Sum of Squares Calculator

Calculate the sum of squared deviations from the mean (SS), sample variance, and population variance for a list of numbers.

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32

Sum of squares

4.57

Sample variance

4

Population variance

Sum of squares (SS)32
Mean5
Sample variance (SS / (n−1))4.57
Sample standard deviation2.14
Population variance (SS / n)4
Population standard deviation2
Count (n)8

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Descriptive Statistics

Sum of squares calculator — SS, sample and population variance

The sum of squares (SS) is the sum of squared deviations from the mean: SS = Σ(xᵢ − x̄)². It is the numerator of variance and the foundation of standard deviation, ANOVA, and regression analysis.

What the sum of squares represents

SS quantifies total variability in a dataset. The larger SS is, the more spread out the values are from the mean. Dividing SS by n gives the population variance; dividing by n − 1 (Bessel's correction) gives the sample variance.

In ANOVA, the total sum of squares is partitioned into between-group SS (explained by the grouping variable) and within-group SS (residual error). Understanding SS is foundational to ANOVA, regression R², and many other inferential methods.

Sample vs population variance

Population variance: σ² = SS / n. Use when your data represents the entire population.

Sample variance: s² = SS / (n − 1). Use when your data is a sample from a larger population. Dividing by n − 1 rather than n corrects for the bias introduced by estimating the population mean from the sample mean.

Frequently asked questions

Why divide by n − 1 for sample variance?

When you estimate the population mean from a sample, using x̄ introduces a slight downward bias in SS. Dividing by n − 1 (Bessel's correction) corrects this bias, making s² an unbiased estimator of σ².

 How is SS related to standard deviation?

Standard deviation is the square root of variance. Sample SD: s = √(SS/(n−1)). Population SD: σ = √(SS/n). SS → variance → SD is the direct chain.

 What is the sum of squares used for in regression?

In simple linear regression, R² = 1 − SS_residual / SS_total, where SS_total = Σ(yᵢ − ȳ)² and SS_residual = Σ(yᵢ − ŷᵢ)². R² measures how much of the total variability in y is explained by the model.

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