4-6 Deviation, variance and standard deviation of a set of values

• The deviation of a value is x − x̄; the deviations always sum to zero, which is why they are squared.
• Variance σ² = Σ(x − x̄)² / n and standard deviation σ = √σ² — the SD returns to the original unit of the data, so it is the more usable number.
• σ = 0 only when all the values are equal; the larger σ, the more scattered the data.
• Shortcut formula: σ² = (Σx²/n) − (x̄)² — often faster and less rounded.

Worked example

• For 4, 6, 7, 9, 14: x̄ = 8; the squared deviations are 16, 4, 1, 1, 36 with sum 58, so σ² = 58/5 = 11.6 and σ ≈ 3.41.