Coefficient of Determination: Definition & Overview

KEY TAKEAWAYS

• The coefficient of determination is the R-squared value.
• The units of measures are the percentage of variability in a data set explainable by a regression model.
• To calculate the coefficient of determination, one must first calculate SST, SSR, and SSE.
• The coefficient of determination can range from 0 to 1, with a value of 1 indicating that the model perfectly explains the data set.
• The coefficient of determination works in conjunction with other measures. 
• These include the F-statistic and R-value, to assess the goodness of fit of a regression model.

What Is the Coefficient of Determination?

Coefficient of determination is also known as the R-squared value. It is a statistical measure that indicates how well a linear regression model fits your data. A higher R-squared value means that the model details more of the variability in data. In other words, it indicates how “good” the fit is.

The R-squared value can assess whether or not a linear regression model is appropriate for a given dataset. If the R-squared value is low, it may be indicative that a linear model is not the best fit for the data.

Coefficient of Determination Formula & Calculation

There are several ways to calculate the coefficient of determination. The most popular method is to use the following formula:

Another method for calculating coefficient of determination is to use the following formula:

You may also calculate the coefficient of determination using Excel. Simply enter your data into a spreadsheet, then use the LINEST function to calculate the R-squared value.

Now that you know how to calculate the coefficient of determination, it’s time to learn how to interpret this value.

As we mentioned earlier, the R-squared value can range from 0-1. A value of 0 indicates that the model doesn’t explain any of the variability in data. A value of 1 indicates that the model details all of the variability in data.

A value near 0 indicates that the linear regression model doesn’t fit the data well. A value near 1 indicates goodness of fit in the linear regression method.

In general, you should only use a linear regression method if the R-squared value is greater than 0.5.

Coefficient of Determination Example

To see how this statistical model works in practice, let’s take a look at an example.

Suppose you have a dataset with 10 observations. The R-squared value for this dataset is 0.75. This means that the linear regression model explains 75% of the variability in data.

In this case, we would say that the linear regression model fits the data well.

It’s important to note that the coefficient of determination is not the same as correlation coefficient. Correlation coefficient measures how strong a linear relationship is between two variables.

It can range from -1 to 1. Here, -1 indicates a perfect negative linear relationship. But 0 indicates no linear relationship. And 1 indicates a perfect positive linear relationship.

Coefficient of determination measures how well a linear regression analyses fit the data. It can range from 0-1. Here, 0 indicates that the linear regression analyses do not explain any of the variability in data. And 1 indicates that the linear regression algorithm explains all of the variability in data.

Coefficient of determination and correlation coefficient are both measures of data. But they are not the same thing.

It’s also important to note that the coefficient of determination is not the same as R-squared. R-squared is simply the coefficient of determination expressed as a percentage.

The R-squared value can range from 0-100. Here, 0 indicates that the linear regression model does not explain any of the variability in data. And 100 indicates that the linear regression model explains all of the variability in data.

So, while R-squared is simply a way to express coefficient of determination as a percentage, they are not the same thing.

Now that you know what the coefficient of determination is and how it’s used, let’s take a look at some examples to see how it’s calculated.

Example 1: Calculating Coefficient of Determination

Suppose you have a dataset with 10 observations. Your sum of squared errors for this dataset is 20. The total sum of squares is 30.

To calculate coefficient of determination, simply plug these values into the formula:

R² = (20/30)

R² = 0.67

This means that the correct regression model explains 67% of the variability in data. In other words, the linear regression model fits the data well.

Example 2: Calculating Coefficient of Determination

Suppose you have a dataset with 10 observations. Your sum of squared error for this dataset is 40. The total sum of squares is 50.

Coefficient of determination would simply plug these values into the formula:

R² = (40/50)

R² = 0.80

This means that the linear regression model explains 80% of the variability in data. In other words, the linear regression model fits the data well.

Example 3: Calculating Coefficient of Determination

Suppose you have a dataset with 10 observations. Your sum of squared errors for this dataset is 60. The total sum of squares is 70.

To calculate coefficient of determination, simply plug these values into the formula:

R² = (60/70)

R² = 0.86

This means that the linear regression model explains 86% of the variability in data. In other words, the linear regression model fits the data well.

As you can see from these examples, the coefficient of determination can range from 0-1. The closer coefficient of determination is to 1, the better the linear regression model fits your data.

Summary

Coefficient of determination illustrates a measure of how well a linear regression model fits your data. The formula of the coefficient of determination method is essential in many industries. You can use it to tell whether you have a bad outcome or an informative outcome. 

For example, in the finance sector, this calculation can predict the performance of investments. With a higher coefficient of determination, it’s more likely that the investment will get impacted by the other variable.