Karl Pearson’s Coefficient of Skewness 📘📉

Karl Pearson developed one of the earliest and most commonly used measures of skewness. It is based on the relationship between Mean, Median, and Mode.

This coefficient helps determine both direction and degree of skewness.

1. Concept

Pearson’s coefficient compares the distance between Mean and Mode (or Median) with the Standard Deviation.

A symmetric distribution has Mean = Median = Mode, giving skewness zero.

2. Formulas

(A) When Mode is known

Sk = (Mean – Mode) / Standard Deviation

(B) When Mode is not known

Pearson suggested the approximation:

Mode ≈ 3 Median – 2 Mean

Then skewness becomes:

Sk = 3(Mean – Median) / Standard Deviation

3. Interpretation of Skewness

• Sk = 0 → Perfectly symmetric distribution
• Sk > 0 → Positively skewed (right‑skewed)
• Sk < 0 → Negatively skewed (left‑skewed)

General guideline:

• 0 to ±0.5 → Mild skewness
• ±0.5 to ±1 → Moderate skewness
• > ±1 → High skewness

4. Solved Examples

Example 1 — Ungrouped Data

Given:

Mean = 60
Median = 55
S.D. = 10

Mode not given, so use:

Sk = 3(Mean – Median) / S.D.
   = 3(60 – 55) / 10
   = 3(5) / 10
   = 15 / 10
Sk = 1.5

Interpretation

Sk = 1.5 → Highly positively skewed distribution.

Example 2 — Mode Known

Given:

Mean = 40
Mode = 48
S.D. = 12

Sk = (Mean – Mode) / S.D.
   = (40 – 48) / 12
   = –8 / 12
Sk = –0.67

Interpretation

Sk = –0.67 → Moderately negatively skewed distribution.

Example 3 — Grouped Data

Class and frequency:

Step 1: Mean

ΣfX = (4×5) + (6×15) + (10×25)
     = 20 + 90 + 250 = 360
Σf = 20
Mean = 360 / 20 = 18

Step 2: Find Median (for Mode approximation)

Cumulative frequencies: 4, 10, 20 N = 20 → N/2 = 10 → Median lies in class 10–20.

Values:

L = 10
C.F_prev = 4
f = 6
h = 10

Median = 10 + [(10 – 4)/6] × 10
       = 10 + (6/6) × 10
       = 20

Step 3: Pearson’s formula

Sk = 3(Mean – Median) / S.D.

But S.D. needed.

Step 4: Standard Deviation (already calculated earlier)

S.D. = 7.72

Step 5: Compute skewness

Sk = 3(18 – 20) / 7.72
   = 3(–2) / 7.72
   = –6 / 7.72
Sk ≈ –0.78

Interpretation

Moderately negatively skewed.

5. Merits ✔️

• Simple and widely used
• Helps identify direction and degree of skewness
• Uses Mean, Median/Mode, and S.D.
• Useful for comparing multiple datasets

6. Limitations ❌

• Assumes existence of Mode (may not be clear)
• Not suitable for highly irregular distributions
• Sensitive to extreme values

7. Business Applications

• Income and salary distribution studies
• Market price analysis
• Customer demand analysis
• Risk estimation in finance
• Quality control and process behaviour

8. Summary ✨

• Pearson’s skewness uses Mean, Median/Mode, and S.D.
• Sk = (Mean – Mode) / S.D.
• If Mode not known → Sk = 3(Mean – Median) / S.D.
• Positive Sk → Right skewed
• Negative Sk → Left skewed

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