Unit 5: Relation Between Moments

I. Orientation: Moments and Distributional Shape

A moment is a numerical expectation that describes a distribution relative to a chosen point. Moments summarize location, dispersion, asymmetry, and tail concentration; their interpretation depends on the order of the moment and the point about which it is measured.

• General moment about a point: For a random variable (X), the (r)-th moment about an arbitrary point (a) is
  

  TEXT

    E[(X - a)^r]

  
  where (E) denotes expectation, (r) is a positive integer, and (a) is the reference point.

• Raw moment: The (r)-th raw moment, or moment about the origin, is
  

  TEXT

    μ′r = E[X^r]

  
  where (\mu'_r) denotes the (r)-th raw moment. In particular, (\mu'_1=E[X]=\mu), the arithmetic mean.

• Central moment: The (r)-th central moment is measured about the mean:
  

  TEXT

    μr = E[(X - μ)^r]

  
  where (\mu=E[X]). Thus, (\mu_1=0) and (\mu_2=\sigma^2), the variance.

• Existence: A moment is defined only when the corresponding expectation is finite; for example, (\mu_4) requires (E[(X-\mu)^4]<\infty).

• Interpretation by order:

  • First order: Indicates location when taken about the origin.
  • Second order: Measures dispersion about the mean.
  • Third order: Detects direction and degree of asymmetry.
  • Fourth order: Describes tail concentration and extremity relative to variance.

• Frequency distributions: For observations (x_i) with frequencies (f_i) and total frequency (N=\sum f_i),
  

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    μ′r = (1/N) Σ fi xi^r
    μr  = (1/N) Σ fi (xi - x̄)^r

  
  where (\bar{x}=(1/N)\sum f_i x_i), and the summation extends over all values or class midpoints.

II. Raw and Central Moments — Converting Between Reference Points

Raw moments are convenient for computation, while central moments are more directly connected with variance and distributional shape.

A. Relation between raw moments and central moments

The relation follows by expanding (X-\mu) with the binomial theorem and taking expectations.

• General conversion to central moments:
  

  TEXT

    μr = Σ[r choose k] (-μ)^(r-k) μ′k,   k = 0, 1, ..., r

  
  Here (\mu'_0=E[1]=1), (\mu=\mu'_1), and ({r\choose k}=r!/[k!(r-k)!]).

• First four central moments:
  

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    μ1 = 0
  
    μ2 = μ′2 - (μ′1)^2
  
    μ3 = μ′3 - 3μ′2μ′1 + 2(μ′1)^3
  
    μ4 = μ′4 - 4μ′3μ′1
               + 6μ′2(μ′1)^2
               - 3(μ′1)^4

  
  The second relation is the familiar identity (\operatorname{Var}(X)=E[X^2]-[E(X)]^2).

• Inverse conversion: Raw moments can similarly be recovered by writing (X=(X-\mu)+\mu):
  

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    μ′1 = μ
  
    μ′2 = μ2 + μ^2
  
    μ′3 = μ3 + 3μμ2 + μ^3
  
    μ′4 = μ4 + 4μμ3 + 6μ^2μ2 + μ^4

  
  In these expressions, the un-subscripted (\mu) is the mean, whereas (\mu_r) is a central moment.

• Odd and even moments: Even central moments satisfy (\mu_{2r}\geq0). Odd central moments may be positive, negative, or zero because deviations on opposite sides of the mean retain opposite signs.

• Symmetry implication: If a distribution is symmetric about its mean and the moments exist, every odd central moment is zero. The converse is not universally valid without additional conditions.

• Worked example: Suppose (\mu'_1=2), (\mu'_2=6), (\mu'_3=20), and (\mu'_4=70). Then
  

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    μ2 = 6 - 2^2 = 2
    μ3 = 20 - 3(6)(2) + 2(2^3) = 0
    μ4 = 70 - 4(20)(2) + 6(6)(2^2) - 3(2^4) = 6

  
  Thus the variance is (2), the third central moment is zero, and the fourth central moment is (6).

III. Transformations of Moments — Shifting and Rescaling Data

A linear transformation changes the numerical moments in predictable ways, allowing inconvenient values or measurement units to be simplified.

A. Effect of change of origin and scale on moments

Changing origin shifts all observations by a constant, while changing scale divides or multiplies their distances by a constant.

• Linear transformation: Let
  

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    Y = (X - a)/h,   h ≠ 0

  
  where (a) is the chosen origin and (h) is the scale factor. Equivalently, (X=a+hY).

• Raw moments after transformation:
  

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    μ′r(Y) = (1/h^r) Σ[r choose k] (-a)^(r-k) μ′k(X),
              k = 0, 1, ..., r

  
  Raw moments are affected by both (a) and (h) because they are measured from zero.

• Mean transformation:
  

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    E[Y] = (E[X] - a)/h

  
  Therefore, shifting origin changes the mean, and rescaling changes both its magnitude and unit.

• Central moment transformation:
  

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    μr(Y) = μr(X)/h^r

  
  Central moments are unaffected by a change of origin because both (X) and its mean shift equally. A change of scale affects the (r)-th central moment by the (r)-th power of the scale factor.

• Variance and standard deviation:
  

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    Var(Y) = Var(X)/h^2
    SD(Y)  = SD(X)/|h|

  
  Variance is always divided by (h^2), while standard deviation uses (|h|) because it cannot be negative.

• Sign under negative scale: If (h<0), odd central moments reverse sign, but even central moments remain unchanged in sign. Reflection therefore reverses skewness but not kurtosis.

• Dimensionless coefficients: Ratios such as (\mu_3/\mu_2^{3/2}) and (\mu_4/\mu_2^2) remain unchanged under positive shifts and rescaling because numerator and denominator acquire matching powers of (h).

• Worked example: If (Y=(X-10)/2), (\mu_2(X)=16), and (\mu_3(X)=24), then
  

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    μ2(Y) = 16/2^2 = 4
    μ3(Y) = 24/2^3 = 3

  
  The shift by (10) has no effect on these central moments.

IV. Moment-Based Shape Measures — Pearson’s Standardized Ratios

Pearsonian coefficients combine central moments into dimensionless quantities, making distributions measured in different units directly comparable.

A. Pearsonian coefficients

Pearson’s moment coefficients quantify asymmetry and kurtosis by standardizing the third and fourth central moments with the variance.

• First Pearsonian coefficient:
  

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    β1 = μ3^2/μ2^3

  
  Since the third moment is squared, (\beta_1\geq0); it measures the magnitude but not the direction of skewness.

• Signed coefficient of skewness:
  

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    γ1 = μ3/μ2^(3/2)

  
  The relation is (\beta_1=\gamma_1^2). A positive (\gamma_1) indicates right skewness, while a negative value indicates left skewness.

• Second Pearsonian coefficient:
  

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    β2 = μ4/μ2^2

  
  This standardized fourth moment measures kurtosis and is nonnegative whenever (\mu_2>0).

• Excess kurtosis coefficient:
  

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    γ2 = β2 - 3

  
  Subtracting (3) uses the normal distribution as the reference, because a normal distribution has (\beta_2=3).

• Invariance: The coefficients (\beta_1) and (\beta_2) are independent of measurement origin and scale. Thus converting centimetres to metres does not alter them.

• Pearson inequality: For distributions possessing the required moments,
  

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    β2 ≥ β1 + 1

  
  This gives a consistency restriction between standardized skewness and kurtosis.

• Degenerate case: If (\mu_2=0), all observations are equal and the standardized coefficients are undefined because their denominators vanish.

V. Distributional Asymmetry — Direction and Degree of Departure from Symmetry

Skewness measures the lack of symmetry around a distribution’s centre and distinguishes elongation toward larger values from elongation toward smaller values.

A. Measures of skewness

Different measures emphasize moments, averages, or positional summaries, so the choice depends on the available data and the distribution’s characteristics.

• Moment coefficient: The most general dimensionless measure is
  

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    γ1 = μ3/σ^3

  
  where (\sigma=\sqrt{\mu_2}). It uses every observation and retains the direction of asymmetry.

• Interpretation:

  • Symmetric distribution: (\gamma_1=0), provided the third moment exists.
  • Positive skewness: (\gamma_1>0); the longer tail extends toward larger values.
  • Negative skewness: (\gamma_1<0); the longer tail extends toward smaller values.

• Karl Pearson’s coefficient:
  

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    SkP = (Mean - Mode)/SD

  
  where (Sk_P) denotes Pearson’s coefficient and (SD) is standard deviation. It is useful when the mode is clearly defined.

• Median approximation: When the mode is unreliable, Pearson’s empirical relation gives
  

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    SkP = 3(Mean - Median)/SD

  
  This is an empirical measure rather than an identity valid for every distribution.

• Bowley’s quartile coefficient:
  

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    SkB = (Q3 + Q1 - 2Q2)/(Q3 - Q1)

  
  Here (Q_1), (Q_2), and (Q_3) are the first quartile, median, and third quartile. It is resistant to extreme values but reflects mainly the middle half of the data.

• Choice of measure: Moment skewness is algebraically powerful; Pearson’s coefficient is intuitive for unimodal data; Bowley’s coefficient is preferable for open-ended classes or highly extreme observations.

• Limitation: Zero skewness does not necessarily prove symmetry. A distribution may have (\mu_3=0) while still possessing asymmetric features not captured by the third moment.

VI. Fourth-Moment Shape — Tail Weight and Extreme Deviations

Kurtosis compares the fourth central moment with squared variance and is best interpreted as sensitivity to tails and extreme observations rather than simply the sharpness of a graph’s peak.

A. Kurtosis

Kurtosis is measured by the standardized fourth central moment and classified relative to the normal distribution.

• Coefficient of kurtosis:
  

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    β2 = μ4/σ^4 = μ4/μ2^2

  
  The fourth power gives unusually large deviations substantial influence, making kurtosis highly sensitive to extreme values.

• Excess kurtosis:
  

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    γ2 = β2 - 3

  
  The benchmark (3) is the kurtosis of every normal distribution, regardless of its mean or variance.

• Three classifications:

  1. Mesokurtic: (\beta_2=3), or (\gamma_2=0); the normal distribution is the standard example.
  2. Leptokurtic: (\beta_2>3), or (\gamma_2>0); the distribution has greater fourth-moment tail weight than the normal benchmark.
  3. Platykurtic: (\beta_2<3), or (\gamma_2<0); the distribution has lower fourth-moment tail weight than the normal benchmark.

• Lower bound: Since (\mu_4\geq\mu_2^2), any non-degenerate distribution satisfies
  

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    β2 ≥ 1

  
  Equality occurs in the limiting two-point case with equal mass at two values equally distant from the mean.

• Transformation property: Kurtosis is unchanged by any nonzero linear rescaling because both (\mu_4) and (\mu_2^2) acquire the factor (h^4).

• Interpretive caution: A larger (\beta_2) does not by itself prove that the density has a visibly sharper central peak. Kurtosis is a global fourth-moment property shaped strongly by tails and outliers.

• Practical limitation: Because deviations are raised to the fourth power, a few extreme observations can dominate sample kurtosis; comparisons should therefore consider data quality and sample size.