Measures of Dispersion (Range, Quartile Deviation, Standard Deviation) — Practice

Question 1

For the following data: 15, 22, 8, 35, 12, 28, 45, 20. Calculate (a) Range (b) If Q₁ = 13 and Q₃ = 31.5, find Quartile Deviation.

Accepted Answer

(a) Range = L − S = 45 − 8 = 37. This means the entire spread of data is 37 units. (b) Quartile Deviation = (Q₃ − Q₁) / 2 = (31.5 − 13) / 2 = 18.5 / 2 = 9.25. Interpretation: The middle 50% of data is spread over a range of 18.5 units, and the typical deviation from the median is about 9.25 units.

Question 2

Calculate the Standard Deviation and Variance for the following data: 4, 8, 12, 16, 20.

Accepted Answer

Step 1: Mean X̄ = (4 + 8 + 12 + 16 + 20) / 5 = 60 / 5 = 12. Step 2: Calculate (X − X̄) and (X − X̄)²: X=4: deviation = 4−12 = −8, (−8)² = 64. X=8: deviation = 8−12 = −4, (−4)² = 16. X=12: deviation = 12−12 = 0, 0² = 0. X=16: deviation = 16−12 = 4, 4² = 16. X=20: deviation = 20−12 = 8, 8² = 64. Step 3: Σ(X − X̄)² = 64 + 16 + 0 + 16 + 64 = 160. Step 4: Variance σ² = 160 / 5 = 32. Step 5: Standard Deviation σ = √32 = 5.66 (approx).

Question 3

Two batsmen's scores across 6 innings are: Batsman A: 48, 52, 55, 60, 42, 55 (X̄ = 52, σ = 5.77). Batsman B: 30, 75, 20, 85, 15, 90 (X̄ = 52.5, σ = 30.4). (a) Calculate the Coefficient of Variation for each. (b) Which batsman is more consistent and why? (c) Why is CV a better measure of comparison than SD here?

Accepted Answer

(a) C.V. for Batsman A = (σ/X̄) × 100 = (5.77/52) × 100 = 11.1%. C.V. for Batsman B = (30.4/52.5) × 100 = 57.9%. (b) Batsman A is more consistent because his CV (11.1%) is far lower than Batsman B's (57.9%). A lower CV means the scores are clustered close to the mean — A scores consistently around 52 runs. B's high CV means his performance is highly variable — sometimes very high, sometimes very low. (c) Why CV is better than SD here: Both batsmen have nearly the same mean (~52 runs). If we only compared SD (5.77 vs 30.4), we'd also conclude A is more consistent. But what if one batsman had a mean of 30 and another had a mean of 80? Comparing SD directly when means differ is misleading — a high SD on a mean of 80 might be proportionally smaller than a lower SD on a mean of 20. CV expresses dispersion as a PERCENTAGE of the mean, making it unit-free and comparable regardless of the magnitude of mean scores.

Practice — Measures of Dispersion (Range, Quartile Deviation, Standard Deviation)

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