with two sets of answers that will satisfy both equations has two points of intersection (thus, two solutions of the system), as shown in the image below.

Converting to Slope-Intercept Form

Identifying solutions on a graph, the substitution method.

• A system of equations is a set of equations that can be solved using a particular set of values.
• The substitution method works by expressing one of the variables in terms of another, then substituting it back into the original equation and simplifying it.
• It is very important to check your work once you have found a set of values for the variables. Do this by substituting the values you found back into the original equations.
• The solution to the system of equations can be written as an ordered pair ( x , y ).
• substitution method : Method of solving a system of equations by putting the equation in terms of only one variable
• In the first equation, solve for one of the variables in terms of the others.
• Substitute this expression into the remaining equations.
• Continue until you have reduced the system to a single linear equation.
• Solve this equation, and then back-substitute until the solution is found.

Solving with the Substitution Method

The elimination method.

• The steps of the elimination method are: (1) set the equations up so the variables line up, (2) modify one equation so both equations share a consistent variable that can be eliminated, (3) add the equations together to eliminate the variable, (4) solve, and (5) back-substitute to solve for the other variable.
• Always check the answer. This is done by plugging both values into one or both of the original equations.
• elimination method : Process of solving a system of equations by eliminating one variable in order to more simply solve for the remaining variable.
• Rewrite the equations so the variables line up.
• Modify one equation so both equations have a variable that will cancel itself out when the equations are added together.
• Add the equations and eliminate the variable.
• Solve for the remaining variable.
• Back-substitute and solve for the other variable.

Solving with the Elimination Method

Inconsistent and dependent systems in two variables.

• Graphically, the equations in a dependent system represent the same line. The equations in an inconsistent system represent parallel lines that never intersect.
• We can use methods for solving systems of equations to identify dependent and inconsistent systems: Dependent systems have an infinite number of solutions. Applying methods of solving systems of equations will result in a true identity, such as [latex]0 = 0[/latex]. Inconsistent systems have no solutions. Applying methods of solving systems of equations will result in a contradiction, such as the statement [latex]0 = 1[/latex].
• independent system : A system of linear equations with exactly one solution pair.
• The system has no solution.
• The system has infinitely many solutions.

Dependent Systems

Dependent system : The equations [latex]3x + 2y = 6[/latex] and [latex]6x + 4y = 12[/latex] are dependent, and when graphed produce the same line.

Inconsistent Systems

In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.

Applications of Systems of Equations

• If you have a problem that includes multiple variables, you can solve it by creating a system of equations.
• Once variables are defined, determine the relationships between them and write them as equations.

Systems of Equations in the Real World

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• Math Article

Linear Equations In Two Variables

Linear equations in two variables, explain the geometry of lines or the graph of two lines, plotted to solve the given equations. As we already know, the linear equation represents a straight line. The plotting of these graphs will help us to solve the equations, which consist of unknown variables. Previously we have learned to solve linear equations in one variable , here we will find the solutions for the equations having two variables.

An equation is said to be linear equation in two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero.

For example, 10x+4y = 3 and -x+5y = 2 are linear equations in two variables.

The solution for such an equation is a pair of values, one for x and one for y which further makes the two sides of an equation equal.

Also, read: 

• Linear Equations In Two Variables Class 9
• Solving Linear Equations
• Applications Of Linear Equations

Solution of Linear Equations in Two Variables

The solution of linear equations in two variables, ax+by = c, is a particular point in the graph, such that when x-coordinate is multiplied by a and y-coordinate is multiplied by b, then the sum of these two values will be equal to c. 

Basically, for linear equation in two variables, there are infinitely many solutions.

In order to find the solution of Linear equation in 2 variables, two equations should be known to us.

Consider for Example:

5x + 3y = 30

The above equation has two variables namely x and y.

Graphically this equation can be represented by substituting the variables to zero.

The value of x when y=0 is

5x + 3(0) = 30

and the value of y when x = 0 is,

5 (0) + 3y = 30

It is now understood that to solve linear equation in two variables, the two equations have to be known and then the substitution method can be followed. Let’s understand this with a few example questions.

Unique Solution

For the given linear equations in two variables, the solution will be unique for both the equations, if and only if they intersect at a single point. 

The condition to get the unique solution for the given linear equations is, the slope of the line formed by the two equations, respectively, should not be equal.

Consider, m 1 and m 2 are two slopes of equations of two lines in two variables. So, if the equations have a unique solution, then:

No Solution

If the two linear equations have equal slope value, then the equations will have no solutions.

This is because the lines are parallel to each other and do not intersect.

System of Linear Equations in Two Variables

Instead of finding the solution for a single linear equation in two variables, we can take two sets of linear equations, both having two variables in them and find the solutions. So, basically the system of linear equations is defined when there is more than one linear equation. 

For example, a+b = 15 and a-b = 5, are the system of linear equations in two variables. Because, the point a = 10 and b = 5 is the solution for both equations, such as:

a+b=10 + 5 = 15

a-b=10-5 = 5

Hence, proved point (10,5) is solution for both a+b=15 and a-b=5.

Problems and Solutions

Question: Find the value of variables which satisfies the following equation:

2x + 5y = 20 and 3x+6y =12.

Using the method of substitution to solve the pair of linear equation, we have:

2x + 5y = 20…………………….(i)

3x+6y =12……………………..(ii)

Multiplying equation (i) by 3 and (ii) by 2, we have:

6x + 15y = 60…………………….(iii)

6x+12y = 24……………………..(iv)

Subtracting equation (iv) from (iii)

Substituting the value of y in any of the equation (i) or (ii), we have

2x + 5(12) = 20

Therefore, x=-20 and y =12 is the point where the given equations intersect.

Now, it is important to know the situational examples which are also known as word problems from linear equations in 2 variables.

Check:  Linear Equations Calculator

Word Problems

Question 1:  A boat running downstream covers a distance of 20 km in 2 hours while for covering the same distance upstream, it takes 5 hours. What is the speed of the boat in still water?

These types of questions are the real-time examples of linear equations in two variables.

In water, the direction along the stream is called downstream. And, the direction against the stream is called upstream.

Let us consider the speed of a boat is u km/h and the speed of the stream is v km/h, then:

Speed Downstream = (u + v) km/h

Speed Upstream = (u – v) km/h

We know that, Speed = Distance/Time

So, the speed of boat when running downstream = (20⁄2) km/h = 10 km/h

The speed of boat when running upstream = (20⁄5) km/h = 4 km/h

From above, u + v = 10…….(1)

u – v = 4 ………. (2)

Adding equation 1 and 2, we get: 2u = 1

Also, v = 3 km/h

Therefore, the speed of the boat in still water = u = 7 km/h

Question 2: A boat running upstream takes 6 hours 30 minutes to cover a certain distance, while it takes 3 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current, respectively?

Solution: If the speed downstream is  a km/hr  and the speed upstream is  b km/hr , then

Speed in still water = (a + b)/2 km/h

Rate of stream = ½ (a − b) kmph

Let the Boat’s rate upstream be  x  kmph and that downstream be  y  kmph.

Then, distance covered upstream in 6 hrs 30 min = Distance covered downstream in 3 hrs.

⇒ x × 6.5 hrs = y × 3hrs

⇒ 13/2x = 3y

⇒ y = 13x/6

Frequently Asked Questions – FAQs

How to solve linear equation in two variables, how many solutions are there for linear equations in two variables, what is the two-variable equation, what are the coefficients of the equation 3x-6y = -13, what is the constant of the equation 3x-6y=-13.

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Course: Algebra 1   >   Unit 4

• Two-variable linear equations intro

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Video transcript

Testing Mindfulness Mechanisms of Action on the Stress and Burnout of Social Workers

• Original Paper
• Open access
• Published: 21 May 2024

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• Alan Maddock   ORCID: orcid.org/0000-0001-5089-1778 1  

Growing evidence suggests that mindfulness-based programmes (MBPs) can reduce stress and burnout among social workers. How MBPs support changes in these outcomes is unclear. This study attempts to identify what some of these mechanisms might be, using the clinically modified Buddhist psychological model (CBPM) as a guiding framework.

This study used data from two randomised controlled trials investigating the impact of MBP participation on social worker stress and burnout. The data from participants ( n = 94) who completed a Mindfulness-based Social Work and Self-Care (MBSWSC) programme, or a mindfulness and self-compassion programme (MSC) were combined. Structural equation models were constructed, and conditional direct and indirect effect models of changes in the CBPM domains (mindfulness, self-compassion, attention regulation, acceptance, non-attachment, non-aversion), mediating variables (rumination and worry) and outcomes (stress and burnout) were tested.

The results suggest that CBPM models, through mediated effects on stress, depersonalisation, and personal accomplishment, as well as direct and mediated effects on stress, emotional exhaustion, and depersonalisation, can provide useful frameworks for explaining how MBPs reduce stress and burnout among social workers. This study also found several other significant conditional direct and indirect effects. The pattern of these relationships indicate that multiple outcomes could be improved through different CBPM domains.

This study provides initial evidence on the potential mechanisms through which MBP participation acts to reduce stress and burnout in social workers.

Preregistration

https://www.clinicaltrials.gov ; Unique identifier: NCT05519267 and NCT05538650.

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Although social work is a very rewarding profession, from which practitioners often attain very high rates of personal accomplishment (McFadden, 2015 ), it is also a very challenging profession which can lead to social workers experiencing concerningly high rates of stress and burnout (Evans et al., 2006 ; Ravalier et al., 2021 ). In a large-scale study, Ravalier et al. ( 2021 ) ( n = 3,421) found that social workers in the United Kingdom (UK) experienced much higher rates of stress than a general English population (Warttig et al., 2013 ), with 60% of their sample highlighting that they planned to leave their current position. In another large-scale study, McFadden ( 2015 ) ( n = 1,359) found, that in comparison to other health and social care workers, social workers in the UK experienced very high rates of emotional exhaustion and depersonalisation. Evans et al. ( 2006 ) ( n = 237) found that mental health social workers in England and Wales experience higher rates of emotional exhaustion than consultant psychiatrists, and rates of emotional exhaustion that are three times higher other mental health practitioners. Practitioner stress and burnout can impact the social work profession negatively, at both a micro, and macro level. On a micro level, it can directly impact on service user care by reducing the social worker’s capacity for attentiveness, empathy, optimism, collaboration, and capacity to connect emotionally with, or risk assess what the service user’s needs might be (Salyers et al., 2017 ; Savaya et al., 2016 ). On a macro level, social worker stress and burnout has been linked to poor job performance, reduced organisational commitment (Savaya et al., 2021 ), productivity, increased training costs and disruption of services, increased employee sick days, and turnover (Ravalier et al., 2021 ). Turley et al. ( 2021 ) highlight that the average length of a social work career in the United Kingdom is less than 8 years, which is significantly shorter than that of nurses (16 years) and medical doctors (25 years).

The evidence that Mindfulness-based programmes (MBP) could help social workers to adapt to and recover from feelings of stress and burnout is growing (Craigie et al., 2016 ; Kinman et al., 2020 ; Lynn & Mensinga, 2015 ; Miller et al., 2020 ). In a randomised controlled trial (RCT) of the Mindfulness-based Social Work and Self-Care programme (MBSWSC), Maddock et al. ( 2023 ) ( n = 62) found that MBSWSC participation led to social workers experiencing large significant reductions in stress and emotional exhaustion versus a mindfulness and self-compassion control group (MSC). MBSWSC group participants also experienced reduced depersonalisation, but there was no significant difference in this outcome versus MSC (Maddock et al. 2023 ). In a follow-up study aiming to replicate these findings among a broader range of professionals, including social work managers, Maddock et al. ( 2024 ) ( n = 60) found that MBSWSC participants experienced large significant reductions in stress, emotional exhaustion, and a moderate significant reduction in depersonalisation versus MSC.

It is currently unclear how MBSWSC improved these outcomes, and over the last two decades, the scientific mindfulness literature has consistently highlighted the need to verify the most important mechanisms of MBPs, which influence changes in mental health and well-being outcomes, such as stress and burnout (Gu et al., 2015 ; Maddock & Blair, 2023 ; Van der Velden et al., 2015 ). The identification of the key mechanisms of action of MBPs would enhance our understanding of how they work for specific groups, such as social workers (Kazdin, 2009 ). It could also support the development of other effective programmes of support, through the enhancement of the most potent mechanisms of action e.g., if self-compassion was found to be a key mechanism, self-compassion activities could be utilised to improve stress and burnout in social work (Miller et al., 2020 ; Svendsen et al., 2017 ). The empirical evaluation of theories, which help to explain the onset, maintenance, and recurrence of stress and burnout in social workers, and how the deleterious effects of these outcomes can be ameliorated through MBP participation, could be an important step in more clearly identifying how MBPs lead to beneficial effects for social workers (Brown, 2015 ; Carlson, 2015 ; Kinman et al., 2020 ; Miller et al., 2020 ). There is only one theory within social work which focusses on how MBPs might improve stress and burnout in social workers, the clinically modified Buddhist psychological model for social work practice and self-care (CBPM) (Maddock, 2023 ). The CBPM is presented in full elsewhere (Maddock, 2023 ). In short, it focusses on the effects of six mindfulness mechanisms of action (CBPM domains): mindfulness, self-compassion, attention regulation, acceptance, non-aversion, and non-attachment, and how they can function both individually, and collectively, to reduce repeated negative thinking patterns (worry and rumination) which often present as cognitive avoidant coping strategies. The development of each CBPM domain leads to increased tendencies in social workers to approach the sources of their stress (e.g., through the fuller engagement with, and processing of difficult emotions which can result from social work practice), which leads to reduced feelings of stress and burnout (emotional exhaustion, depersonalisation, and increased personal accomplishment), both directly, and indirectly through reduced worry and rumination (Maddock, 2023 ).

This study has two aims: (1) To provide a greater understanding of how MBPs improve stress and burnout in social workers, using pre and post RCT data from a sample of social workers who had completed a MBP in Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ), using the CBPM as a theoretical framework; (2) investigate if changes in the CBPM domain and mediating variable scores after engaging in a MBP predicted the stress and burnout levels of social workers. To achieve these aims, the study tested hypotheses related to the CBPM's structural equation models' fit and the direct and indirect effects of specific mindfulness mechanisms on stress and burnout .

Participants

The participants were from two randomised controlled trials examining the effects of mindfulness-based programmes (MBPs) on stress and burnout among social workers in Northern Ireland (Maddock et al., 2023 ; Maddock et al., 2024 ). To examine the changes in each CBPM domain, mediating and outcome variable (i.e., stress and burnout) more rigorously in each MBP, only participants ( n = 94) who completed both the experimental MBSWSC ( n = 56) and control MSC ( n = 38) programmes in Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) and provided post programme data were included in this study. A total of 83 female (88%) (51 in the MBSWSC programme, and 32 in the MSC programme), and 11 male (12%) (5 in the MBSWSC, and 6 in the MSC programme) social work professionals completed the MBP that they were allocated to, and the study’s measures, across both RCTs. This gender split is in line with the overall gender profile of social workers in Northern Ireland (Hughes, 2022 ). The sample included social work professionals in Northern Ireland who engaged with service users as part of their role, including social workers, senior social work practitioners, and social work service managers. This sample size is sufficient to power structural equation models of fit, as the sample contained more than ten participants per estimated variable (Bentler & Chou, 1987 ). The sample size for this study also means that Hypothesis 2 and 3 are powered sufficiently to detect large moderated mediation effects at a power of 80% (Fritz & Mackinnon, 2007 ).

MBSWSC and MSC follow a protocol, and were delivered by the same facilitators in both Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ). Fidelity to each MBPs protocol was assessed after each session, with an adapted treatment fidelity tool, which is based on Kechter et al. ( 2019 ) recommendations for reporting treatment fidelity in MBP trials.

MBSWSC is based on the CBPM (Maddock, 2023 ), and theories of cognitive and emotion regulation and stress coping. MBSWSC is a unique group-based, online MBP (delivered via MS Teams), in that it focusses on social work practice and self-care, with a key goal of the programme being to support social workers to reduce their levels of stress and burnout, through increased introspective awareness of the interconnections between thoughts, emotions, physical sensations, and behaviours. MBSWSC does this through three main teaching modalities, delivered each week, over a 6-week period: (1) psychoeducation on what key concepts such as mindfulness, attention regulation, acceptance, self-compassion, non-aversion, non-attachment, worry, rumination, avoidant and approach stress coping are, and what their potential roles are in reducing the onset, maintenance and recurrence of stress and burnout, (2) experiential learning through engagement with, and reflection on participation in CBPM domain focussed body scan exercises. The goal being to allow participants to learn experientially, and to better integrate and embody the programme’s didactic content, and (3) apply their conceptual and experiential learning to social work case contexts through the engagement with role play scenarios which are common in social work e.g., dealing with a challenging service user. The participants engaged with several different body scan meditations over the 6-week duration of the programme. The goal of these body scan mediations was to support participants to develop a personal mindfulness practice which focussed on improving each CBPM domain, with a view to reducing repeated negative thinking (worry and/or rumination), stress, and burnout, in line with the CBPM theory (Maddock, 2023 ). These were: (1) a general mindfulness body scan meditation, (2) a attention regulation body scan, (3) a acceptance body scan meditation, (4) a non-aversion and non-attachment body scan, and (5) a self-compassion body scan meditation (including the writing of a self-compassionate letter) to help participants develop, refine and practice mindfulness skills/techniques. To support participants to develop, refine and practice these mindfulness skills/techniques in their day-to-day lives and social work practice, participants were requested each week to complete homework activities, which incorporated these body scans. This homework took 20-30 min to complete, 6 out of the 7 days per week over the programme’s 6-week duration. The MBSWSC programme was facilitated by two professionally qualified social workers, who are trained mindfulness facilitators. More details on the MBSWSC programme, and its delivery are published in Maddock et al. ( 2023 ).

The mindfulness and self-compassion programme (MSC) was adapted from the Mindfulness Based Living Course programme, and delivered in an online group format (via MS Teams) over the same 6-week period (MBLC: Choden & Regan-Addis, 2018 ). The aim of the MSC programme was to introduce participants to the concepts of mindfulness and self-compassion, and related experiential practices. Sessions included key learning about: (1) how body scans can shift the attention away from the mind to the body, (2) that this process can support the development of embodied reflexivity, supporting social work practice, (3) mindfulness, and (4) self-compassion’s potential role in reducing stress, burnout, and improving social work practice. The participants were introduced to three experiential practices which were deemed to be relevant to social work practice: (1) the body scan, (2) the 3-min breathing space (3MBS), and (3) the self-compassion break (SCB). In order to embed the key learning from these exercises, and support their use in the participants everyday lives, visual mnemonics for the 3MBS and SCB were provided. To support participants to develop, refine and practice these mindfulness skills/techniques, they were requested to complete homework activities. These homework activities also took approximately 20–30 min to complete, 6 days per week over the same 6-week period. The MSC programme was also facilitated by two professionally qualified social workers, who were trained mindfulness facilitators. More details on the MSC programme, and its delivery is published in Maddock et al. ( 2023 ).

Demographic information was collected at the baseline of both RCTs. In order to empirically examine the relationships set out in the CBPM, self-report measures of each CBPM domain, mediating and outcome variable were collected pre and post MBP completion. Scale reliabilities were calculated using the responses provided.

The Perceived Stress Scale (PSS) is a widely used valid and reliable 10-item measure of perceived stress (Cohen et al., 1983 ). The PSS is scored on a 5-point Likert scale (0= never  to 4= very often ), with higher scores indicating higher perceived stress. The reliability of the scores on the PSS in both RCTs with social workers was deemed to be acceptable (Cronbach’s α = 0.88) (Maddock et al., 2023 ), and (Cronbach’s α = 0.87) (Maddock et al., 2024 ). McDonald's omega for the PSS in this study was 0.86 (Hayes & Coutts, 2020 ).

The Maslach Burnout Inventory (MBI) is reliable, valid, and the most widely used measure of occupational burnout (Maslach et al., 1996 ; Maslach et al., 2001 ). The MBI is a 22-item measure and is scored on a 7-point Likert scale (0= never  to 6= everyday ), with 3 subscales measuring: emotional exhaustion, depersonalisation/loss of empathy, and personal accomplishment. Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) confirmed moderate to high reliability on the MBI subscales reporting Cronbach’s α = 0.91, 0.69 and 0.75, and α = 0.92, 0.81 and 0.84 for emotional exhaustion, depersonalisation, and personal accomplishment, respectively. McDonald's omega for the emotional exhaustion, depersonalisation, and personal accomplishment in the subscales in this study were 0.90, 0.64, and 0.75, respectively (Hayes & Coutts, 2020 ).

The Southampton Mindfulness Questionnaire (SMQ) is a valid and reliable 16-item measure that assesses the degree to which a person can remain mindful in response to unpleasant thoughts and images (Chadwick et al., 2008 ). The SMQ contains 8 positively and negatively worded items, on a 7-point Likert scale (0= disagree totally to 6= agree totally ). Scores on this measure range from 0-96. High scores on this measure indicate higher levels of mindfulness. Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) reported good internal consistency among social workers ( α =0.87, and 0.87, respectively). This scale also delves into the components of mindfulness comprising 2 subscales which measure components of the CBPM; Letting Go/Non-attachment (SMQ-LG); Non-aversion (SMQ-AV). Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) reported acceptable Cronbach’s alpha of 0.76 and 0.74 for the SMQ-LG, and good to acceptable Cronbach’s α of 0.82 and 0.60 for the SMQ-AV, respectively. McDonald's omega for the SMQ in this study was 0.60 for SMQ-AV, and 0.70 for SMQ-LG (Hayes & Coutts, 2020 ).

The Experiences Questionnaire – Decentering (EQ-D; Fresco et al., 2007 ) is a valid and reliable 11-item measure of the capacity to regulate attention through decentering i.e., an individual’s ability to objectively observe their thoughts and emotions (Fresco et al., 2007 ). The EQ-D is scored on a 5-point Likert scale (1= never ; 5= always ), and scores range from 11-55, with higher scores indicative of higher levels of attention regulation. The reliability of the scores on the EQ-D in both RCTs with social workers was deemed to be acceptable (Cronbach’s α = 0.84) (Maddock et al., 2023 ), and (Cronbach’s α = 0.84) (Maddock et al., 2024 ). McDonald's omega for the EQ-D in this study was 0.88 (Hayes & Coutts, 2020 ).

The Philadelphia Mindfulness – Acceptance Subscale (PHLSMS-A) is a valid and reliable 10-item measure of acceptance (Cardaciotto et al., 2008 ). It is scored on a 5-point Likert scale (1= never ; 5= very often ); with total scores ranging from 10-50. Higher levels of acceptance are indicated by lower scores. Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) reported good scale reliabilities among social workers, with a Cronbach’s alphas of 0.89 and 0.88 respectively. McDonald's omega for the PHLMS-A in this study was 0.89 (Hayes & Coutts, 2020 ).

The Self-Compassion Scale-Short Form (SCS-SF; Raes et al., 2011 ) is a valid and reliable 12-item measure of self-compassion, which is scored on a 5-point Likert scale (1= almost never ; 5= almost always ). Higher scores indicate higher levels of self-compassion. The reliability of the scores on the SCS was found to be acceptable in both Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) (Cronbach’s α = 0.87, and 0.86). McDonald's omega for the SCS in this study was 0.86 (Hayes & Coutts, 2020 ).

The Penn State Worry Questionnaire (PSWQ; Meyer et al., 1990 ) is a valid and reliable 16-item measure of the intensity, pervasiveness, and uncontrollability of worry (Startup & Erickson, 2006 ). PSWQ scores range from 16-80, with higher scores indicating higher levels of worry (Startup & Erickson, 2006 ). The reliability of the scores on the PSWQ in in both Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) was found to be high (Cronbach’s α = 0.95, and 0.94). McDonald's omega for the PSWQ in this study was 0.94 (Hayes & Coutts, 2020 ).

The Rumination Reflection Questionnaire (RRQ; Trapnell & Campbell, 1999 ) is valid and reliable 12-item measure of rumination. Scores range from 12-60 on this 5-point Likert scale (1= strongly disagree ; 5= strongly agree ), with higher scores indicating higher rates of rumination. Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) both highlighted the scales high internal consistency with social workers (Cronbach’s α = 0.94, and 0.94). McDonald's omega for the RRQ in this study was 0.92 (Hayes & Coutts, 2020 ).

Data Analyses

To test if mediated effect, and direct and mediated effect CBPM models of stress, emotional exhaustion, depersonalisation, and personal accomplishment would be good fits to the data, structural equation modelling (SEM) was used. SEM is a multicomponent statistical process which can allow researchers to construct theoretical models, such as the CBPM, and test to see if the relationships set out in such theories fit a dataset, providing evidence for, or against, the theory as being a useful explanatory framework (Malaeb et al., 2000 ). To ensure that the SEM analyses were statistically valid, through having enough participants to support the number of variables included in each SEM model, the total score for the SMQ was used to assess levels of mindfulness. This means that each CBPM SEM model included four (mindfulness, self-compassion, attention regulation, and acceptance) of the six CBPM domains. Data were screened for errors, missing values, and outliers using the interquartile rule (Hoaglin et al., 1986 ) on SPSS 27 (IBM, Armonk, NY), with whisker and box plots. A complete dataset is needed to construct SEMs, so any missing data was dealt with using the expectation-maximization method (Schumacker & Lomax, 2016 ). Measurement and structural models based on the CBPM were developed (model specification and identification). Four mediated effect, and four direct and mediated effect models representing different potential explanatory frameworks of how the CBPM might help to explain changes in stress, emotional exhaustion, depersonalisation, and personal accomplishment were specified and estimated using LISREL 10.3.4.4 (Joreskog & Sorbom, 2009 ). A covariance and an asymptotic weight matrix were computed with the parameters being estimated using maximum likelihood (Joreskog & Sorbom, 2009 ). The fit of each model to the data was then assessed (model estimation and testing) (Schumacker & Lomax, 2016 ). When reporting the assessment of model fit in SEM, it is important to use several different indices, as there are no consensus rules within the literature that researchers can follow (Crowley & Fan, 1997 ; Hooper et al., 2008 ). In line with Hooper et al. ( 2008 ) and Kline ( 2005 ), this study includes the Chi-Square statistic (where a non-significant chi-square indicates model fit), its degrees of freedom and p value. It also includes a number of relative fit indices: the Standardised Root Mean Square Residual ( SRMR ) (< 0.05, Hooper et al., 2008 ), the Comparative Fit Index ( CFI ) (≥ 0.95, Hu & Bentler, 1999 ), and the parsimony fit index - PNFI (< 0.05, Hooper et al., 2008 ), and the RMSEA , which is an absolute fit index, where values which are ≥0.06 are taken indicate good model fit (Hu & Bentler, 1999 ). The Akaike Information Criterion ( AIC ; Akaike, 1987 ) was used for the purposes of model comparison, with the smallest value being indicative of the most parsimonious model (Byrne, 2012 ). In this study, if the Chi-Square test, more than one relative fit index ( SRMR, CFI, PNFI ), and the RMSEA are deemed to indicate model fit in this study, the model will be deemed to be a good fit to the data. If only one fit index (the chi-square , more than one relative fit index such as  SRMR, CFI, PNFI , or the RMSEA ) deems the model to fit the data, the model will be deemed to be an acceptable fit. If none deem the model to fit the data, the model will be deemed to be a poor model fit.

To examine the direct and indirect effect (through worry and/or rumination) of each CBPM domain (mindfulness, attention regulation, acceptance, self-compassion, non-attachment, non-aversion) on social worker stress and burnout (emotional exhaustion, depersonalisation, and personal accomplishment), several conditional indirect and direct effect models were tested. These analyses were performed using SPSS 27.0 (IBM, Armonk, NY), with the SPSS PROCESS macro (Hayes, 2018 ). Preacher and Hayes’ bias-corrected non-parametric bootstrapping techniques were used, with 5000 bootstrapped samples (Hayes, 2018 ). The models were considered statistically significant if the point estimates of the 95% confidence intervals (CIs) did not contain zero (Furr, 2018 ; Hayes, 2018 ). To further examine that any significant effects found where due to MBP participation and not merely associations between the variables of interest, these tests were also run on the participants baseline data, and compared. If significant differences were found i.e., significant direct or indirect effects were found on the change score dataset, and not in the baseline dataset, these significant effects were attributed to MBP participation.

SEM Model Fit Statistics

The model fit statistics for all 8 mediated, and direct and mediated CBPM models are presented in Table 1 .

The fit indices in Table 1 indicate that the mediated effects CBPM model has a good model fit to the data on stress change scores. The Chi-Square is non-significant ( p = 0.76), which is > 0.05; the RMSEA = 0.00, which is < 0.06 (Hu & Bentler, 1999 ), and the CFI = 0.99, which is > 0.95 (Hu & Bentler, 1999 ), and the SRMR = 0.02, which is < 0.05 (Hooper et al., 2008 ). The direct and mediated effects CBPM model was also found to be a good model fit to the data on stress change scores. The Chi-Square is non-significant ( p = 0.40), which is > 0.05; the RMSEA = 0.00, which is < 0.06 (Hu & Bentler, 1999 ), and the CFI = 0.99, which is > 0.95 (Hu & Bentler, 1999 ), and the SRMR = 0.01, which is < 0.05 (Hooper et al., 2008 ).

Burnout-Emotional Exhaustion

The fit indices in Table 1 indicate that the mediated effects CBPM model was not a good model fit to the data on Burnout-emotional exhaustion change scores. The direct and mediated effects CBPM model was found to be a good model fit to the data on Burnout-emotional exhaustion change scores. The Chi-Square is non-significant ( p = 0.39), the RMSEA = 0.00, the CFI = 0.99, and the SRMR = 0.01.

Burnout-Depersonalisation

The fit indices in Table 1 indicate that the mediated effects CBPM model has a good model fit to the data on Burnout-depersonalisation. The Chi-Square is non-significant ( p = 0.2), the RMSEA = 0.07, the CFI = 0.99, and the SRMR = 0.04. The direct and mediated effects CBPM model was found to be a good fit to the data on Burnout-depersonalisation change scores. The Chi-Square is non-significant ( p = 0.4), the RMSEA = 0.00, the CFI = 0.99, and the SRMR = 0.01.

Burnout-Personal Accomplishment

The fit indices in Table 1 indicate that the mediated effects CBPM model has a good model fit to the data on Burnout-personal accomplishment change scores. The Chi-Squar e is non-significant ( p = 0.58), the RMSEA = 0.00, the CFI = 0.99, and the SRMR = 0.03. The direct and mediated effects CBPM model was found to be a good fit to the data on Burnout-personal accomplishment change scores. The Chi-Square is non-significant ( p = 0.40), the RMSEA = 0.00, the CFI = 0.99, and the SRMR = 0.01.

Conditional Direct and Indirect Effects

The conditional direct effects of the CBPM domains on the stress and burnout (emotional exhaustion, depersonalisation, personal accomplishment) of social workers is presented in Table 2 , with the significant direct effects italicised. Changes in mindfulness ( β = -0.26, SE = 0.07: 95% CI -0.40, -0.11), attention regulation ( β = -0.38, SE = 0.14: 95% CI -0.66, -0.09), and non-aversion ( β = -0.58, SE = 0.19: 95% CI -0.96, -0.19) scores post MBP participation were found to have a conditional direct effect on emotional exhaustion. Changes in mindfulness ( β = -0.15, SE = 0.06: 95% CI -0.3, -0.03), self-compassion ( β = -0.18, SE = 0.09: 95% CI -0.36, -0.01), and non-aversion ( β = -0.35, SE = 0.18: 95% CI -0.7, -0.1), scores post MBP participation were found to have a conditional direct effect on depersonalisation. Changes in mindfulness scores were also found to have significant conditional direct effect on both stress and personal accomplishment, however these significant relationships were also present within the baseline data set. Changes in attention regulation were also found to have significant conditional direct effect on both depersonalisation and personal accomplishment, however these significant relationships were also present at baseline. Changes in non-attachment scores were found to have significant conditional direct effect on personal accomplishment, however this significant relationship was also found within the baseline dataset.

The conditional indirect effects of the CBPM domains on the stress and burnout (emotional exhaustion, depersonalisation, personal accomplishment) of social workers, when mediated by worry or rumination are presented in Table 3 , with the significant indirect effects italicised. Changes in self-compassion ( β = 0.08, SE = 0.05: 95% CI 0.01, 0.18) scores were significantly associated with perceived stress when mediated by reduced rumination scores. Changes in non-attachment, self-compassion, mindfulness, attention regulation, and acceptance scores were significantly associated with perceived stress when mediated by reduced worry scores, however these significant relationships were present within the baseline dataset. Changes in non-attachment, non-aversion, and attention regulation were significantly associated with perceived stress when mediated by reduced rumination scores, however these significant relationships were found in the dataset at baseline.

There were no significant indirect effects found on emotional exhaustion due to MBP participation alone. Self-compassion, and acceptance were significantly associated with Burnout-emotional exhaustion when mediated by reduced worry scores, but these relationships were also found at baseline.

Changes in attention regulation ( β = -0.34, SE = 0.11: 95% CI -0.56, -0.11) was significantly associated with Burnout-depersonalisation when mediated by reduced worry scores. Self-compassion ( β = -0.07, SE = 0.04: 95% CI -0.16, -0.01), and mindfulness change scores ( β = -0.09, SE = 0.04: 95% CI -0.18, -0.01) were significantly associated with Burnout-depersonalisation when mediated by reduced rumination scores. Non-attachment, non-aversion, and acceptance were significantly associated with Burnout-depersonalisation when mediated by reduced rumination scores, but these relationships were also present in the baseline data.

Non-aversion ( β = 0.17, SE = 0.09: 95% CI 0.02, 0.36), self-compassion ( β = 0.08, SE = 0.04: 95% CI 0.01, 0.17), attention regulation ( β = 0.17, SE = 0.09: 95% CI 0.02, 0.36), acceptance ( β = -0.05, SE = 0.03: 95% CI -0.11, -0.01) were significantly associated with Burnout-personal accomplishment when mediated by reduced rumination scores.

One of the main aims of this paper was to provide a greater understanding of how MBPs alleviate stress and burnout in social workers, using pre and post RCT data from a sample of social work professionals who had completed a MBP, using the CBPM as a guiding theoretical framework (Maddock, 2023 ). This paper provides promising preliminary evidence that the mediated effects, and direct and mediated effects CBPM models of stress, depersonalisation and personal accomplishment are good explanatory frameworks of how engaging in a MBP might improve these outcomes for social workers. This paper also provides initial evidence that the direct and mediated effects CBPM model is a good explanatory framework of how changes in emotional exhaustion may occur for social workers due to MBP participation.

The other key aim of this paper is to investigate if changes in each CBPM domain (mindfulness, attention regulation, acceptance, self-compassion, non-attachment, and non-aversion) and mediating variable (worry and rumination) both directly and indirectly predicted the stress and burnout levels of the social workers who completed the MBPs in Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ). Several of the significant direct and indirect effects found in the analysis of the change scores from Maddock et al. ( 2023 ) and Maddock et al. ( 2024 ) were also found within the baseline data, meaning that we cannot attribute these significant effects to MBP participation. Several other significant direct and indirect effects were found to exist in the change score data, which were not found in the baseline data, meaning that we can attribute these effects to MBP participation. These significant findings are in line with the results of several systematic reviews (Alsubaie et al., 2017 ; Maddock & Blair, 2023 ; Van der Velden et al., 2015 ), which have highlighted the potentially complex direct and mediated change processes in outcomes such as stress, emotional exhaustion, depersonalisation, and personal accomplishment following MBP participation. The results from this paper highlighted how different outcomes (e.g., emotional exhaustion) were significantly predicted directly by changes in individual CBPM domains (e.g., attention regulation), that individual outcomes were also directly predicted by multiple CBPM domains (e.g., changes in non-aversion, and mindfulness, and self-compassion scores directly predicted changes in depersonalisation scores), and that a number of CBPM domains predicted changes in a number of different outcomes (e.g., changes in mindfulness and non-aversion predicted changes in both emotional exhaustion and depersonalisation). It is also clear that this complexity applies to the results of the conditional indirect effect analyses, where improvements in individual CBPM domains (e.g., self-compassion) were found to have a significant indirect effect through reduced rumination on individual outcomes (e.g., stress), and changes in a number of CBPM domains (e.g., attention regulation, self-compassion) were found to significantly predict changes in a number of CBPM domains when mediated by either change in worry and/or rumination (e.g., depersonalisation and personal accomplishment). These findings support Kazdin ( 2009 ) who highlighted how multiple outcomes, such as stress, emotional exhaustion, depersonalisation, and personal accomplishment can be improved through multiple pathways i.e., that by improving self-compassion, stress might improve both directly and indirectly through reduced rumination.

The results from this study indicate that MBP participation appears to reduce the emotional exhaustion of social workers through improvements in mindfulness, attention regulation, and non-aversion. The finding that attention regulation had a conditional direct effect on emotional exhaustion is supported by Crowder and Sears ( 2017 ), who in a small-scale mixed methods exploratory study examining the effects of MBP participation on social workers, highlighted the potential for improved attention regulation skills to be an important protective factor against burnout in social work. In an RCT study, Hülsheger et al. ( 2013 ) found that increased mindfulness post MBP participation predicted reduced emotional exhaustion in Dutch public sector employees, supporting the results from this study. The findings that non-aversion (which in Western mental health literature most closely resembles experiential avoidance) significantly directly predicted emotional exhaustion is in line with Iglesias et al. ( 2010 ) who found that experiential avoidance was negatively associated with the emotional exhaustion of critical care nurses.

Depersonalisation in social workers appears to reduce through both direct and mediated pathways post MBP participation. Supporting this paper’s findings, Maddock and McCusker ( 2022 ) found that changes in mindfulness was a significant predictor of change in depersonalisation in social work students who completed an MBSWSC programme. The findings that non-aversion significantly directly predicted depersonalisation is also in line with Maddock and McCusker ( 2022 ), and with Iglesias et al. ( 2010 ) who both found that experiential avoidance was negatively associated with the depersonalisation of critical care nurses. The finding that changes in self-compassion post MBP participation had a conditional direct effect on the levels of depersonalisation experienced by the social workers in this study is also supported by Maddock and McCusker ( 2022 ). The findings relating to the potential roles that mindfulness, self-compassion, and non-aversion may play in reducing depersonalisation not only has important implications for our understanding of how social worker burnout may be reduced through MBP participation, but they could also help us to understand how MBP participation may improve social work practice. Depersonalisation, often conceptualised as a loss of empathy, can lead to social workers developing negative attitudes towards their service users, which can lead to increased cynicism and insensitivity being shown to the recipients of their service (Frieiro Padin et al., 2021 ). The results from this study indicate that MBP participation improves social worker’s competencies in being mindful, self-compassionate, and their ability to observe their thinking processes non-judgementally and without defence. It appears that if social workers were to engage in an MBP, the development of these three CBPM domains would allow them to develop increased feelings of empathy for their service users, which is a core therapeutic process in social work (Rogers, 1967 ; Tanner, 2020 ), and a key ingredient of effective social work practice (Gerdes & Segal, 2011 ).

The finding that changes in four (self-compassion, attention regulation, acceptance, and non-aversion) out of the six CBPM domains post MBP participation had conditional indirect effects on the personal accomplishment of the social workers when mediated by rumination, and three (self-compassion, mindfulness, and attention regulation) out of the six CBPM domains post MBP participation had conditional indirect effects on the depersonalisation of the social workers when mediated by either rumination or worry is in line with the CBPM (Maddock, 2023 ). These results provide further empirical support that the CBPM could be a useful explanatory framework for how experiences of depersonalisation and feelings of personal accomplishment may change due to MBP participation (Maddock, 2023 ). The important role that reducing rumination may have played in reducing the stress, depersonalisation and personal accomplishment of the social workers in this study, is in line with systematic reviews conducted by Gu et al. ( 2015 ) and Alsubaie et al. ( 2017 ) with clinical and non-clinical populations respectively, which highlighted the key role that reduced rumination post MBP participation appears to play in reducing the risk of onset, maintenance and recurrence of stress and burnout.

This paper provides some initial preliminary evidence, and greater theoretical clarity on how MBPs might support improvements in stress and burnout in social workers. This theoretical clarity is particularly important, as there is an increasing awareness about the need reduce social worker stress, burnout and subsequent attrition within the profession (Kinman et al., 2020 ; Romero-Martín et al., 2024 ; Turley et al., 2021 ), and the potentially important role that MBPs could play in social work education and practice to meet this need (Beer et al., 2020 ; Beer et al., 2021 ; Maddock, 2023 ). This literature highlights that MBPs could help to reduce stress and burnout, by supporting social workers to develop self-care competencies (Beer et al., 2020 , 2021 ; Maddock, 2023 ; Romero-Martín et al., 2024 ) which are increasingly becoming recognised by social work regulatory bodies and international social work associations as key standards of proficiency that help to ensure safe and effective social work practice (NASW, 2021 ; NISCC, 2015 ; SWRB, 2019 ). It has been argued that the significant time commitments that accompany participation in MBSR and MBCT may not be suitable for health and social care professionals (Craigie et al., 2016 ). There has been calls in this literature for tailored MBPS, which are refined and adapted to meet different professional occupational needs (Calcagni et al., 2021 ). Social work focussed MBPs, which are refined and adapted to meet the needs of already busy and stressed social workers are needed (Beer et al., 2020 ; Maddock, 2023 ). The identification of each significant conditional direct and indirect effect of changes in each CBPM domain and mediating variable on the stress and burnout of social workers post MBP programme in this study, helps to highlight which mechanisms of action could be focussed on, or intensified when MBPs for social workers are developed or modified.

The development of each CBPM domain, through MBP participation, would likely support improved social work practice through the enhancement of a number of social work skills and competencies, including: (1) increased resilience and self-awareness, through reduced fear of the negative thoughts and emotions that are commonly avoided in social work e.g., shame (Gibson, 2016 ). The development of each CBPM domain would support an increased capacity to approach and recognise when difficult thoughts and emotions are triggered in practice, and how to process these adaptively, using mindfulness-based skills and practices e.g., acceptance and self-compassion (Maddock, 2023 ); (2) increased empathy and compassion, which would allow social workers to more clearly understand what the service user’s needs and wishes are, facilitating improved assessment and relationship-based practice (Gibson, 2016 ; Klinger et al., 2012 ); (3) improved anti-oppressive practice through increased skills in reflection (Schon, 1983 ), and reflexivity (McCusker, 2022 ). The development of each CBPM domain would support social workers to develop a clearer understanding of how negative thoughts and emotions arise e.g., shame, and corrosive impact they can have on their own lives, if they are not regulated. This would likely support the still incomplete, but more accurate assessment of the impact of the service user’s cultural and structural environment on their lived experience (Thompson, 2012 ). For example, if a service user in recovery from a substance use problem is experiencing feelings of shame due to stigma and/or social exclusion, through enhanced capacities in reflection and reflexivity, the social worker may be in a better position to recognise that there is an increased risk of the service user re-initiating substance use, particularly if they are not given the opportunity to emotionally ventilate (Elison et al., 2006 ).

Limitations and Future Research

The results are preliminary and should be interpreted with caution due to several limitations. The external validity of this study’s findings is limited by self-selection bias, as the social workers in this study wanted to take part in this study, and were randomly allocated to and completed one of the MBPs. This makes it difficult to establish how representative the participants in this study are of wider populations of social workers. The use of a single purposive and convenience sample also means that these findings cannot be generalised to a larger population of social workers (Unrau & Grinnell, 2011 ). The nature of SEM means that the well-fitting CBPM models for each outcome may be one of many possible models that also fit the data. The patterns of relationships in the data appear to be consistent with the theoretical relationships set out in the CBPM, but do not definitively prove that the relationships exist (Schumacker & Lomax, 2016 ). Both the MBSWSC programme and the MSC control group occurred within supportive group environments with trained facilitators. The impact of these environments, in which participants could share and learn from each other, and have a positive social experience is a potential confounding variable that may have impacted both MBSWSC’s and MSC’s effects. The use of one data collection point, which the change scores function as, means that causality cannot be asserted (Kazdin, 2007 ; Mathieu & Taylor, 2006 ). This study is only powered to control for type II error for large moderated mediation effects. This study was underpowered to detect for small to medium moderated mediation effects (Fritz & MacKinnon, 2007 ). The number of pathways tested likely reduced the study’s power further, increasing the potential risk of type II error, and of potentially statistically significant pathways not being detected. The results from this study indicate that future research that aims to investigate potential mindfulness mechanisms of action, particularly those within social work research, might benefit from using the CBPM as a guiding theoretical framework. The CBPM would benefit from having its validity and reliability established over time, across other groups of social workers, and in different cultural contexts.

In conclusion, this study provides some promising initial preliminary evidence for the CBPM as being a useful explanatory framework of how social worker stress and burnout might be improved through MBP participation. The study’s results also suggest that if social workers engage in MBPs which positively change each CBPM domain and mediating variable, they are likely to experience reduced stress and burnout.

Data Availability

The data associated with this paper is available upon reasonable request.

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Open Access funding provided by the IReL Consortium This work was supported by the Office of Social Services, within the Department of Health, Northern Ireland.

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This study was performed in line with the principles of the Declaration of Helsinki. Approval was granted by the Research Ethics Committee, School of Social Sciences, Education and Social Work at Queen’s University Belfast (REF_204_2021 and REF_167_2122).

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4.2: Calculus of Functions of Two Variables

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Now that you have some familiarity with functions of two variables, it's time to start applying calculus to help us solve problems with them. In Chapter 2, we learned about the derivative for functions of two variables. Derivatives told us about the shape of the function, and let us find local max and min – we want to be able to do the same thing with a function of two variables.

First let's think. Imagine a surface, the graph of a function of two variables. Imagine that the surface is smooth and has some hills and some valleys. Concentrate on one point on your surface. What do we want the derivative to tell us? It ought to tell us how quickly the height of the surface changes as we move… Wait, which direction do we want to move? This is the reason that derivatives are more complicated for functions of several variables – there are so many (in fact, infinitely many) directions we could move from any point.

It turns out that our idea of fixing one variable and watching what happens to the function as the other changes is the key to extending the idea of derivatives to more than one variable.

Partial Derivatives

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Suppose that \(z = f(x, y)\) is a function of two variables.

The partial derivative of \(f\) with respect to \(x\) is the derivative of the function \(f(x,y)\) where we think of \(x\) as the only variable and act as if \(y\) is a constant.

The partial derivative of \(f\) with respect to \(y\) is the derivative of the function \(f(x,y)\) where we think of \(y\) as the only variable and act as if \(x\) is a constant.

The with respect to \(x\) or with respect to \(y\) part is really important – you have to know and tell which variable you are thinking of as THE variable.

Geometrically

Geometrically the partial derivative with respect to \(x\) gives the slope of the curve as you travel along a cross-section, a curve on the surface parallel to the \(x\)-axis. The partial derivative with respect to \(y\) gives the slope of the cross-section parallel to the \(y\)-axis.

Notation for the Partial Derivative

The partial derivative of \(z = f(x,y)\) with respect to \(x\) is written as \[ f_x(x,y) \nonumber \] or simply \[ f_x \quad\text{or}\quad z_x. \nonumber \]

The Leibniz notation is \[ \frac{\partial f}{\partial x} \nonumber \] or \[ \frac{\partial z}{\partial x}. \nonumber \]

We use an adaptation of the \( \frac{\partial z}{\partial x} \) notation to mean find the partial derivative of \(f(x,y)\) with respect to \(x\) : \[ \frac{\partial}{\partial x}\left(f(x,y)\right)=\frac{\partial f}{\partial x} \nonumber \]

To estimate a partial derivative from a table or contour diagram

The partial derivative with respect to \(x\) can be approximated by looking at an average rate of change, or the slope of a secant line, over a very tiny interval in the \(x\)-direction (holding \(y\) constant). The tinier the interval, the closer this is to the true partial derivative.

To compute a partial derivative from a formula

If \(f(x,y)\) is given as a formula, you can find the partial derivative with respect to \(x\) algebraically by taking the ordinary derivative thinking of \(x\) as the only variable (holding \(y\) fixed).

Of course, everything here works the same way if we're trying to find the partial derivative with respect to \(y\) – just think of \(y\) as your only variable and act as if \(x\) is constant.

The idea of a partial derivative works perfectly well for a function of several variables: you focus on one variable to be THE variable and act as if all the other variables are constants.

Example \(\PageIndex{1}\)

Here is a contour diagram for a function \(g(x,y)\).

Use the diagram to answer the following questions:

• Estimate \( g_x(3,5) \) and \( g_y(3,5) \).
• Where on this diagram is \( g_x \) greatest? Where is \( g_y \) greatest?

Now we can find the average rate of change: \[ \begin{align*} \text{Average rate of change } & = \frac{\text{(change in output)}}{\text{(change in input)}} \\ & = \frac{\Delta g}{\Delta x}\\ & = \frac{0.7-0.6}{4.2-3}\\ & = \frac{1}{12}\approx 0.083 \end{align*} \nonumber \] We can do the same thing by going to the next point we can read to the left, which is \(g(2.4,5) = 0.5\). Then the average rate of change is \[ \frac{\Delta g}{\Delta x}=\frac{0.5-0.6}{2.4-3}=\frac{1}{6}\approx 0.167.\nonumber \]

Either of these would be a fine estimate of \( g_x(3,5) \) given the information we have, or we could take their average. We can estimate that \( g_x(3,5)\approx 0.125 \).

Estimate \( g_y(3,5) \) the same way, but moving on the vertical line. Using the next point up, we get the average rate of change is \[ \frac{\Delta g}{\Delta y}=\frac{0.7-0.6}{5.8-5}=\frac{1}{8}=0.125.\nonumber \] Using the next point down, we get \[ \frac{\Delta g}{\Delta y}=\frac{0.5-0.6}{4.5-5}=\frac{1}{5}=0.2.\nonumber \] Taking their average, we estimate \( g_y(3,5)\approx 0.1625 \).

• \( g_x \) means \(x\) is our only variable, and we're thinking of \(y\) as a constant. So we're thinking about moving across the diagram on horizontal lines. \( g_x \) will be greatest when the contour lines are closest together, i.e., when the surface is steepest – then the denominator in \( \frac{\Delta g}{\Delta x} \) will be small, so \( \frac{\Delta g}{\Delta x} \) will be big. Scanning the graph, we can see that the contour lines are closest together when we head to the left or to the right from about (0.5, 8) and (9, 8). So \( g_x \) is greatest at about (0.5, 8) and (9, 8). For \( g_y \), we want to look at vertical lines. \( g_y \) is greatest at about (5, 3.8) and (5, 12).

Example \(\PageIndex{2}\)

Cold temperatures feel colder when the wind is blowing. Windchill is the perceived temperature, and it depends on both the actual temperature and the wind speed – a function of two variables! You can read more about windchill at www.nws.noaa.gov/om/windchill/. Below is a table that shows the perceived temperature for various temperatures and windspeeds.

Note that they also include the formula, but for this example we'll use the information in the table.

• What is the perceived temperature when the actual temperature is 25\(^{\circ}\)F and the wind is blowing at 15 miles per hour?
• Suppose the actual temperature is 25\(^{\circ}\)F. Use information from the table to describe how the perceived temperature would change if the wind speed increased from 15 miles per hour?
• Reading the table, we see that the perceived temperature is 13\(^{\circ}\)F

What are the units? \(W\) is measured in \(^{\circ}\)F and \(V\) is measured in mph, so the units here are \(^{\circ}\)F/mph. And that lets us describe what happens: The perceived temperature would decrease by about 0.4\(^{\circ}\)F for each mph increase in wind speed.

Example \(\PageIndex{3}\)

Find \( f_x \) and \( f_y \) at the points (0, 0) and (1, 1) if \( f(x,y)=x^2-4xy+4y^2 \).

To find \( f_x \), take the ordinary derivative of \(f\) with respect to \(x\), acting as if \(y\) is constant: \[ f_x(x,y)=2x-4y. \nonumber \]

Note that the derivative of the \( 4y^2 \) term with respect to \(x\) is zero because it's a constant (as far as \( x \) is concerned).

Similarly, \[ f_y(x,y)=-4x+8y. \nonumber \]

Now we can evaluate these at the points:

\( f_x(0,0)=0 \) and \( f_y(0,0)=0 \); this tells us that the cross sections parallel to the \(x\)- and \(y\)- axes are both flat at (0,0).

\( f_x(1,1)=-2 \) and \( f_y(1,1)=4 \); this tells us that above the point (1, 1), the surface decreases if we move to more positive \(x\) values and increases if we move to more positive \(y\) values.

Example \(\PageIndex{4}\)

Find \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) if \( f(x,y)=\frac{e^{x+y}}{y^3+y}+y\ln(y) \).

\( \frac{\partial f}{\partial x} \) means \(x\) is our only variable, we're thinking of \(y\) as a constant. Then we'll just find the ordinary derivative. From \(x\)'s point of view, this is an exponential function, divided by a constant, with a constant added. The constant pulls out in front, the derivative of the exponential function is the same thing, and we need to use the chain rule, so we multiply by the derivative of that exponent (which is just 1): \[ \frac{\partial f}{\partial x}=\frac{1}{y^3+y}e^{x+y}. \nonumber \]

\(\frac{\partial f}{\partial y}\) means that we're thinking of \(y\) as the variable, acting as if \(x\) is constant. From \(y\)'s point of view, \(f\) is a quotient plus a product, so we'll need the quotient rule and the product rule: \[ \begin{align*} \frac{\partial f}{\partial y} & = \frac{(\ )(\ )-(\ )(\ )}{(\ )^2}+(\ )(\ )+(\ )(\ ) \\ & = \frac{\left( e^{x+y}(1) \right)\left( y^3+y \right)-\left( e^{x+y} \right)\left( 3y^2+1 \right)}{\left( y^3+y \right)^2}+\left( 1 \right)\left( \ln(y) \right)+\left( y \right)\left( \frac{1}{y} \right) \end{align*} \nonumber \]

Example \(\PageIndex{5}\)

Find \( f_z \) if \( f(x,y,z,w)=35x^2w-\frac{1}{z}+yz^2 \).

\( f_z \) means we act as if \(z\) is our only variable, so we'll act as if all the other variables (\(x\), \(y\), and \(w\)) are constants and take the ordinary derivative: \[ f_z(x,y,z,w)=\frac{1}{z^2}+2yz. \nonumber \]

Using Partial Derivatives to Estimate Function Values

We can use the partial derivatives to estimate values of a function. The geometry is similar to the tangent line approximation in one variable. Recall the one-variable case: if \(x\) is close enough to a known point \(a\), then \[ f(x)\approx f(a)+f'(a)(x-a). \nonumber \] In two variables, we do the same thing in both directions at once:

Approximating Function Values with Partial Derivatives

To approximate the value of \(f(x, y)\), find some point \((a, b)\) where

• \((x, y)\) and \((a, b)\) are close, that is, \(x\) and \(a\) are close and \(y\) and \(b\) are close.
• You know the exact values of \(f(a, b)\) and both partial derivatives there.

Then \[ f(x,y)\approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b). \nonumber \]

Notice that the total change in \(f\) is being approximated by adding the approximate changes coming from the \(x\) and \(y\) directions. Another way to look at the same formula: \[ \Delta f\approx f_x\Delta x+f_y\Delta y. \nonumber \]

How close is close? It depends on the shape of the graph of \(f\). In general, the closer the better.

Example \(\PageIndex{6}\)

Use partial derivatives to estimate the value of \( f(x,y)=x^2-4xy+4y^2 \) at (0.9, 1.1).

Note that the point (0.9, 1.1) is close to an easy point, (1, 1). In fact, we already worked out the partial derivatives at (1, 1): \( f_x(x,y)=2x-4y \) so \( f_x(1,1)=-2 \), and \( f_y(x,y)=-4x+8y \) so \( f_y(1,1)=4 \). We also know that \( f(1,1)=1 \).

So, \[ f(0.9,1.1)\approx 1-2(-0.1)+4(0.1)=1.6. \nonumber \]

Note that in this example it would have been possible to simply compute the exact answer: \[ f(0.9,1.1)=(0.9)^2-4(0.9)(1.1)+4(1.1)^2=1.69. \nonumber \] Our estimate is not perfect, but it's pretty close.

Example \(\PageIndex{7}\)

Here is a contour diagram for a function \(g(x,y)\). Use partial derivatives to estimate the value of \(g(3.2, 4.7)\).

This is the same diagram from before, so we already estimated the value of the function and the partial derivatives at the nearby point (3,5). \(g(3, 5)\) is 0.6, our estimate of \( g_x(3,5)\approx 0.125 \), and our estimate of \( g_y(3,5)\approx 0.1625 \). So \[ g(3.2,4.7)\approx 0.6+(0.125)(0.2)+(0.1625)(-0.3)=0.57625. \nonumber \] Note that in this example we have no way to know how close our estimate is to the actual value.

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These 2 EV Stocks Are Still Under the Radar. What Do You Need to Know About Them?

May 20, 2024 — 06:15 am EDT

Written by Daniel Miller for The Motley Fool  ->

For investors who jumped into the start-up electric vehicle (EV) industry hoping to find the next Tesla , it's been a rough ride, to say the least. Many start-ups are suffering from a cash crunch, slowing demand, and high costs, among other variables. It has forced some to file bankruptcy, and others, such as Fisker , seem to be nearing the end of their rope.

So are these two new EV stocks the next big thing, or stocks for investors to be wary of? Let's dive in and see.

You likely haven't heard of Zeekr Intelligent Technology (NYSE: ZK) before. That's because the China-based, Geely -owned sub-brand was only created three years ago and only recently went public. Zeekr is focused on EVs and was intended to compete with the likes of NIO and Tesla .

The brand has caught on fairly quickly in China. It achieved a total delivery of 10,000 units of its Zeekr 001 in less than four months after its initial delivery, which is one of the fastest for EV models in China, according to company filings.

Its product lineup focuses primarily on the 001, a five-seater crossover hatchback; the 009, a luxury six-seater multi-purpose vehicle (MPV); the X, a compact SUV; and an upscale sedan model. At the end of 2023, Zeekr had delivered a total of over 196,000 vehicles since its first vehicle delivery in October 2021, almost all of which were delivered in China.

One advantage the company has being a former sub-brand of Geely is the synergies between the two companies. Zeekr develops its EVs on a set of open-source platforms owned by Geely Holding, which are compatible with sedans, SUVs, MPVs, hatchbacks, roadsters, pickup trucks, and a robotaxi. This shared platform improves cost efficiency and simplifies the vehicle development process, giving the company competitive advantages in the market.

There's certainly optimism surrounding the company as it topped its initial public offering (IPO) estimates, and shares rose almost 35% above the offering price on the day of its IPO. It was the largest U.S. listing by a China-based company since 2021, and currently the stock is valued with a market capitalization of just under $7 billion.

VinFast Auto (NASDAQ: VFS) is another foreign-based EV maker but has recently been focused on getting a foothold in the U.S. market. VinFast is a member of Vingroup JSC in Vietnam and was founded in 2017. It manufactures and exports anything from electric SUVs to scooters to buses across Vietnam -- where it dominates -- as well as North America and, soon, Europe.

The young EV maker has a couple of intriguing qualities to it, including a state-of-the-art plant in Hai Phong that boasts up to 90% automation with annual production capacity of 300,000 units. It also has the backing of the larger Vingroup which has poured in over $9 billion to help fund the company.

Further, Vietnamese billionaire Pham Nhat Vuong recently pledged to invest at least another $1 billion of personal wealth into VinFast as the company continued to bleed cash. However, for investors, that kind of backing and support is reassuring at a time when a handful of start-up EVs are on the brink of bankruptcy.

Currently, VinFast is making inroads into the valuable U.S. EV market using a growing dealership network. It recently added 12 franchised dealerships to sell its EV crossovers, which brings its total count to 18 dealerships across seven states. It's the first phase of what the automaker eventually expects to become a network of more than 100 U.S. dealers by the end of 2024.

The next big thing?

Both of these EV stocks have an intriguing investment thesis. VinFast dominates its home market and is prioritizing highly valuable U.S. and European markets while having the financial backing of a much larger Vingroup and the wealthy billionaire Vuong. On the other hand, Zeekr is focused on China and has proven itself capable of quick growth.

Both companies are losing money, highly speculative, and have a long way to go on the path to profitability. However, they are certainly worth keeping on your watchlist as investors try to find the next out-of-the-box idea in the EV industry.

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• Open access
• Published: 14 May 2024

Pregnancy-related complications in patients with endometriosis in different stages

• Khadijeh Shadjoo 1 ,
• Atefeh Gorgin 2 ,
• Narges Maleki 2 ,
• Arash Mohazzab 3 ,
• Maryam Armand 2 ,
• Atiyeh Hadavandkhani 2 ,
• Zahra Sehat 2 &
• Aynaz Foroughi Eghbal 4  

Contraception and Reproductive Medicine volume  9 , Article number:  23 ( 2024 ) Cite this article

133 Accesses

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Endometriosis is one of the most common and costly diseases among women. This study was carried out to investigate pregnancy outcomes in women with endometriosis because of the high prevalence of endometriosis in reproductive ages and its effect on pregnancy-related complications outcomes.

This was a cross-sectional study performed on 379 pregnant women with endometriosis who were referred to the endometriosis clinic of the Avicenna Infertility Treatment Center from 2014 to 2020. Maternal and neonatal outcomes were assessed for the endometriosis group and healthy mothers. The group with endometriosis was further divided into two groups: those who underwent surgery and those who either received medication alone or were left untreated before becoming pregnant. The analysis of the data was done using SPSS 18.

The mean age of the patients was 33.65 ± 7.9 years. The frequency of endometriosis stage ( P  = 0.622) and surgery ( P  = 0.400) in different age groups were not statistically significant. The highest rates of RIF and infertility were in stages 3 ( N  = 46, 17.2%) ( P  = 0.067), and 4 ( N  = 129, 48.3%) ( P  = 0.073), respectively, but these differences were not statistically different, and the highest rate of pregnancy with ART/spontaneous pregnancy was observed in stage 4 without significant differences ( P  = 0.259). Besides, the frequency of clinical/ectopic pregnancy and cesarean section was not statistically different across stages ( P  > 0.05). There is no significant relationship between endometriosis surgery and infertility ( P  = 0.089) and RIF ( P  = 0.232). Most of the people who had endometriosis surgery with assisted reproductive methods got pregnant, and this relationship was statistically significant ( P  = 0.002) in which 77.1% ( N  = 138) of ART and 63% ( N  = 264) of spontaneous pregnancies were reported in patients with endometriosis surgery. The rate of live births (59.4%) was not statistically significant for different endometriosis stages ( P  = 0.638). There was no stillbirth or neonatal death in this study. All cases with preeclampsia ( N  = 5) were reported in stage 4. 66.7% ( N  = 8) of the preterm labor was in stage 4 and 33.3% ( N  = 4) was in stage 3 ( P  = 0.005). Antepartum bleeding, antepartum hospital admission, preterm labor, gestational diabetes, gestational hypertension, abortion, placental complications and NICU admission were higher in stage 4, but this difference had no statistical difference.

Endometriosis is significantly correlated with infertility. The highest rates of RIF and infertility are observed in stages 3 and 4 of endometriosis. The rate of pregnancy with ART/spontaneous pregnancy, preterm labor, preeclampsia and pregnancy-related complications is higher in stage 4. Most of the people who had endometriosis surgery with assisted reproductive methods got significantly pregnant. Clinical/ectopic pregnancy, cesarean sections, and live birth were not affected by the endometriosis stages.

Introduction

The presence of endometrial-like glandular tissue, stroma, or endometrial tissue outside the uterine cavity is known as endometriosis, a chronic gynecological disease that affects 30 to 50% of infertile women [ 1 ]. Endometriosis commonly affects various parts of the female reproductive system, including the pelvic area, ovaries, posterior cul-de-sac, uterine ligaments, pelvic peritoneum, rectovaginal septum, cervix, vulva, vagina, as well as the intestines and urinary system. Endometriosis can cause symptoms like infertility, dysmenorrhea, and chronic pelvic inflammatory disease, which can worsen pain, dyspareunia, and painful bowel movements, ultimately lowering the quality of life for the affected woman [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 ]. Laparoscopic surgery is both the standard surgical procedure and the best treatment for endometriosis [ 14 ]. However, endometriosis remains a problematic issue due to its negative impact on ovarian reserve and the recurrence rate of 40–50% after 5 years of surgery [ 15 , 16 ]. Numerous studies have shown the negative effects of endometriosis on pregnancy, including the increase in preterm labor, placental abruption and cesarean delivery, preeclampsia, placental problems and postpartum hemorrhage, premature rupture of membranes (PROM), preterm birth, small for gestational age (SGA), NICU admission, neonatal mortality and morbidity, and hypertensive disorders of pregnancy (HDP) with low birth weight (LBW) [ 2 , 5 , 6 , 17 ]. Since the effects of endometriosis on the course of pregnancy are still controversial, this work aimed to first identify the negative effects of endometriosis on pregnancy and then determine whether laparoscopic surgery or other drug interventions before pregnancy were beneficial.

Materials and methods

This cross-sectional study was carried out on 379 pregnant women with a history of endometriosis and pregnancy who were referred to the endometriosis clinic of the Avicenna Infertility Treatment Center between January 2014 and January 2020. This study was approved by the Ethics Committee of Avicenna Infertility Treatment Center (IR.ACECR.AVICENNA.REC.1398.031) in accordance with the tents of the Declaration of Helsinki, and the patient’s oral and written consent was obtained to ensure that they participated in the study voluntarily. Specific means of identifying endometriosis were approved after laparoscopic surgery with pathologic confirmation, magnetic resonance imaging (MRI), ultrasound imaging, and clinically confirmed presence of symptoms. Exclusion criteria were less than 22 weeks of gestation at the time of delivery, fetal malformations, and incomplete medical files. Maternal and neonatal outcomes were assessed for the endometriosis group and healthy mothers. The group with endometriosis was further divided into two groups: those who underwent surgery and those who either received medication alone or were left untreated before becoming pregnant. A history of laparoscopic surgery or other surgeries and hormonal therapies (oral contraceptive pills, progestin, and gonadotropin-releasing hormone agonists) were obtained from the patient’s medical files. Maternal characteristics in this study included maternal age, parity, pre-pregnancy weight and BMI, pre-pregnancy blood pressure, chronic hypertension, diabetes mellitus (DM), cholestasis, and assisted reproductive technology (ART). Outcomes evaluated included gestational age, ectopic pregnancy, clinical pregnancy, mode of delivery, antepartum hemorrhage, antepartum hospitalization, preterm labor (< 37 weeks of gestation), labor dystocia, gestational diabetes mellitus (GDM), gestational hypertension, gestational cholestasis, placental abruption and placenta previa, PROM, and abortion. Neonatal characteristics included birth weight, height, SGA, stillbirth, neonatal death, and NICU admission.

Statistical analysis

The data were analyzed using SPSS 18. Normality was checked using the Kolmogorov-Smirnov test. Continuous variables with a normal distribution were summarized as mean and standard deviation and compared between the two groups using an independent t-test. Categorical variables were presented as frequency and percentage to be compared between the two groups using either the Fisher’s exact test or the chi-square ( x 2 ) test. The significance level was defined as p  < 0.05.

During the study period, all 379 women with a mean age of 33.65 ± 7.9 years underwent treatment and were followed until a negative pregnancy test or the end of the pregnancy. The mean marriage duration was 9.72 ± 4.71 years. In this study, 16.1% of the people were in the age group of 25–30 years, 35.6% were in the age group of 30–35 years, and the rest (92.3%) belonged to the age group of more than 40 years. The age group with the highest number of surgeries for endometriosis is 35–40 years. The age group of 25–30 years experiences the highest incidence of stage 1 endometriosis, while the age group of 30–35 years has the highest occurrence of stage 2. Additionally, the age group of 30–35 years also has the highest number of individuals with stage 3, while the age group of 35–40 years has the highest number of people with stage 4 (Table  1 ). The majority of patients in stage 4 needed surgery (89.9%) (Table  2 ).

According to the information in Table  3 , the highest rate of RIF and infertility was in stage 3 ( N  = 46, 17.2%) ( P  = 0.067), and 4 ( N  = 129, 48.3%) ( P  = 0.073), respectively but these differences were not statistically significant. Also, the highest rate of pregnancy with ART/spontaneous pregnancy was observed in stage 4 without significant differences ( P  = 0.259). Besides, the frequency of clinical/ectopic pregnancy and cesarean sections was not statistically different across stages ( P  > 0.05) (Table  4 ).

There is no significant relationship between endometriosis surgery and infertility ( P  = 0.089) and RIF ( P  = 0.232). Most of the people who had endometriosis surgery with assisted reproductive methods got pregnant, and this relationship was statistically significant ( P  = 0.002) in which 77.1% ( N  = 138) of ART and 63% ( N  = 264) of spontaneous pregnancies were reported in patients with endometriosis surgery (Table  3 ).

The rate of live births (59.4%) was not statistically significant by different endometriosis stages ( P  = 0.638) (Table  5 ).

There was no stillbirth or neonatal death in this study. All cases with preeclampsia ( N  = 5) were reported in stage 4. Additionally, 66.7% ( N  = 8) of the preterm labor were in stage 4 and 33.3% ( N  = 4) were in stage 3 in which this difference was statistically significant ( P  = 0.005). Antepartum bleeding (70%), antepartum hospital admission (75.9%), preterm labor (66.7%), gestational diabetes (80%), gestational hypertension (85.7%), abortion (71.4%), placental complications (66.7%) and NICU admission (71%) were higher in stage 4 but this difference had no statistical difference (Table  6 ).

Women with endometriosis have lower fertility rates than ever before, but many of them are still able to give birth because of advancements in IVF and intracytoplasmic sperm injection (ICSI) technology. This cross-sectional research was conducted to examine maternal and neonatal outcomes in endometriosis patients with a history of pregnancy referred to the Avicenna Infertility Treatment Center between January 2014 and January 2020. Patients with endometriosis had a live birth rate of 54.9% Endometriosis is a common cause of infertility, and ART can help patients become pregnant. Despite these interventions, some studies have shown poor pregnancy outcomes in patients with endometriosis. Poor oocyte and embryo quality and impaired endometrial receptivity have been suggested as potential causes of poor clinical outcomes. Burghaus et al. Endometriosis risk factors have been identified as age at menarche, length of each menstrual cycle, length of menstrual years, number of pregnancies, miscarriages, and smoking [ 7 ].

Hardiman et al. concluded that premenstrual spotting lasting more than two days is significantly associated with endometriosis, with a higher predictive rate than painful menstruation and painful intercourse [ 8 ]. It may be more difficult to distinguish between the effects of endometriosis on pregnancy complications and the assisted reproductive process if many endometriosis-affected women use ART techniques during their pregnancies [ 9 ]. According to studies, there is no established association between endometriosis and preeclampsia, meaning that some studies report an increased risk of preeclampsia after endometriosis, while other research reports no change and other research reports a decreasing pattern [ 5 ].

Pérez-López et al. found a significant association between endometriosis and gestational diabetes mellitus [ 10 ]. Maggiore et al. found in 2016 that there is a significant connection between endometriosis and placenta previa. Furthermore, this association is not related to spontaneous insemination or laboratory-assisted reproductive techniques and occurs in both cases. In this context, fetal malformations and cesarean sections can be attributed to placenta previa [ 11 ]. There is a significant association between endometriosis, and cesarean sections and low birth weight in spontaneous fertilization, but no association has been found in ART pregnancies [ 6 ]. Also, Lim et al. found that women diagnosed with endometriosis exhibited a significantly higher incidence of unfavorable pregnancy outcomes in comparison to their counterparts who did not have endometriosis. These unfavorable outcomes associated with endometriosis encompassed preterm labor, preterm birth, preeclampsia, fetal growth restriction, placenta previa, placental abruption, stillbirth, antepartum, and postpartum bleeding. Furthermore, they also demonstrated an augmented risk of blood transfusion, uterine artery embolization, and cesarean hysterectomy in the group of women with endometriosis as opposed to the group without this condition [ 18 ]. Besides, Miura et al. disclosed that there was a heightened incidence of postpartum hemorrhage and placenta previa in the group diagnosed with endometriosis. Nonetheless, the other maternal and neonatal consequences exhibited no significant disparity among patients with/without endometriosis [ 19 ]. Borisova et al. reported that even though patients with endometriosis may achieve pregnancy after undergoing assisted reproductive technologies, they still face a significantly elevated risk of obstetric complications. These complications include, but are not limited to, miscarriage, preterm birth, preeclampsia, placental abnormalities, hemorrhage during labor, the birth of infants who are small for their gestational age, stillbirth, and a higher incidence of cesarean section. Furthermore, it is important to note that acute complications specific to endometriosis can manifest during pregnancy, and in most cases, surgical intervention becomes necessary to address this condition [ 20 ].

Based on the aforementioned studies, the findings of our study were consistent in the majority of respects, and the novelty of our investigation lies in the evaluation of various stages of endometriosis, which holds significance as a considerable number of patients seek the assistance of pertinent clinics during the final stages. Consequently, understanding the adverse effects at the stage of interest can provide clinicians with valuable insights into effectively addressing the patients’ status.

Endometriosis is significantly correlated with infertility. The highest rates of RIF and infertility are observed in stages 3 and 4 of endometriosis. The rate of pregnancy with ART/spontaneous pregnancy, preterm labor, preeclampsia, and pregnancy-related complications is higher in stage 4. Most of the people who had endometriosis surgery with assisted reproductive methods got significantly pregnant. Clinical/ectopic pregnancy, cesarean sections and live birth were not affected by endometriosis stages.

Data availability

The data used in this study can be send after formal and reasonable request to the corresponding author.

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Acknowledgements

We acknowledge all staffs who worked in Avicenna Fertility Center for their great help to perform this study.

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Khadijeh Shadjoo

Infertility Clinic, Avicenna Research Institute, ACECR, Tehran, Iran

Atefeh Gorgin, Narges Maleki, Maryam Armand, Atiyeh Hadavandkhani & Zahra Sehat

Department of Epidemiology, School of Public Health, Iran University of Medical Sciences, Tehran, Iran

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KS and AG wrote the main manuscript text and NM, AM and AFE prepared the data. AM, MA, AHK and ZS analyzed the data and prepared their interpretation. All authors contributed in the writing of the draft. All authors reviewed the manuscript before submission.

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The research conducted in accordance with the tents of the Declaration of Helsinki. The present study was approved by the ethical committee of Avicenna Research Institute, Tehran, Iran. Written informed consent was obtained from all the participants (IR.ACECR.AVICENNA.REC.1398.031).

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Shadjoo, K., Gorgin, A., Maleki, N. et al. Pregnancy-related complications in patients with endometriosis in different stages. Contracept Reprod Med 9 , 23 (2024). https://doi.org/10.1186/s40834-024-00280-0

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