Moments, Skewness and Kurtosis

Averages describe where a distribution sits and dispersion how wide it is. Two more features complete the picture: whether it leans to one side (skewness) and whether it is sharply peaked or flat-topped (kurtosis). Moments are the single family of measures that delivers all four. This chapter defines raw and central moments, converts one into the other, and builds the standard measures of skewness and kurtosis from them.

Raw and central moments

The r-th moment about a point A is μ′ᵣ = Σf(x − A)ʳ ÷ N; when A = 0 these are the raw moments. The r-th central moment is taken about the mean: μᵣ = Σf(x − x̄)ʳ ÷ N. Two values are fixed: μ₀ = 1 and μ₁ = 0 (deviations from the mean sum to zero). Then μ₂ is the variance, μ₃ carries the lean of the distribution and μ₄ its peakedness.

Central moments from moments about A
MomentRelation
μ₂μ′₂ − μ′₁²
μ₃μ′₃ − 3μ′₂μ′₁ + 2μ′₁³
μ₄μ′₄ − 4μ′₃μ′₁ + 6μ′₂μ′₁² − 3μ′₁⁴

Worked: about A the first four moments are 1, 4, 10 and 46. Then μ₂ = 4 − 1 = 3, μ₃ = 10 − 3(4)(1) + 2(1) = 0, μ₄ = 46 − 4(10)(1) + 6(4)(1) − 3(1) = 27. Also μ′₁ = x̄ − A, so the mean is A + 1. Central moments are unaffected by a change of origin; under a change of scale y = x/h, μᵣ(y) = μᵣ(x)/hʳ.

Skewness: which way the distribution leans

A distribution is symmetric when its two halves mirror each other; then mean = median = mode. If the right tail is longer it is positively skewed and mean > median > mode; if the left tail is longer it is negatively skewed and mean < median < mode. Income distributions are the classic positive case: a few very high incomes drag the mean up.

Measures of skewness
MeasureFormulaLimits
Karl Pearson (first)(Mean − Mode) ÷ σusually −3 to +3
Karl Pearson (second)3(Mean − Median) ÷ σ−3 to +3
Bowley (quartile)(Q₃ + Q₁ − 2Q₂) ÷ (Q₃ − Q₁)−1 to +1
Kelly (percentile)(P₉₀ + P₁₀ − 2P₅₀) ÷ (P₉₀ − P₁₀)−1 to +1
Momentβ₁ = μ₃² ÷ μ₂³, γ₁ = μ₃ ÷ σ³β₁ ≥ 0; γ₁ carries the sign

Worked: mean 50, mode 45, σ = 10 gives Pearson's Sk = (50 − 45)/10 = 0.5, a moderate positive skew. With Q₁ = 20, Q₂ = 30, Q₃ = 50, Bowley's Sk = (50 + 20 − 60)/(50 − 20) = 10/30 ≈ 0.33: the upper quartile is farther from the median than the lower one.

⚠️ Exam trap
β₁ = μ₃²/μ₂³ is a square, so it can never show the direction of skew. If a question asks whether the skew is positive or negative, look at the sign of μ₃ (or γ₁), not at β₁.

Kurtosis: how peaked the distribution is

Kurtosis compares the peak and tails of a distribution with the normal curve. Its moment measure is β₂ = μ₄ ÷ μ₂², with γ₂ = β₂ − 3 (excess kurtosis). The normal curve has β₂ = 3 and is mesokurtic; β₂ > 3 is leptokurtic (tall, thin peak and heavy tails); β₂ < 3 is platykurtic (flat-topped). In the worked example β₂ = 27/3² = 3, so that distribution is mesokurtic, and μ₃ = 0 makes it symmetric.

A quantile measure also exists: the percentile coefficient of kurtosis κ = Q.D. ÷ (P₉₀ − P₁₀), which is about 0.263 for a normal distribution.

🧠 Memory trick
Lepto = slender (tall peak), platy = flat (like a plate), meso = middle (normal, β₂ = 3).

Exam traps and a quick revision

  • μ₁ = 0 and μ₂ = σ² always; μ₂ = μ′₂ − μ′₁².
  • Positive skew: mean > median > mode; negative skew: the reverse.
  • Bowley's and Kelly's coefficients lie between −1 and +1; Bowley ignores the outer quarters.
  • β₂ = 3 mesokurtic, > 3 leptokurtic, < 3 platykurtic.
  • For any distribution β₂ ≥ β₁ + 1, so β₂ is never less than 1.

Key takeaways

  • Central moments are moments about the mean: μ₁ = 0, μ₂ = variance.
  • μ₃ = μ′₃ − 3μ′₂μ′₁ + 2μ′₁³ and μ₄ = μ′₄ − 4μ′₃μ′₁ + 6μ′₂μ′₁² − 3μ′₁⁴.
  • Pearson Sk = (Mean − Mode)/σ ≈ 3(Mean − Median)/σ; Bowley Sk = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁).
  • β₁ = μ₃²/μ₂³ measures the size of skew; the sign comes from μ₃.
  • β₂ = μ₄/μ₂²: 3 for the normal curve; above 3 leptokurtic, below 3 platykurtic.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. A distribution with β₂ = 3 is called:

    1. leptokurtic
    2. mesokurtic
    3. platykurtic
    4. positively skewed
    Show answer

    Answer: B — mesokurtic

    β₂ = 3 is the kurtosis of the normal curve, the reference level called mesokurtic. Kurtosis says nothing about skewness, which needs μ₃.

  2. In a positively skewed distribution:

    1. mean > median > mode
    2. median > mean > mode
    3. mode > median > mean
    4. mean = median = mode
    Show answer

    Answer: A — mean > median > mode

    The long right tail pulls the mean furthest to the right, the median follows part of the way, and the mode stays at the peak.

  3. The second central moment μ₂ of a distribution is its:

    1. variance
    2. mean
    3. mean deviation
    4. standard deviation
    Show answer

    Answer: A — variance

    μ₂ = Σf(x − x̄)²/N, which is the variance by definition. The SD is its square root, and the first central moment is always zero.

  4. The first two moments of a distribution about the value 5 are 2 and 20. The variance is:

    1. 18
    2. 20
    3. 16
    4. 4
    Show answer

    Answer: C — 16

    μ₂ = μ′₂ − μ′₁² = 20 − 4 = 16. The mean is A + μ′₁ = 7, which is not needed for the variance.

  5. About a point A the first four moments are 1, 4, 10 and 46. The fourth central moment μ₄ is:

    1. 46
    2. 3
    3. 27
    4. 30
    Show answer

    Answer: C — 27

    μ₄ = μ′₄ − 4μ′₃μ′₁ + 6μ′₂μ′₁² − 3μ′₁⁴ = 46 − 40 + 24 − 3 = 27. With μ₂ = 3, β₂ = 27/9 = 3, so the distribution is mesokurtic.

  6. A distribution has mean 50, mode 45 and standard deviation 10. Karl Pearson's coefficient of skewness is:

    1. 0.5
    2. −0.5
    3. 1.5
    4. 5
    Show answer

    Answer: A — 0.5

    Sk = (Mean − Mode)/σ = (50 − 45)/10 = 0.5. It is positive because the mean exceeds the mode.

  7. For a distribution Q₁ = 20, median = 30 and Q₃ = 50. Bowley's coefficient of skewness is:

    1. −1/3
    2. 2/3
    3. 1/3
    4. 1/2
    Show answer

    Answer: C — 1/3

    Sk = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁) = (50 + 20 − 60)/30 = 10/30 = 1/3. Q₃ is farther from the median than Q₁, so the skew is positive.

  8. Bowley's coefficient of skewness always lies between:

    1. −3 and +3
    2. −1 and +1
    3. 0 and 3
    4. 0 and 1
    Show answer

    Answer: B — −1 and +1

    Because Q₃ + Q₁ − 2Q₂ can never exceed Q₃ − Q₁ in size, Bowley's measure is bounded by ±1. Pearson's measure is the one usually quoted as lying within ±3.

  9. A distribution has mean 60, median 57 and standard deviation 9. Its coefficient of skewness based on the median is:

    1. 1
    2. 3
    3. 1/3
    4. −1
    Show answer

    Answer: A — 1

    Pearson's second coefficient = 3(Mean − Median)/σ = 3 × 3/9 = 1. Forgetting the factor 3 gives 1/3.

  10. For a distribution μ₂ = 4 and μ₄ = 60. The distribution is:

    1. platykurtic, since β₂ = 3.75
    2. leptokurtic, since β₂ = 15
    3. mesokurtic, since β₂ = 3
    4. leptokurtic, since β₂ = 3.75
    Show answer

    Answer: D — leptokurtic, since β₂ = 3.75

    β₂ = μ₄/μ₂² = 60/16 = 3.75, which exceeds 3, so the curve is more peaked than the normal: leptokurtic. Dividing by μ₂ instead of μ₂² gives the wrong 15.