Measurement of Skewness: Karl Pearson & Bowley Methods

1. Absolute vs. Relative Measures of Skewness

To quantify the degree of tilt in a distribution, statisticians calculate both absolute and relative measures:

2. Karl Pearson’s Coefficient of Skewness (Sk_p)

Formulated by the father of modern mathematical statistics, Karl Pearson, this coefficient scales the distance between the Mean and Mode by the Standard Deviation:

Properties & Interpretation of Pearson’s Sk_p:

• Limits: Theoretically lies between -3 and +3 (in practice, rarely exceeds -1 to +1).
• If Sk_p = 0: The distribution is perfectly symmetrical.
• If Sk_p > 0 (Positive): The distribution is positively skewed (tail stretches to the right).
• If Sk_p < 0 (Negative): The distribution is negatively skewed (tail stretches to the left).

Solved Exam Example (Pearson’s Method):

Problem: In an exam, the Mean marks of students are 65, Median is 62, and Standard Deviation is 12. Calculate Karl Pearson’s coefficient of skewness.

• Solution:
  Since Mode is not given, we use Formula 2:
  Sk_p = [ 3 × (Mean - Median) ] ÷ σ
Sk_p = [ 3 × (65 - 62) ] ÷ 12
Sk_p = [ 3 × 3 ] ÷ 12 = 9 ÷ 12 = +0.75
• Conclusion: The data is moderately positively skewed (+0.75).

3. Bowley’s Quartile Coefficient of Skewness (Sk_b)

Formulated by Arthur Bowley, this measure is based entirely on the Quartiles (Q1, Median, and Q3).

Why Is Bowley’s Method Unique?

1. Works with Open-Ended Classes: If a distribution has open ends (e.g., “Below ₹1,000” or “Above ₹10,000”), the Mean and Standard Deviation cannot be calculated. But Quartiles and Median depend only on positions! Therefore, Bowley’s method is the ONLY method that can measure skewness for open-ended tables.
2. Strict Limits: Bowley’s coefficient strictly lies between -1 and +1 (-1 ≤ Sk_b ≤ +1).

Solved Exam Example (Bowley’s Method):

Problem: For a wage distribution, Lower Quartile Q1 = 30, Median M = 40, and Upper Quartile Q3 = 55. Calculate Bowley’s coefficient of skewness.

• Solution:
  Numerator = Q3 + Q1 - 2(Median)
Numerator = 55 + 30 - 2(40) = 85 - 80 = +5
Denominator = Q3 - Q1 = 55 - 30 = 25
Sk_b = 5 ÷ 25 = +0.20
• Conclusion: The wage distribution is slightly positively skewed (+0.20).

4. Kelley’s Coefficient of Skewness (Sk_k)

Truman L. Kelley refined Bowley’s measure by utilizing Percentiles (10th and 90th percentiles) or Deciles (1st and 9th deciles) to cover 80% of observations instead of just 50%:

• Formula Using Percentiles:
  Sk_k = (P90 + P10 - 2 × P50) ÷ (P90 - P10)
• Formula Using Deciles:
  Sk_k = (D9 + D1 - 2 × D5) ÷ (D9 - D1)

5. Master Comparison Table for Exam Preparation