Sum of Squares Calculator
This sum of squares calculator turns your numbers into the sum of squared deviations, the sum of raw squares and the variance. Example: for 3, 5, 7, 9, 11 the result is Σ(x − x̄)² = 40 and Σx² = 285.
Source: Wikipedia – Quadratsumme. Updated: .
How it is calculated
The sum of squares measures the total spread of a data set around its mean by adding up all the squared deviations. In statistics it is also written SS or SST and shows up everywhere, from variance to regression.
Formulas
The sum of squares about the mean is SS = Σ(x − x̄)². The computational formula SS = Σx² − (Σx)² / n is more convenient because it needs only the sum and the sum of squares. Dividing SS by n − 1 gives the sample variance, dividing by n the population variance. The calculator shows both routes and compares them as a check.
Example
For 3, 5, 7, 9, 11 the mean is 7. The deviations are −4, −2, 0, 2, 4, their squares 16, 4, 0, 4, 16 with sum 40. Check: Σx² = 9 + 25 + 49 + 81 + 121 = 285, and 285 − 35² / 5 = 285 − 245 = 40. The variance is 40 / 4 = 10 for a sample and 40 / 5 = 8 for a population.
- The sum of raw squares Σx² is not the same as the sum of squares about the mean; the latter is always smaller or equal.
- In analysis of variance, SS is partitioned: total SS = SS between groups + SS within groups.
Frequently asked questions
What is the sum of squares?
The sum of the squared deviations of all values from the mean: Σ(x − x̄)². It measures the total spread of the data.
What is the difference between Σx² and Σ(x − x̄)²?
Σx² squares the raw values, Σ(x − x̄)² squares the deviations from the mean. They are linked by SS = Σx² − (Σx)² / n.
How do I get from the sum of squares to the variance?
Divide by n − 1 for a sample or by n for a population. With 40 and n = 5 that gives 10 and 8.
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