sum of squares
Also called: sums of squares, SS
The sum of squared deviations from the mean, Σ(xᵢ − x̄)². For 6, 9, 4, 7 and 14 it is 58. Divided by n − 1 it gives the variance; in ANOVA it splits into between and within parts.
Example
Values 6, 9, 4, 7, 14; mean 8.
- Deviations: −2, +1, −4, −1, +6 (sum 0).
- Squares: 4, 1, 16, 1, 36; SS = 58.
- Variance s² = 58 ÷ 4 = 14.5; SD = √14.5 ≈ 3.81.
Common mistakes
- Squaring the sum instead of summing the squares. (Σ(x − x̄))² is always 0.
- Dividing by n for a sample. The sample variance divides by n − 1.
- Reporting SS or variance in squared units where people expect the SD.
Learn it in the course
- Why we square: variance, sums of squares and n − 1 · Why spread formulas square deviations, variance s² and σ², sums of squares split into between and within, λ as a sum of squared loadings, and the idea behind n − 1 and degrees of freedom.
- Splitting a total · Split a total into parts that add up exactly: sums of squares and R², plan vs actual as volume + mix + price, and a change in savings as contributions + growth.
- The Σ sign: sums, limits and running totals · Read Σ with its limits, see why deviations from the mean add up to zero, use sums of products, running totals and double indices, and two shortcuts: n(n + 1)/2 and k(k − 1)/2.
- Pythagoras in statistics: distances and errors in quadrature · Euclidean and Manhattan distance, why independent errors add as squares, the pooled SE of a z-test vs the unpooled SE of an interval, and how to read the Wilson and Newcombe interval formulas.
- Several variables · Formulas with several inputs: ŷ = b₀ + Σ bⱼxⱼ, 0/1 variables and interactions, a driver-based forecast, ±10% sensitivity and scenarios.