What Is Mean Square?
The mean square is the average of the squared values, the sum of x² divided by n. Its square root is the root mean square, or RMS, a common measure of the typical size of values regardless of sign.
In ANOVA and regression, mean square has a related meaning: a sum of squares divided by its degrees of freedom. About the mean, with n − 1 degrees of freedom, that is the sample variance.
Paste your numbers and you get both versions, with a table of the squares.
How to Use the Mean Square Calculator
- Paste your numbers.
- Read the mean square and the RMS.
- Use the SS and the two mean sums of squares if you need a variance.
Mean Square Formula
Mean square
Mean sum of squares (about the mean)
Mean Square Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x | Each value in the data | |
| n | Number of values | |
| Σx² | Sum of the squared values | |
| SS | Sum of squared deviations from the mean | Σ(x − x̄)² |
Mean Square Worked Example
With the example values loaded in the calculator above, the steps are:
- n = 6. Square each value and add: Σx² = 191.
- Mean square = Σx² ÷ n = 191 ÷ 6 = 31.8333. Its square root is the RMS, 5.6421.
- Mean = 5.16667. Sum of squared deviations SS = Σ(x − x̄)² = 30.8333.
- Mean sum of squares about the mean: SS ÷ (n − 1) = 6.16667 (sample), or SS ÷ n = 5.13889 (population).
Result: Mean square (Σx² ÷ n) = 31.8333; Root mean square (RMS) = 5.6421.
Worked example: 2, 4, 4, 5, 7, 9
Squares: 4, 16, 16, 25, 49, 81. Their sum is 191, so the mean square is 191 ÷ 6 = 31.8333 and the RMS is 5.6421.
The mean is 5.1667. The sum of squared deviations is 191 − 6 × 5.1667² = 30.8333, so SS ÷ (n − 1) = 6.1667.
Mean square versus variance
Variance measures spread around the mean. Mean square measures the average squared size around zero. They agree only when the mean is 0.
Where Mean Square Is Used
Signal and error size
Find the RMS of a set of measurements or errors.
ANOVA coursework
See how a sum of squares becomes a mean square.
Variance check
Confirm that mean square equals variance plus mean squared.
Common Mean Square Mistakes to Avoid
- Use RMS when the sign does not matter, for example for signal strength.
- Divide by n − 1, not n, for a sample variance.
Mean Square Terms and Definitions
- RMS
- The square root of the mean square.
- Mean sum of squares
- A sum of squares divided by its degrees of freedom.
Mean Square FAQs
What is the mean square?
The average of the squared values, Σx² ÷ n.
What is the mean sum of squares?
A sum of squares divided by its degrees of freedom. About the mean, with n − 1, it is the sample variance.
How is mean square related to variance?
Mean square equals the population variance plus the mean squared.
What is RMS used for?
It measures the typical size of values regardless of sign, such as an AC voltage or an error.