Variance and standard deviation

Standard deviation measures how far data is spread around the mean, using every value. A small standard deviation means the data is bunched close to the mean.

Key facts

  • Variance = mean of the squares − square of the mean: Σx²/n − x̄².
  • Standard deviation σ = √variance.
  • Adding a constant to every value doesn’t change σ; multiplying by k multiplies σ by k.
  • Your calculator’s statistics mode does this quickly.

Worked examples

Example 1

Find the standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

  1. Mean: 40 ÷ 8 = 5.
  2. Σx² = 4 + 16 + 16 + 16 + 25 + 25 + 49 + 81 = 232
  3. Variance: 232 ÷ 8 − 25 = 4 σ = 2

Answer: 2