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Moments, Skewness & Kurtosis

Paper-II Statistics (Post-specific) · Statistics Syllabus

Useful for: SSC CGL

Overview

Moments summarise the shape of a distribution; skewness measures asymmetry and kurtosis measures peakedness.

Key points

μ₁ about the mean is 0; μ₂ is the variance.
Karl Pearson’s skewness = (mean − mode)/SD, or 3(mean − median)/SD.
Bowley’s skewness = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁).
Kurtosis β₂ = μ₄/μ₂²: β₂ = 3 mesokurtic (normal), > 3 leptokurtic, < 3 platykurtic.
Positive skew: mean > median > mode; negative skew: mean < median < mode.

Explanation

β₁ = μ₃²/μ₂³ measures skewness without sign.
Bowley’s measure lies between −1 and +1.
Excess kurtosis = β₂ − 3.
Raw moments can be converted to central moments with standard relations.

Step by step

Identify the measure or test asked and its formula.
Organise the data (table, totals) before computing.
Interpret the result (sign, size, significance).

Examples

Mean 50, mode 45, SD 10 → skewness = 0.5 (positive).
μ₂ = 4, μ₄ = 48 → β₂ = 48/16 = 3 → mesokurtic.
Q₁ = 20, Q₂ = 30, Q₃ = 50 → Bowley = (50 + 20 − 60)/30 = 0.33.

Traps and advanced points

Trap: Mixing signs of skewness.
Trap: Using μ₄/μ₂ instead of μ₄/μ₂².
Trap: Calling β₂ = 3 leptokurtic.

Quick revision

Pearson: (mean − mode)/SD.
β₂ = μ₄/μ₂²; 3 is normal.
Mean > median > mode → positive skew.

Practice MCQs

The first central moment of any distribution is always
1. 0
2. 1
3. the variance
4. the mean
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Answer: (a) 0. The sum of deviations from the mean is zero.
The second central moment equals the
1. skewness
2. mean
3. variance
4. kurtosis
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Answer: (c) variance. μ₂ = σ².
A distribution with a longer right tail is
1. mesokurtic
2. negatively skewed
3. symmetric
4. positively skewed
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Answer: (d) positively skewed. For it, mean > median > mode.
For a symmetric distribution
1. median = 0
2. mean > mode
3. mode > mean
4. mean = median = mode
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Answer: (d) mean = median = mode. There is no skewness.
For a normal distribution, β₂ (Pearson's measure of kurtosis) equals
1. 1
2. 0
3. 3
4. 2
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Answer: (c) 3. Such a curve is mesokurtic.
A curve more peaked than the normal curve is called
1. platykurtic
2. leptokurtic
3. mesokurtic
4. bimodal
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Answer: (b) leptokurtic. Its β₂ is greater than 3.

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Notes and questions are written by Kundali IQ for practice. They are not official papers; always check the latest official notification for the syllabus and pattern.