The Correlation Coefficient – Meaning & Calculations

The correlation coefficient is a statistical measure that quantifies the degree to which two variables are related to each other. It is a key concept in statistics and is used widely in fields like social sciences, finance, and medicine. The correlation coefficient ranges from -1 to +1. A coefficient of +1 indicates a perfect positive correlation, a coefficient of -1, on the other hand, indicates a perfect negative correlation and a correlation of 0 implies no relationship. Learn more in this article.

Definition: The correlation coefficient

The correlation coefficient is a measure of the strength of the relationship between two variables. There are several types of correlation coefficients, but the most commonly used is Pearson’s (r).

Understanding the correlation coefficient

This coefficient is used as inferential statistics, meaning it can be used to test statistical hypotheses. It also works as descriptive statistics and can be used to summarize findings in research.

Here are other names for this coefficient:

• Pearson’s r
• Pearson’s product-moment correlation coefficient (PPMCC)
• Bivariate correlation

Correlation coefficient: Graphical representation

The line of best fit is a line that goes through scatter plots of data points, and it helps to show the relationship between variables.

If the variables have a correlation coefficient of zero, it will not be possible to draw a line of best fit. If there is a perfect correlation between the two variables, you will get all points on the line of best fit.

When to use the correlation coefficient

You should use this coefficient in the following cases:

• When both variables are quantitative
• When the variables are normally distributed
• When the data has no outliers
• When the relationship between the variables is linear

Other types of correlation coefficients

Spearman’s rank correlation coefficient can be used for monotonic relationships.

It is appropriate for measurements taken from ordinal scales. You should also use it if the variables aren’t normally distributed and if it includes outliers.

Calculating the correlation coefficient

You can calculate this value using a formula or software. Here is the formula used:

The following shows a sample of a dataset to demonstrate how the correlation coefficient can be calculated.

Step 1: x and y variables

The independent variable should be on the x-axis, while the dependent variable should be on the y-axis.

• In this case, age will be considered the x variable.
• The person’s income will be considered the y variable.

Step 2: x² and y²

The first step is to square the x and y variables. This will give you the following values:

Step 3: Cross product and the sum

Next, you have to multiply x and y together. This will give you the following values:

Step 4: Calculate the correlation coefficient

You can add up the x values, y values, xy values, x² values, and y² values. You can now plug the numbers into the formula.

• This will give you a correlation coefficient of 0.9938.

That means age and income are strongly correlated.

Testing correlation coefficient significance

To determine whether the relationship between two variables is significant, you have to find the value of r and n, which is the sample size. Your goal is to determine whether Pearson’s correlation coefficient of the population (p) is significantly different from 0.

• Null hypothesis (H₀): p = 0
• Alternative hypothesis (H_a): p ≠ 0

Aside from using the following steps, you can test the hypotheses using software like Stata. As an example, we will assume you have the following figures:

Step 3: Compare t-value to critical value

For this step, you have to use the absolute value of t, meaning you should ignore any negative signs. Determine whether the t-value is higher or lower than the critical value of t.

Step 4: To reject or not to reject

If the absolute value of t is greater than the critical value, you will reject the null hypothesis. If it is less, you will fail to reject the null hypothesis.

In our example, we will reject the null hypothesis.

Reporting the correlation coefficient

In a research paper, the correlation coefficient is included in the results section. If you are using APA style, you should italicize “r“ and round the correlation coefficient to two decimal places.

The p-value should be rounded off to three decimal places. You also need to drop the leading 0 from these two values.