Skewness: Karl Pearson's and Bowley's coefficients

1. Meaning

Skewness means lack of symmetry. A distribution is symmetrical when the frequencies rise and fall in the same way on both sides of the centre; the curve folded at the centre would fit on itself. Croxton and Cowden: “When a series is not symmetrical it is said to be asymmetrical or skewed.” Averages tell the centre and dispersion the spread; skewness tells the shape.

2. Types

• Symmetrical: Mean = Median = Mode; the curve is bell-shaped (the normal curve).
• Positively skewed (skewed to the right): the longer tail is on the right; Mean > Median > Mode. Typical of incomes: most earn modest amounts, a few earn very large ones and pull the mean up.
• Negatively skewed (skewed to the left): the longer tail is on the left; Mean < Median < Mode. Typical of marks in an easy paper.

Tests of skewness (skewness is present if): the three averages differ; quartiles are not equidistant from the median (Q₃ − Md ≠ Md − Q₁); the curve is not bell-shaped; the sum of positive deviations from the median does not equal the sum of negative deviations; frequencies are not equal at points equidistant from the mode.

Dispersion versus skewness: dispersion measures the spread of items around the average; skewness measures the direction in which they are spread. Dispersion is about the amount of variation, skewness about its balance. Two series can have the same SD and opposite skewness.

3. Measures

• Karl Pearson’s coefficient: Sk_P = (x̄ − Mo) / σ. When the mode is ill-defined (bimodal data), use Sk_P = 3(x̄ − Md) / σ. In theory it lies between −3 and +3; in practice usually between −1 and +1.
• Bowley’s coefficient (quartile coefficient): Sk_B = (Q₃ + Q₁ − 2Md) / (Q₃ − Q₁). It lies between −1 and +1. Useful for open-end classes and where only the middle 50% matters; its weakness is that it ignores the top and bottom quarters.

4. Worked: Karl Pearson, discrete (January 2025)

Values 6, 12, 18, 24, 30, 36, 42 with f = 4, 7, 9, 18, 15, 10, 5. Mode = 24 (highest frequency 18). Take A = 24, h = 6.

• x̄ = 24 + 15/68 × 6 = 24 + 1.32 = 25.32.
• σ = 6 × √(173/68 − (15/68)²) = 6 × √(2.5441 − 0.0487) = 6 × √2.4955 = 6 × 1.5797 = 9.48.
• Sk_P = (25.32 − 24) / 9.48 = +0.14: slight positive skewness.

5. Worked: Karl Pearson, grouped (100-mark January 2026)

0–10: 12, 10–20: 18, 20–30: 28, 30–40: 26, 40–50: 16. A = 25, h = 10; d′ = −2 to +2; fd′ = −24, −18, 0, 26, 32 (Σ = 16); fd′² = 48, 18, 0, 26, 64 (Σ = 156); N = 100.

• x̄ = 25 + 16/100 × 10 = 26.6.
• σ = 10 × √(1.56 − 0.0256) = 10 × √1.5344 = 12.39.
• Mode: modal class 20–30 (f₁ = 28, f₀ = 18, f₂ = 26): Mo = 20 + (28 − 18)/(56 − 18 − 26) × 10 = 20 + 10/12 × 10 = 28.33.
• Sk_P = (26.6 − 28.33) / 12.39 = −1.73 / 12.39 = −0.14: slight negative skewness.

6. Worked: Bowley (January 2026)

Weight 90–100: 20, 100–110: 25, 110–120: 40, 120–130: 35, 130–140: 30. N = 150; c.f. 20, 45, 85, 120, 150.

• Q₁ (37.5th): 100 + (37.5 − 20)/25 × 10 = 107.00.
• Median (75th): 110 + (75 − 45)/40 × 10 = 117.50.
• Q₃ (112.5th): 120 + (112.5 − 85)/35 × 10 = 127.86.
• Sk_B = (127.86 + 107.00 − 235.00) / (127.86 − 107.00) = −0.14 / 20.86 = −0.007.

The distribution is practically symmetrical, with a negligible negative skew: Q₃ − Md = 10.36 and Md − Q₁ = 10.50 are almost equal.

7. Worked: the reverse problem (August 2024)

x̄ = 50, CV = 40%, coefficient of skewness = −0.4. Find SD, mode and median.

• CV = σ/x̄ × 100 ⇒ 40 = σ/50 × 100 ⇒ σ = 20.
• Sk_P = (x̄ − Mo)/σ ⇒ −0.4 = (50 − Mo)/20 ⇒ 50 − Mo = −8 ⇒ Mode = 58.
• Mode = 3Md − 2x̄ ⇒ 58 = 3Md − 100 ⇒ Median = 52.67.

Check: Mean (50) < Median (52.67) < Mode (58), as a negative skew requires.

How this is examined

Bowley’s coefficient: January 2026 (ten marks). Karl Pearson’s coefficient: January 2025 (discrete, six marks) and the 100-mark January 2026 paper (grouped). The reverse problem: August 2024. Theory: “What is skewness? Explain the types and measures” (June 2025), “Write a note on skewness” (August 2024, and a short note in the 100-mark January 2026 paper). Draw the three curves roughly by hand in a theory answer.