Variance and standard deviation

Variance σ² is the mean of the squared deviations from the mean: σ² = (1/n) Σ (xᵢ − x̄)².

Variance is in squared units (runs², cm²), so the spread is reported as its positive square root, the standard deviation: σ = √[(1/n) Σ (xᵢ − x̄)²], in the same unit as the data.

Arjun (made up): squared deviations 64, 4, 0, 4, 25, 25, 0, 64 add to 186, so σ² = 186/8 = 23.25 runs² and σ = √23.25 ≈ 4.82 runs.

Kabir: squared deviations add to 12386, so σ² = 12386/8 = 1548.25 runs² and σ ≈ 39.35 runs. With equal means, the smaller σ marks the more consistent batsman.

Equivalent working form: σ² = (Σxᵢ²)/n − x̄². For Arjun, Σxᵢ² = 20186, so σ² = 20186/8 − 50² = 2523.25 − 2500 = 23.25, the same answer.

Big deviations dominate after squaring: Arjun's 42 and 58 (deviation 8) contribute 64 each, sixteen times what a deviation of 2 contributes.

NCERT's worked value: 6, 8, 10, …, 24 (ten values) have mean 15, variance 33 and σ = √33 ≈ 5.74.