Sum of Squares Calculator — SS, Variance & Standard Deviation
Paste a list of numbers to compute SS = Σ(xᵢ − x̄)² — the core building block of variance, standard deviation and ANOVA — along with sample and population variance and standard deviation.
SS = Σ(xᵢ − x̄)²
- 1
Mean x̄
Σxᵢ ÷ 5 = 12.2 - 2
Sum of squares SS
Σ(xᵢ − 12.2)² = 186.8000Each deviation is squared so positives and negatives do not cancel, and larger gaps are weighted more.
How does this calculator work?
SS = Σ(xᵢ − x̄)² measures total variability by summing squared deviations from the mean. Sample variance = SS/(n−1); population variance = SS/n; standard deviation = √variance. Enter any list of numbers (minimum 2) separated by commas or spaces to get SS, both variances and both standard deviations instantly.
Formula
How this is calculated
The sum of squares (SS) measures total variability in a dataset by squaring each value's deviation from the mean and summing all the squared deviations. Squaring serves two purposes: it removes the sign (so positive and negative deviations both add to variability) and it penalises large deviations more heavily than small ones.
Dividing SS by (n − 1) gives the sample variance s². The (n − 1) denominator — Bessel's correction — compensates for the fact that using the sample mean slightly underestimates deviations from the true population mean, producing an unbiased estimator of σ². Dividing by n instead gives the population variance σ², which is correct only when the data represents the full population. Taking the square root of either gives the corresponding standard deviation.
SS is also the numerator of the F-statistic in ANOVA and appears in regression as the residual sum of squares (RSS). Interpretation depends heavily on the unit of the data — SS grows with n and with the scale of the values, while variance and SD normalise the measure for comparison across datasets.
Frequently asked questions
Sum of squares (SS) is the raw total of squared deviations from the mean. Variance divides SS by n − 1 (sample) or n (population) to get a per-observation average. SS grows with sample size; variance stays on the same scale regardless of n, making it suitable for comparing spread across datasets.
Use sample variance when your data is a sample drawn from a larger population and you want an unbiased estimate of the true population variance. Use population variance only when your data represents the entire population — for example, the complete set of exam scores for a single class.
In ANOVA the total SS is partitioned into between-group SS (variance explained by the grouping factor) and within-group SS (unexplained residual variance). The ratio of these, each divided by its degrees of freedom, gives the F-statistic used to test whether group means differ significantly.
Also known as
TG we-Calculate Editorial Team. (2026). Sum of Squares Calculator — SS, Variance & Standard Deviation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sum-of-squares-calculator
TG we-Calculate Editorial Team. "Sum of Squares Calculator — SS, Variance & Standard Deviation." TG we-Calculate. 2026. https://we-calculate.com/calculator/sum-of-squares-calculator.
TG we-Calculate Editorial Team, "Sum of Squares Calculator — SS, Variance & Standard Deviation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sum-of-squares-calculator
@misc{wecalculate_sum_of_squares_calculator, title = {Sum of Squares Calculator — SS, Variance & Standard Deviation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sum-of-squares-calculator}}, year = {2026}, note = {TG we-Calculate} }
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