Lesson 6 of 11 · 7 min
From absolute values to squares
NCERT §13.4.3, §13.5
The coach is happy with mean deviation, but the school's maths teacher warns him: it works for one table, but it cannot be combined or pushed through formulas. What else can remove the signs of the deviations?
The lesson in notes
In short
Limits of mean deviation: where the data vary a great deal, the median is not a representative centre, so the mean deviation about it cannot be fully trusted.
The sum of absolute deviations about the mean is more than the sum about the median, so mean deviation about the mean is not a very sound measure either.
Most important: absolute values do not fit further algebra. Mean deviation cannot be combined or manipulated through formulas, so another measure is needed.
Squaring also removes signs, since every (xᵢ − x̄)² ≥ 0. If Σ(xᵢ − x̄)² = 0, every observation equals the mean and there is no spread; a small sum means low spread and a large one high spread.
The plain sum of squares is not enough, because it grows with the number of observations. NCERT's example: six values 5, 15, …, 55 give Σ(xᵢ − 30)² = 1750, while 31 values 15, 16, …, 45 give 2480, even though the first set is more spread out.
Dividing by n fixes this: 1750/6 ≈ 291.67 against 2480/31 = 80, so the six-value set does have the larger spread. The mean of the squared deviations is the proper measure.