How to Calculate SST, SSR, and SSE in R

We often use three different sum of squares values to measure how well a regression line actually fits a dataset:

1. Sum of Squares Total (SST) – The sum of squared differences between individual data points (y_i) and the mean of the response variable (y).

• SST = Σ(y_i – y)²

2. Sum of Squares Regression (SSR) – The sum of squared differences between predicted data points (ŷ_i) and the mean of the response variable(y).

• SSR = Σ(ŷ_i – y)²

3. Sum of Squares Error (SSE) – The sum of squared differences between predicted data points (ŷ_i) and observed data points (y_i).

• SSE = Σ(ŷ_i – y_i)²

The following step-by-step example shows how to calculate each of these metrics for a given regression model in R.

Step 1: Create the Data

First, let’s create a dataset that contains the number of hours studied and exam score received for 20 different students at a certain college:

#create data frame
df <- data.frame(hours=c(1, 1, 1, 2, 2, 2, 2, 2, 3, 3,
                         3, 4, 4, 4, 5, 5, 6, 7, 7, 8),
                 score=c(68, 76, 74, 80, 76, 78, 81, 84, 86, 83,
                         88, 85, 89, 94, 93, 94, 96, 89, 92, 97))

#view first six rows of data frame
head(df)

  hours score
1     1    68
2     1    76
3     1    74
4     2    80
5     2    76
6     2    78

Step 2: Fit a Regression Model

Next, we’ll use the lm() function to fit a simple linear regression model using score as the response variable and hours as the predictor variable:

#fit regression model
model <- lm(score ~ hours, data = df)

#view model summary
summary(model)

Call:
lm(formula = score ~ hours, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6970 -2.5156 -0.0737  3.1100  7.5495 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  73.4459     1.9147  38.360  < 2e-16 ***
hours         3.2512     0.4603   7.063 1.38e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.289 on 18 degrees of freedom
Multiple R-squared:  0.7348,	Adjusted R-squared:  0.7201 
F-statistic: 49.88 on 1 and 18 DF,  p-value: 1.378e-06

Step 3: Calculate SST, SSR, and SSE

We can use the following syntax to calculate SST, SSR, and SSE:

#find sse
sse <- sum((fitted(model) - df$score)^2)
sse

[1] 331.0749

#find ssr
ssr <- sum((fitted(model) - mean(df$score))^2)
ssr

[1] 917.4751

#find sst
sst <- ssr + sse
sst

[1] 1248.55

The metrics turn out to be:

• Sum of Squares Total (SST): 1248.55
• Sum of Squares Regression (SSR): 917.4751
• Sum of Squares Error (SSE): 331.0749

We can verify that SST = SSR + SSE:

• SST = SSR + SSE
• 1248.55 = 917.4751 + 331.0749

We can also manually calculate the R-squared of the regression model:

• R-squared = SSR / SST
• R-squared = 917.4751 / 1248.55
• R-squared = 0.7348

This tells us that 73.48% of the variation in exam scores can be explained by the number of hours studied.

Additional Resources

You can use the following calculators to automatically calculate SST, SSR, and SSE for any simple linear regression line:

SST Calculator
SSR Calculator
SSE Calculator