Linear regression is used to find a line that best “fits” a dataset.
We often use three different sum of squares values to measure how well the regression line actually fits the data:
1. Sum of Squares Total (SST) – The sum of squared differences between individual data points (yi) and the mean of the response variable (y).
- SST = Σ(yi – y)2
2. Sum of Squares Regression (SSR) – The sum of squared differences between predicted data points (ŷi) and the mean of the response variable(y).
- SSR = Σ(ŷi – y)2
3. Sum of Squares Error (SSE) – The sum of squared differences between predicted data points (ŷi) and observed data points (yi).
- SSE = Σ(ŷi – yi)2
The following relationship exists between these three measures:
SST = SSR + SSE
Thus, if we know two of these measures then we can use some simple algebra to calculate the third.
SSR, SST & R-Squared
R-squared, sometimes referred to as the coefficient of determination, is a measure of how well a linear regression model fits a dataset. It represents the proportion of the variance in the response variable that can be explained by the predictor variable.
The value for R-squared can range from 0 to 1. A value of 0 indicates that the response variable cannot be explained by the predictor variable at all. A value of 1 indicates that the response variable can be perfectly explained without error by the predictor variable.
Using SSR and SST, we can calculate R-squared as:
R-squared = SSR / SST
For example, if the SSR for a given regression model is 137.5 and SST is 156 then we would calculate R-squared as:
R-squared = 137.5 / 156 = 0.8814
This tells us that 88.14% of the variation in the response variable can be explained by the predictor variable.
Calculate SST, SSR, SSE: Step-by-Step Example
Suppose we have the following dataset that shows the number of hours studied by six different students along with their final exam scores:
Using some statistical software (like R, Excel, Python) or even by hand, we can find that the line of best fit is:
Score = 66.615 + 5.0769*(Hours)
Once we know the line of best fit equation, we can use the following steps to calculate SST, SSR, and SSE:
Step 1: Calculate the mean of the response variable.
The mean of the response variable (y) turns out to be 81.
Step 2: Calculate the predicted value for each observation.
Next, we can use the line of best fit equation to calculate the predicted exam score () for each student.
For example, the predicted exam score for the student who studied one hours is:
Score = 66.615 + 5.0769*(1) = 71.69.
We can use the same approach to find the predicted score for each student:
Step 3: Calculate the sum of squares total (SST).
Next, we can calculate the sum of squares total.
For example, the sum of squares total for the first student is:
(yi – y)2 = (68 – 81)2 = 169.
We can use the same approach to find the sum of squares total for each student:
The sum of squares total turns out to be 316.
Step 4: Calculate the sum of squares regression (SSR).
Next, we can calculate the sum of squares regression.
For example, the sum of squares regression for the first student is:
(ŷi – y)2 = (71.69 – 81)2 = 86.64.
We can use the same approach to find the sum of squares regression for each student:
The sum of squares regression turns out to be 279.23.
Step 5: Calculate the sum of squares error (SSE).
Next, we can calculate the sum of squares error.
For example, the sum of squares error for the first student is:
(ŷi – yi)2 = (71.69 – 68)2 = 13.63.
We can use the same approach to find the sum of squares error for each student:
We can verify that SST = SSR + SSE
- SST = SSR + SSE
- 316 = 279.23 + 36.77
We can also calculate the R-squared of the regression model by using the following equation:
- R-squared = SSR / SST
- R-squared = 279.23 / 316
- R-squared = 0.8836
This tells us that 88.36% 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:
Perfectly explained. Thank you
I’m seeing in several other sources the y and y-hat switched in the SSE equation. Wondering if there is a typo in this article or some other explanation.
Since the difference is squared the order does not matter.
Thanks, this is like 10 hours of wothless lectures summarized into something that is good and comprehensible and actually makes sense
Great work! Thank you for this. God bless you! You made that super simple to understand
Hi Ana…You are very welcome! Thank you for your kind words!
If the coefficient of determination R^2 is zero then sse=? I don’t understand how to tell or figure out how the amount of the y variance that can be explained by the model, relates to the sum of squares error. Can anyone explain that to me?
Absolutely — let’s break it down **step by step** using the relationships between the **sum of squares** and **R² (coefficient of determination)**.
—
### 🎯 **Definitions Recap**
For any regression model:
* **SST (Total Sum of Squares):**
Total variation in the observed data
$SST = \sum (y_i – \bar{y})^2$
* **SSR (Regression Sum of Squares):**
Explained variation (by the model)
$SSR = \sum (\hat{y}_i – \bar{y})^2$
* **SSE (Error Sum of Squares):**
Unexplained variation (left in the residuals)
$SSE = \sum (y_i – \hat{y}_i)^2$
And they relate as:
$$
SST = SSR + SSE
$$
—
### 💡 Coefficient of Determination $R^2$
$$
R^2 = \frac{SSR}{SST} = 1 – \frac{SSE}{SST}
$$
It tells us **what proportion of the total variation in $y$** is explained by the model.
—
### 🔍 Your Specific Question:
> “If $R^2 = 0$, then what is SSE?”
Let’s plug into the formula:
$$
R^2 = 1 – \frac{SSE}{SST}
$$
If $R^2 = 0$, then:
$$
0 = 1 – \frac{SSE}{SST}
\Rightarrow \frac{SSE}{SST} = 1
\Rightarrow SSE = SST
$$
—
### ✅ **Interpretation:**
If $R^2 = 0$:
* The model explains **none** of the variability in the data.
* That means all the variation in $y$ remains **unexplained**.
* So the error (SSE) is **equal** to the total variation (SST).
In other words, the model predictions $\hat{y}$ are **no better than just using the mean** $\bar{y}$ for every data point.
—
### 🔄 Summary:
| Metric | Meaning if $R^2 = 0$ |
| —— | ———————– |
| SSR | 0 |
| SSE | = SST |
| Model | Explains 0% of variance |
Thank you. That helps me out a lot. It was the relationship between them, that I wasn’t grasping until I saw this explanation.
You are very welcome, Philip! Always happy to help!
thanks so much, my test prepararions has been eased up. am grateful
You are very welcome Ajuna! We appreciate the feedback and support!