Variance and standard deviation

Mean deviation gets rid of signs by simply dropping them. A neater way is to square the deviations: squares are never negative, and they work well in algebra.

Just adding the squares is not fair, because a bigger data set gives a bigger total. So we average them:

• Variance σ² = Σ(xᵢ – x̄)² / n: the mean of the squared deviations.
• Standard deviation σ = √(variance), the positive root. It is back in the same units as the data (marks, cm, rupees), which makes it the most used measure of spread.

Example: 2, 4, 6, 8, 10. Mean = 6.

• Deviations: –4, –2, 0, 2, 4. Squares: 16, 4, 0, 4, 16. Sum 40.
• Variance = 40/5 = 8, σ = √8 ≈ 2.83.

A quicker formula: σ² = Σx²/n – x̄² (mean of the squares minus the square of the mean). Here Σx² = 4 + 16 + 36 + 64 + 100 = 220, so σ² = 44 – 36 = 8, the same.

For a frequency table: σ² = Σf(x – x̄)² / N.

• N = 6, Σfx = 30, mean 5.
• f(x – 5)²: 9, 2, 2, 9, total 22.
• Variance = 22/6 ≈ 3.67, σ ≈ 1.91.

Short-cut for big numbers. Shift and shrink the values with y = (x – A)/h, find the spread of y, then scale back: σₓ = h × σᵧ, so σₓ² = h²[Σfy²/N – (Σfy/N)²].

For the class table 0-10, ..., 40-50 with frequencies 5, 8, 14, 8, 5 (A = 25, h = 10):

• y = –2, –1, 0, 1, 2. Σfy = 0 and Σfy² = 20 + 8 + 0 + 8 + 20 = 56.
• σᵧ² = 56/40 = 1.4, so σₓ² = 100 × 1.4 = 140 and σₓ ≈ 11.83.

Square roots like √140 are found with a calculator or a table; the long-division method also works if you have to do it by hand.