Shortcut method for variance

Large values or mid-points make the squares tedious. Step deviations shrink them: yᵢ = (xᵢ − A)/h, so xᵢ = A + h yᵢ, where A is an assumed mean and h the class width.

Then x̄ = A + h ȳ, where ȳ = (Σfᵢyᵢ)/N.

Because every deviation xᵢ − x̄ equals h(yᵢ − ȳ), the variances are linked by σₓ² = h² σᵧ², that is σₓ = h σᵧ.

Working formula: σₓ² = (h²/N²)[N Σfᵢyᵢ² − (Σfᵢyᵢ)²] and σₓ = (h/N)√[N Σfᵢyᵢ² − (Σfᵢyᵢ)²].

Practice match (made up) with A = 35, h = 10: yᵢ runs from −3 to 3, Σfᵢyᵢ = −20 and Σfᵢyᵢ² = 112. Mean = 35 + 10 × (−20/50) = 31. Variance = (100/2500)(50 × 112 − 400) = (1/25)(5200) = 208, matching the long method.

Any A gives the right answer; choosing it near the middle keeps yᵢ small. The h in σₓ = h σᵧ must be remembered, because the variance of the y values alone is only 208/100 = 2.08.

NCERT's worked shortcut: with A = 65 and h = 10 its continuous example gives mean 62, variance 201 and σ ≈ 14.18, the same as its long method.