In statistics, moments are arithmetic averages of certain powers of deviations of observations from a fixed point. Moments help us describe the centre, spread, asymmetry and peakedness of a distribution.
The \(r^{th}\) moment about the point \(A\) is
\[ \mu'_r(A) \;=\; \dfrac{1}{n}\sum_{i=1}^{n}(x_i - A)^r \]When \(A = 0\), \(\mu'_r\) is the \(r^{th}\) moment about the origin; when \(A = \bar{x}\), it becomes the central moment \(\mu_r\).
Important values of central moments:
(Here \(\mu'_r\) are taken about an arbitrary point \(A\).)
Write each deviation from the point \(A\) as \(d = x - A\), so \(\mu'_r = \overline{d^{\,r}}\) and the mean sits at \(\bar x = A + \mu'_1\). A deviation from the mean is then \(x - \bar x = d - \mu'_1\), and every central moment is just the binomial expansion of \((d - \mu'_1)^r\) averaged:
\[ \mu_2 = \overline{(d-\mu'_1)^2} = \overline{d^2} - 2\mu'_1\overline{d} + (\mu'_1)^2 = \mu'_2 - 2\mu'_1(\mu'_1) + (\mu'_1)^2 = \mu'_2 - (\mu'_1)^2, \]
using \(\overline{d} = \mu'_1\). The \(\mu_3\) and \(\mu_4\) formulae are the same expansion of \((d-\mu'_1)^3\) and \((d-\mu'_1)^4\). Note that when \(A = \bar x\) we have \(\mu'_1 = 0\), and the formulae collapse to \(\mu_r = \mu'_r\) — as they must. The \(\mu_2\) identity is exactly the variance shortcut \(\sigma^2 = \overline{x^2} - \bar x^2\) from Unit 4, shifted to the point \(A\).
Data: 2, 4, 6, 8, 10 (n = 5). \(\bar x = 6\).
Deviations \(d = x - 6\): −4, −2, 0, 2, 4.
| x | d | d² | d³ | d⁴ |
|---|---|---|---|---|
| 2 | −4 | 16 | −64 | 256 |
| 4 | −2 | 4 | −8 | 16 |
| 6 | 0 | 0 | 0 | 0 |
| 8 | 2 | 4 | 8 | 16 |
| 10 | 4 | 16 | 64 | 256 |
| Σ | 0 | 40 | 0 | 544 |
\(\mu_1 = 0,\;\; \mu_2 = 40/5 = 8,\;\; \mu_3 = 0,\;\; \mu_4 = 544/5 = 108.8\). Distribution is symmetric.
Suppose \(\mu'_1 = 2,\;\mu'_2 = 20,\;\mu'_3 = 40,\;\mu'_4 = 200\) about origin.
\(\mu_2 = 20 - 4 = 16\)
\(\mu_3 = 40 - 3(20)(2) + 2(2)^3 = 40 - 120 + 16 = -64\)
\(\mu_4 = 200 - 4(40)(2) + 6(20)(4) - 3(2)^4 = 200 - 320 + 480 - 48 = 312\)
When data is grouped into class intervals of width \(h\), all observations within a class are represented by the mid-point. This grouping introduces a small error in the moments. W. F. Sheppard suggested corrections for moments computed from grouped continuous data.
Odd-order central moments need no correction.
For a grouped distribution, \(\mu_2 = 25.5\) and \(h = 5\).
Corrected \(\mu_2 = 25.5 - 25/12 = 25.5 - 2.083 = \mathbf{23.417}\).
For a grouped distribution \(\mu_2 = 16,\; \mu_4 = 800,\; h = 4\).
Correction to \(\mu_4\): \(800 - (16/2)(16) + 7(256)/240 = 800 - 128 + 7.467 = \mathbf{679.467}\).
Skewness is the lack of symmetry in a distribution. A symmetric distribution has equal tails on both sides of the mean; an asymmetric one is "skewed" to the left or right.
If mode is ill-defined, use the empirical relation:
\[ S_k \;=\; \dfrac{3(\text{Mean} - \text{Median})}{\sigma} \]Range of \(S_k\): theoretically \(\pm 3\); practically lies between \(\pm 1\).
Mean = 50, Mode = 45, SD = 8. \(S_k = (50 - 45)/8 = 0.625\) (positively skewed).
Mean = 25, Median = 28, SD = 5. \(S_k = 3(25 - 28)/5 = -1.8\) (strongly negatively skewed — use with caution since exceeds ±1).
Range: \(-1 \le S_B \le 1\). Used when extreme values exist or distribution is open-ended.
\(Q_1 = 20,\; Q_2 = 30,\; Q_3 = 50\).
\(S_B = (50 + 20 - 60)/(50 - 20) = 10/30 = 0.333\) → moderately right-skewed.
\(Q_1 = 25,\; Q_2 = 40,\; Q_3 = 50\).
\(S_B = (50 + 25 - 80)/(50 - 25) = -5/25 = -0.20\) → slightly left-skewed.
From Section 4 Example 1: \(\mu_2 = 8,\; \mu_3 = 0\).
\(\beta_1 = 0,\; \gamma_1 = 0\) → symmetric.
If \(\mu_2 = 16\) and \(\mu_3 = -64\), \(\beta_1 = (-64)^2/16^3 = 4096/4096 = 1\). \(\gamma_1 = -64/16^{1.5} = -64/64 = -1\) → strongly negatively skewed.
Kurtosis measures the peakedness or flatness of a frequency distribution relative to the normal curve.
| Type | β₂ | Shape | Memory |
|---|---|---|---|
| Leptokurtic | > 3 | tall & thin | "Lepto" = leaping (high jump) |
| Mesokurtic | = 3 | normal bell | "Meso" = middle |
| Platykurtic | < 3 | flat & wide | "Platy" = plate (flat) |
From Section 4 Example 1: \(\mu_2 = 8,\;\mu_4 = 108.8\).
\(\beta_2 = 108.8 / 64 = 1.7\) → Platykurtic (flatter than normal). \(\gamma_2 = -1.3\).
If \(\mu_2 = 4\) and \(\mu_4 = 90\), \(\beta_2 = 90/16 = 5.625\) → Leptokurtic (sharper peak, heavier tails). \(\gamma_2 = 2.625\).
For a complete description of a distribution we report:
Additional worked problems with step-by-step procedures to support self-study, matching this unit's topics.
Daily earnings (₹) of 7 workers: 126, 121, 124, 122, 125, 124, 123. Compute the first four raw moments (about \(A=123\)) and central moments, with coefficients of skewness and kurtosis.
Raw moments about \(A=123\) (\(\sum(x-A)=4, \sum(x-A)^2=20, \sum(x-A)^3=28, \sum(x-A)^4=116\)):
\(\mu_1' = \tfrac{4}{7} = 0.57,\quad \mu_2' = \tfrac{20}{7} = 2.86,\quad \mu_3' = \tfrac{28}{7} = 4,\quad \mu_4' = \tfrac{116}{7} = 16.57.\)
Mean \(= 865/7 = 123.57\). Central moments: \(\mu_1 = 0,\ \mu_2 = 2.53,\ \mu_3 = -0.52,\ \mu_4 = 12.70\). Median \(= 124\), Mode \(= 124\), \(\sigma = 1.59\).
Karl Pearson's skewness: based on median \(= \dfrac{3(\bar x - M_d)}{\sigma} = \dfrac{3(123.57-124)}{1.59} = -0.81\); based on mode \(= \dfrac{\bar x - M_o}{\sigma} = -0.27\).
Bowley's skewness (\(Q_1=122, Q_2=124, Q_3=125\)): \(S_b = \dfrac{Q_3+Q_1-2Q_2}{Q_3-Q_1} = \dfrac{125+122-248}{3} = -0.33\).
Kurtosis: \(\beta_2 = \dfrac{\mu_4}{\mu_2^2} = \dfrac{12.70}{2.53^2} = 1.98\); \(\gamma_2 = \beta_2 - 3 = -1.02\). The curve is negatively skewed and platykurtic.
Milk yield (kg): classes 4–6, 6–8, …, 16–18 with cows 8, 10, 27, 38, 25, 20, 7 (\(N=135\)); moments computed about \(A=11\).
Mean \(= 11.22\), Median \(= 11.18\), Mode \(= 10.92\), \(\sigma = 3.01\).
Karl Pearson's skewness: based on median \(= \dfrac{3(11.22-11.18)}{3.01} = 0.04\); based on mode \(= \dfrac{11.22-10.92}{3.01} = 0.10\).
Bowley's skewness (\(Q_1=9.17, Q_2=11.18, Q_3=13.46\)): \(S_b = \dfrac{13.46+9.17-2(11.18)}{13.46-9.17} = 0.06\).
Kurtosis: \(\beta_2 = \dfrac{\mu_4}{\mu_2^2} = \dfrac{207.05}{9.05^2} = 2.53\); \(\gamma_2 = -0.47\).
Reading the three skewness measures together. Pearson's and Bowley's coefficients are small and positive, but the moment coefficient is small and negative: from the same moments, \(\mu_3 = -3.02\) and \(\gamma_1 = \mu_3/\mu_2^{3/2} = -0.11\). When the measures disagree in sign and all are this close to zero, the honest conclusion is that the curve is very nearly symmetric and platykurtic.
The first three moments about the value 2 are \(\mu_1'=1,\ \mu_2'=16,\ \mu_3'=-40\).
Mean \(= A + \mu_1' = 2 + 1 = 3\).
Variance \(= \mu_2 = \mu_2' - (\mu_1')^2 = 16 - 1 = 15\).