What is the difference between Correlation vs. Regression?

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Correlation and regression are two terms in statistics that are related, but not quite the same.

In this tutorial, we’ll provide a brief explanation of both terms and explain how they’re similar and different.

What is Correlation?

Correlation measures 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

For example, suppose we have the following dataset that contains two variables: (1) Hours studied and (2) Exam Score received for 20 different students:

If we created a scatterplot of hours studied vs. exam score, here’s what it would look like:

Just from looking at the plot, we can tell that students who study more tend to earn higher exam scores. In other words, we can visually see that there is a positive correlation between the two variables.

Using a calculator, we can find that the correlation between these two variables is r = 0.915. Since this value is close to 1, it confirms that there is a strong positive correlation between the two variables.

What is Regression?

Regression is a method we can use to understand how changing the values of the x variable affect the values of the y variable.

A regression model uses one variable, x, as the predictor variable, and the other variable, y, as the . It then finds an equation with the following form that best describes the relationship between the two variables:

ŷ = b₀ + b₁x

where:

• ŷ: The predicted value of the response variable
• b₀: The y-intercept (the value of y when x is equal to zero)
• b₁: The regression coefficient (the average increase in y for a one unit increase in x)
• x: The value of the predictor variable

Using a , we find that the following equation best describes the relationship between these two variables:

Predicted exam score = 65.47 + 2.58*(hours studied)

The way to interpret this equation is as follows:

• The predicted exam score for a student who studies zero hours is 65.47.
• The average increase in exam score associated with one additional hour studied is 2.58.

We can also use this equation to predict the score that a student will receive based on the number of hours studied.

For example, a student who studies 6 hours is expected to receive a score of 80.95:

Predicted exam score = 65.47 + 2.58*(6) = 80.95.

We can also plot this equation as a line on a scatterplot:

We can see that the regression line “fits” the data quite well.

Recall earlier that the correlation between these two variables was r = 0.915. It turns out that we can square this value and get a number called “r-squared” that describes the total proportion of in the response variable that can be explained by the predictor variable.

In this example, r² = 0.915² = 0.837. This means that 83.7% of the variation in exam scores can be explained by the number of hours studied.

Correlation vs. Regression: Similarities & Differences

Here is a summary of the similarities and differences between correlation and regression:

Similarities:

• Both quantify the direction of a relationship between two variables.
• Both quantify the strength of a relationship between two variables.

Differences:

• Regression is able to show a cause-and-effect relationship between two variables. Correlation does not do this.
• Regression is able to use an equation to predict the value of one variable, based on the value of another variable. Correlation does not does this.
• Regression uses an equation to quantify the relationship between two variables. Correlation uses a single number.

The following tutorials offer more in-depth explanations of topics covered in this post.