What are Correlation Coefficients?
Correlation coefficients measure the strength of the relationship between two variables. A correlation between variables indicates that as one variable changes in value, the other variable tends to change in a specific direction. Understanding that relationship is useful because we can use the value of one variable to predict the value of the other variable. For example, height and weight are correlated—as height increases, weight also tends to increase. Consequently, if we observe an individual who is unusually tall, we can predict that his weight is also above the average.
In statistics, correlation coefficients are a quantitative assessment that measures both the direction and the strength of this tendency to vary together. There are different types of correlation coefficients that you can use for different kinds of data. In this post, I cover the most common type of correlation—Pearson’s correlation coefficient.
Before we get into the numbers, let’s graph some data first so we can understand the concept behind what we are measuring.
Graph Your Data to Find Correlations
Scatterplots are a great way to check quickly for correlation between pairs of continuous data. The scatterplot below displays the height and weight of pre-teenage girls. Each dot on the graph represents an individual girl and her combination of height and weight. These data are actual data that I collected during an experiment.
At a glance, you can see that there is a correlation between height and weight. As height increases, weight also tends to increase. However, it’s not a perfect relationship. If you look at a specific height, say 1.5 meters, you can see that there is a range of weights associated with it. You can also find short people who weigh more than taller people. However, the general tendency that height and weight increase together is unquestionably present—a correlation exists.
Pearson’s correlation coefficient takes all of the data points on this graph and represents them as a single number. In this case, the statistical output below indicates that the Pearson’s correlation coefficient is 0.694.
What do the Pearson correlation coefficient and p-value mean? We’ll interpret the output soon. First, let’s look at a range of possible correlation coefficients so we can understand how our height and weight example fits in.
How to Interpret Pearson Correlation Coefficients
Pearson’s correlation coefficient is represented by the Greek letter rho (ρ) for the population parameter and r for a sample statistic. This correlation coefficient is a single number that measures both the strength and direction of the linear relationship between two continuous variables. Values can range from -1 to +1.
The greater the absolute value of the Pearson correlation coefficient, the stronger the relationship.
- The extreme values of -1 and 1 indicate a perfectly linear relationship where a change in one variable is accompanied by a perfectly consistent change in the other. For these relationships, all of the data points fall on a line. In practice, you won’t see either type of perfect relationship.
- A coefficient of zero represents no linear relationship. As one variable increases, there is no tendency in the other variable to either increase or decrease.
- When the value is in-between 0 and +1/-1, there is a relationship, but the points don’t all fall on a line. As r approaches -1 or 1, the strength of the relationship increases and the data points tend to fall closer to a line.
The sign of the Pearson correlation coefficient represents the direction of the relationship.
- Positive coefficients indicate that when the value of one variable increases, the value of the other variable also tends to increase. Positive relationships produce an upward slope on a scatterplot.
- Negative coefficients represent cases when the value of one variable increases, the value of the other variable tends to decrease. Negative relationships produce a downward slope.
Statisticians consider Pearson’s correlation coefficients to be a standardized effect size because they indicate the strength of the relationship between variables using unitless values that fall within a standardized range of -1 to +1. Effect sizes help you understand how important the findings are in a practical sense. To learn more about unstandardized and standardized effect sizes, read my post about Effect Sizes in Statistics.
Learn how to calculate correlation in my post, Correlation Coefficient Formula Walkthrough.
Covariance is an unstandardized form of correlation. Learn about it in my post, Covariance: Definition, Formula & Example.
Examples of Positive and Negative Correlation Coefficients
A positive correlation example is the relationship between the speed of a wind turbine and the amount of energy it produces. As the turbine speed increases, electricity production also increases.
A negative correlation example is the relationship between outdoor temperature and heating costs. As the temperature increases, heating costs decrease.
Graphs for Different Correlation Coefficients
Graphs always help bring concepts to life. The scatterplots below represent a spectrum of different Pearson correlation coefficients. I’ve held the horizontal and vertical scales of the scatterplots constant to allow for valid comparisons between them.
Discussion about the Scatterplots
For the scatterplots above, I created one positive correlation between the variables and one negative relationship between the variables. Then, I varied only the amount of dispersion between the data points and the line that defines the relationship. That process illustrates how correlation measures the strength of the relationship. The stronger the relationship, the closer the data points fall to the line. I didn’t include plots for weaker correlation coefficients that are closer to zero than 0.6 and -0.6 because they start to look like blobs of dots and it’s hard to see the relationship.
A common misinterpretation is assuming that negative Pearson correlation coefficients indicate that there is no relationship. After all, a negative correlation sounds suspiciously like no relationship. However, the scatterplots for the negative correlations display real relationships. For negative correlation coefficients, high values of one variable are associated with low values of another variable. For example, there is a negative correlation coefficient for school absences and grades. As the number of absences increases, the grades decrease.
Earlier I mentioned how crucial it is to graph your data to understand them better. However, a quantitative measurement of the relationship does have an advantage. Graphs are a great way to visualize the data, but the scaling can exaggerate or weaken the appearance of a correlation. Additionally, the automatic scaling in most statistical software tends to make all data look similar.
Fortunately, Pearson’s correlation coefficients are unaffected by scaling issues. Consequently, a statistical assessment is better for determining the precise strength of the relationship.
Graphs and the relevant statistical measures often work better in tandem.
Pearson’s Correlation Coefficients Measure Linear Relationship
Pearson’s correlation coefficients measure only linear relationships. Consequently, if your data contain a curvilinear relationship, the Pearson correlation coefficient will not detect it. For example, the correlation for the data in the scatterplot below is zero. However, there is a relationship between the two variables—it’s just not linear.
This example illustrates another reason to graph your data! Just because the coefficient is near zero, it doesn’t necessarily indicate that there is no relationship.
Spearman’s correlation is a nonparametric alternative to Pearson’s correlation coefficient. Use Spearman’s correlation for nonlinear, monotonic relationships and for ordinal data. For more information, read my post Spearman’s Correlation Explained!
Hypothesis Test for Correlation Coefficients
Correlation coefficients have a hypothesis test. As with any hypothesis test, this test takes sample data and evaluates two mutually exclusive statements about the population from which the sample was drawn. For Pearson correlations, the two hypotheses are the following:
- Null hypothesis: There is no linear relationship between the two variables. ρ = 0.
- Alternative hypothesis: There is a linear relationship between the two variables. ρ ≠ 0.
Correlation coefficients that equal zero indicate no linear relationship exists. If your p-value is less than your significance level, the sample contains sufficient evidence to reject the null hypothesis and conclude that the Pearson correlation coefficient does not equal zero. In other words, the sample data support the notion that the relationship exists in the population.
Related post: Overview of Hypothesis Tests
Interpreting our Height and Weight Correlation Example
Now that we have seen a range of positive and negative relationships, let’s see how our Pearson correlation coefficient of 0.694 fits in. We know that it’s a positive relationship. As height increases, weight tends to increase. Regarding the strength of the relationship, the graph shows that it’s not a very strong relationship where the data points tightly hug a line. However, it’s not an entirely amorphous blob with a very low correlation. It’s somewhere in between. That description matches our moderate correlation coefficient of 0.694.
For the hypothesis test, our p-value equals 0.000. This p-value is less than any reasonable significance level. Consequently, we can reject the null hypothesis and conclude that the relationship is statistically significant. The sample data support the notion that the relationship between height and weight exists in the population of preteen girls.
Correlation Does Not Imply Causation
I’m sure you’ve heard this expression before, and it is a crucial warning. Correlation between two variables indicates that changes in one variable are associated with changes in the other variable. However, correlation does not mean that the changes in one variable actually cause the changes in the other variable.
Sometimes it is clear that there is a causal relationship. For the height and weight data, it makes sense that adding more vertical structure to a body causes the total mass to increase. Or, increasing the wattage of lightbulbs causes the light output to increase.
However, in other cases, a causal relationship is not possible. For example, ice cream sales and shark attacks have a positive correlation coefficient. Clearly, selling more ice cream does not cause shark attacks (or vice versa). Instead, a third variable, outdoor temperatures, causes changes in the other two variables. Higher temperatures increase both sales of ice cream and the number of swimmers in the ocean, which creates the apparent relationship between ice cream sales and shark attacks.
In statistics, you typically need to perform a randomized, controlled experiment to determine that a relationship is causal rather than merely correlation.
Learn more about Correlation vs. Causation: Understanding the Differences.
How Strong of a Correlation is Considered Good?
What is a good correlation? How high should correlation coefficients be? These are commonly asked questions. I have seen several schemes that attempt to classify correlations as strong, medium, and weak.
However, there is only one correct answer. A Pearson correlation coefficient should accurately reflect the strength of the relationship. Take a look at the correlation between the height and weight data, 0.694. It’s not a very strong relationship, but it accurately represents our data. An accurate representation is the best-case scenario for using a statistic to describe an entire dataset.
The strength of any relationship naturally depends on the specific pair of variables. Some research questions involve weaker relationships than other subject areas. Case in point, humans are hard to predict. Studies that assess relationships involving human behavior tend to have correlation coefficients weaker than +/- 0.6.
However, if you analyze two variables in a physical process, and have very precise measurements, you might expect correlations near +1 or -1. There is no one-size fits all best answer for how strong a relationship should be. The correct values for correlation coefficients depend on your study area.
Taking Correlation to the Next Level with Regression Analysis
Wouldn’t it be nice if instead of just describing the strength of the relationship between height and weight, we could define the relationship itself using an equation? Regression analysis does just that. That analysis finds the line and corresponding equation that provides the best fit to our dataset. We can use that equation to understand how much weight increases with each additional unit of height and to make predictions for specific heights. Read my post where I talk about the regression model for the height and weight data.
Regression analysis allows us to expand on correlation in other ways. If we have more variables that explain changes in weight, we can include them in the model and potentially improve our predictions. And, if the relationship is curved, we can still fit a regression model to the data.
Additionally, a form of the Pearson correlation coefficient shows up in regression analysis. R-squared is a primary measure of how well a regression model fits the data. This statistic represents the percentage of variation in one variable that other variables explain. For a pair of variables, R-squared is simply the square of the Pearson’s correlation coefficient. For example, squaring the height-weight correlation coefficient of 0.694 produces an R-squared of 0.482, or 48.2%. In other words, height explains about half the variability of weight in preteen girls.
If you’re learning about statistics and like the approach I use in my blog, check out my Introduction to Statistics book! It’s available at Amazon and other retailers.