flowchart LR
S[Symmetric<br/>Mean=Median=Mode] --- P[Positive<br/>Right Tail<br/>Mean>Median>Mode]
S --- N[Negative<br/>Left Tail<br/>Mean<Median<Mode]
classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;
40 Measures of skewness
40.1 Concept of Skewness
Skewness measures the degree and direction of departure from symmetry in a frequency distribution. A symmetric distribution has zero skewness — the two tails mirror each other. A positively skewed (right-skewed) distribution has a long right tail; a negatively skewed (left-skewed) distribution has a long left tail. Skewness tells us where the mass of the distribution lies relative to the mean. Kurtosis, a related concept, measures peakedness — whether the distribution is leptokurtic (peaked), mesokurtic (normal), or platykurtic (flat).
40.2 Symmetry, Positive Skewness, Negative Skewness
TipThree Types — Position of Mean, Median, Mode
| Shape | Mean-Median-Mode | Tail |
|---|---|---|
| Symmetric | Mean = Median = Mode | Both equal |
| Positive (right) skewness | Mean > Median > Mode | Long right tail; outliers high |
| Negative (left) skewness | Mean < Median < Mode | Long left tail; outliers low |
40.3 Measures of Skewness
40.3.1 1. Karl Pearson’s Coefficient
\[S_{kp} = \frac{Mean - Mode}{SD}\]
or (when mode is ill-defined) using the empirical relation:
\[S_{kp} = \frac{3(Mean - Median)}{SD}\]
Range: typically between −3 and +3 (theoretically); practically usually within ±1.
40.3.2 2. Bowley’s Coefficient (Quartile Coefficient)
\[S_{kb} = \frac{Q_3 + Q_1 - 2 \cdot Median}{Q_3 - Q_1}\]
Range: between −1 and +1. Robust to outliers.
40.3.3 3. Kelly’s Coefficient
\[S_{kk} = \frac{P_{90} + P_{10} - 2 \cdot P_{50}}{P_{90} - P_{10}}\]
Range: between −1 and +1. Uses tail percentiles.
40.3.4 4. Moment-Based (Beta-One Coefficient)
\[\beta_1 = \frac{\mu_3^2}{\mu_2^3}, \quad \gamma_1 = \sqrt{\beta_1} = \frac{\mu_3}{\sigma^3}\]
where μ_k = k-th central moment. γ₁ = standardised third moment; positive → right skew; negative → left skew.
40.4 Kurtosis
Kurtosis measures peakedness/tailedness of a distribution.
TipThree Types of Kurtosis
| Type | β₂ value | Shape |
|---|---|---|
| Leptokurtic | β₂ > 3 | More peaked, fatter tails than normal |
| Mesokurtic | β₂ = 3 | Normal distribution |
| Platykurtic | β₂ < 3 | Flatter than normal |
Excess kurtosis = β₂ − 3 = γ₂.
NoteDistractor warning
PYQ trap: Positive skewness has Mean > Mode; Negative skewness has Mean < Mode. The Mode sits at the peak; the Mean is pulled toward the long tail.
40.5 Practice Questions
Q 01SymmetricEasy
For a perfectly symmetric distribution:
View solution
Correct Option: A
Symmetry ⇒ all three coincide.
Q 02SkewEasy
Positive skewness implies:
View solution
Correct Option: A
Right tail pulls Mean up.
Q 03Karl PearsonMedium
Karl Pearson's coefficient of skewness equals:
View solution
Correct Option: A
**Karl Pearson** — (Mean − Mode)/SD.
Q 04BowleyMedium
Bowley's coefficient of skewness uses:
View solution
Correct Option: B
Bowley = (Q₃ + Q₁ − 2 Median)/(Q₃ − Q₁).
Q 05Bowley rangeMedium
Bowley's coefficient of skewness lies between:
View solution
Correct Option: A
Bowley range: **−1 to +1**.
Q 06KellyMedium
Kelly's coefficient uses:
View solution
Correct Option: B
Kelly uses P₁₀, P₅₀, P₉₀.
Q 07KurtosisMedium
If β₂ > 3, the distribution is:
View solution
Correct Option: C
β₂ > 3 → **leptokurtic** (peaked + fat tails).
Q 08NormalMedium
For a normal distribution, β₂ equals:
View solution
Correct Option: C
Normal: β₂ = **3** (mesokurtic). Excess kurtosis γ₂ = 0.
Q 09EmpiricalMedium
Mean = 50; Median = 48; SD = 10. Karl Pearson's coefficient (using empirical relation):
View solution
Correct Option: A
3(50 − 48)/10 = 6/10 = **+0.6** (positive skew).
Q 10SkewnessMedium
Match:
| Distribution | Skewness | ||
| (i) | Income | (a) | Negative |
| (ii) | Age at death | (b) | Positive |
View solution
Correct Option: A
Income — long right tail (positive); age at death — long left tail (negative).
Q 11SkewnessEasy
A negative skewness implies:
View solution
Correct Option: B
Negative → long left tail.
Q 12γ₁Hard
Moment-based skewness γ₁ equals:
View solution
Correct Option: B
γ₁ = μ₃/σ³ = standardised third moment.
Q 13β₂ formulaHard
β₂ (kurtosis) equals:
View solution
Correct Option: B
**β₂ = μ₄/μ₂²**.
Q 14UseMedium
When Mode is ill-defined, the substitute Karl Pearson formula uses:
View solution
Correct Option: A
Use empirical relation: Mode ≈ 3 Median − 2 Mean → S_kp = 3(Mean − Median)/SD.
Q 15PlatyEasy
A "platykurtic" distribution is:
View solution
Correct Option: B
Platykurtic = flat-topped; β₂ < 3.
Q 16PearsonMedium
Mean = 30; Mode = 24; SD = 8. Karl Pearson coefficient:
View solution
Correct Option: A
(30 − 24)/8 = **+0.75**.
Q 17AuthorsMedium
Match the skewness measure with its author:
| Measure | Author | ||
| (i) | (Mean − Mode)/SD | (a) | Bowley |
| (ii) | (Q₃+Q₁−2Med)/(Q₃−Q₁) | (b) | Karl Pearson |
| (iii) | (P₉₀+P₁₀−2P₅₀)/(P₉₀−P₁₀) | (c) | Kelly |
View solution
Correct Option: A
Karl Pearson, Bowley, Kelly.
Q 18ApplicationHard
Stock-return distributions are typically *leptokurtic* because they have:
View solution
Correct Option: B
Leptokurtic → fatter tails — important for risk management.
Q 19CompareMedium
Bowley's coefficient is preferred over Karl Pearson when:
View solution
Correct Option: A
Bowley uses quartiles → robust to outliers.
Q 20Symmetric β₁Hard
For a symmetric distribution, β₁ equals:
View solution
Correct Option: A
μ₃ = 0 ⇒ β₁ = 0 in symmetric distributions.
40.6 Quick Recall
ImportantQuick recall
- Skewness — departure from symmetry. Symmetric → Mean = Median = Mode. Positive → Mean > Median > Mode (right tail). Negative → Mean < Median < Mode (left tail).
- Karl Pearson: (Mean − Mode)/SD; or 3(Mean − Median)/SD when mode ill-defined; range ≈ ±3.
- Bowley: (Q₃ + Q₁ − 2 Median)/(Q₃ − Q₁); range ±1; outlier-robust.
- Kelly: (P₉₀ + P₁₀ − 2 P₅₀)/(P₉₀ − P₁₀); range ±1.
- Moment-based: γ₁ = μ₃/σ³; β₁ = μ₃²/μ₂³.
- Kurtosis: β₂ = μ₄/μ₂². Normal → β₂ = 3 (mesokurtic). β₂ > 3 — leptokurtic; < 3 — platykurtic.
- Stock returns — typically leptokurtic (fat tails).