Quick answer
The sum of squares about the mean is SS = Σ(x − x̄)². For 1 to 5 it is 10, and Σx² = 55.
Enter a list of numbers. The calculator returns SS about the mean and the raw sum of squares.
Related tools: Standard Deviation Calculator: Sample or population standard deviation with variance and mean. Z-Score Calculator: Standardize a value using a mean and standard deviation. Mean Median Mode Range Calculator: Mean, median, mode and range of a data set.
How to use the Sum of Squares Calculator
- Type data set, for example 4, 8, 15, 16, 23, 42.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Data set | Text, such as 4, 8, 15, 16, 23, 42 | — |
Formula and method
Formula
SS = Σ(x − x̄)²
IEEE-754 double precision; shown to at most 6 decimal places.
Squared deviations from the mean, summed.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Data set = 1, 2, 3, 4, 5.
| Result | Value |
|---|---|
| Sum of squares (about the mean) | 10 |
| Σx² | 55 |
Example 2
Inputs: Data set = 4.
| Result | Value |
|---|---|
| Sum of squares (about the mean) | 0 |
| Σx² | 16 |
The first line is the headline result (sum of squares (about the mean)). Below it the sum of Squares Calculator lists σx², which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Relation to variance
Variance is SS ÷ (n − 1): 10 ÷ 4 = 2.5. The shortcut formula SS = Σx² − (Σx)²/n gives 55 − 225/5 = 10, the same.
Where it appears
SS is the building block of ANOVA (SSB and SSW) and regression (SSE and SST). A single value has an SS of 0.
Shortcut and link to variance
For 1 to 5 the mean is 3 and the deviations 4 + 1 + 0 + 1 + 4 = 10. The shortcut Σx² − (Σx)² ÷ n gives the same: 55 − 225 ÷ 5 = 10.
Dividing by n gives the population variance (2) and by n − 1 the sample variance (2.5).
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Data set: left empty shows "Enter data set." beside that field and no result.
Choosing between the Sum of Squares Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Standard Deviation Calculator](/statistics/standard-deviation-calculator/) | data set and data type | Population standard deviation (σ) |
| [Z-Score Calculator](/statistics/z-score-calculator/) | value, mean and standard deviation | Z-score |
| [Mean Median Mode Range Calculator](/statistics/mean-median-mode-range-calculator/) | data set | Mean |
| [Median Calculator](/statistics/median-calculator/) | data set | Median |
| [Relative Standard Deviation Calculator](/statistics/relative-standard-deviation-calculator/) | data set and data type | Relative standard deviation |
Background: descriptive statistics
Population statistics divide by n and sample statistics divide by n − 1 (Bessel's correction), because a sample tends to understate variability. Use the population version when your data is every value in the group, and the sample version when it is a subset.
Defaults differ between tools, and so do quartile conventions, which can give slightly different values for small data sets; this page states the method it uses.
Assumptions and limitations
- One value gives 0
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for data set to confirm this.
Frequently asked questions
What is the SS of 2, 4?
For 2 and 4 the mean is 3, so SS = (−1)² + 1² = 2.
Why square?
To prevent cancellation.
What is SST?
The total sum of squares.
How is the sum of squares related to variance?
Variance is the sum of squares divided by n for a population or n − 1 for a sample.
Why can't the sum of squares be negative?
Each squared deviation is zero or positive, so their sum is too.
What does the Sum of Squares Calculator calculate?
Sum of squared deviations from the mean, and sum of squared values. It returns Sum of squares (about the mean) and Σx².
What formula does the Sum of Squares Calculator use?
SS = Σ(x − x̄)². Squared deviations from the mean, summed.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. One value gives 0.
What is an example of a sum of Squares result?
With data set 1, 2, 3, 4, 5, the sum of squares (about the mean) is 10.
Related calculators
- Standard Deviation Calculator — Sample or population standard deviation with variance and mean.
- Z-Score Calculator — Standardize a value using a mean and standard deviation.
- Mean Median Mode Range Calculator — Mean, median, mode and range of a data set.
- Median Calculator — Median of a data set.
- Relative Standard Deviation Calculator — Relative standard deviation (RSD) as a percentage.
- Variance Calculator — Sample or population variance.
Sources
- Standard statistics definition (e.g. NIST/SEMATECH e-Handbook of Statistical Methods)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.