Central Moments Overview
Central moments are statistical measures calculated around the mean (or any fixed point) rather than around zero. They describe various characteristics of a distribution’s shape.
The r-th central moment is defined as:
$$m_r = \frac{\sum(x_i - \bar{x})^r}{n}$$
where $\bar{x}$ is the mean and n is the number of observations.
The Four Key Central Moments
First Central Moment (m₁)
Always equals zero by definition:
$$m_1 = \frac{\sum(x_i - \bar{x})}{n} = 0$$
This is because the sum of deviations from the mean is always zero.
Second Central Moment (m₂)
Equals the variance:
$$m_2 = \frac{\sum(x_i - \bar{x})^2}{n} = \sigma^2$$
Property: Measures spread or dispersion around the mean
Third Central Moment (m₃)
Related to skewness:
$$m_3 = \frac{\sum(x_i - \bar{x})^3}{n}$$
Property: Measures asymmetry; used to calculate skewness coefficient
$$\text{Skewness} = \frac{m_3}{s^3}$$
Fourth Central Moment (m₄)
Related to kurtosis:
$$m_4 = \frac{\sum(x_i - \bar{x})^4}{n}$$
Property: Measures tailedness; used to calculate kurtosis
$$\text{Kurtosis} = \frac{m_4}{s^4}$$
Central Moments for Ungrouped Data
Example 1: Complete Calculation
Data: 12, 14, 16, 18, 20
Calculate all central moments.
Solution:
Step 1: Calculate mean
$$\bar{x} = \frac{12 + 14 + 16 + 18 + 20}{5} = \frac{80}{5} = 16$$
Step 2: Create deviation table
| $x_i$ | $(x_i - \bar{x})$ | $(x_i - \bar{x})^2$ | $(x_i - \bar{x})^3$ | $(x_i - \bar{x})^4$ |
|---|---|---|---|---|
| 12 | -4 | 16 | -64 | 256 |
| 14 | -2 | 4 | -8 | 16 |
| 16 | 0 | 0 | 0 | 0 |
| 18 | 2 | 4 | 8 | 16 |
| 20 | 4 | 16 | 64 | 256 |
| 40 | 0 | 544 |
Step 3: Calculate central moments
$$m_1 = \frac{0}{5} = 0$$
$$m_2 = \frac{40}{5} = 8 \text{ (variance)}$$
$$m_3 = \frac{0}{5} = 0$$
$$m_4 = \frac{544}{5} = 108.8$$
Step 4: Calculate skewness and kurtosis
$$\text{Skewness} = \frac{m_3}{(\sqrt{m_2})^3} = \frac{0}{(2.83)^3} = 0$$
(Symmetric distribution)
$$\text{Kurtosis} = \frac{m_4}{(\sqrt{m_2})^4} = \frac{108.8}{(8)^2} = 1.7$$
$$\text{Excess Kurtosis} = 1.7 - 3 = -1.3$$
(Platykurtic - flatter than normal)
Example 2: Skewed Distribution
Data: 10, 15, 20, 25, 50
Calculate central moments.
Solution:
$$\bar{x} = \frac{120}{5} = 24$$
| $x_i$ | $(x_i - \bar{x})$ | $(x_i - \bar{x})^2$ | $(x_i - \bar{x})^3$ | $(x_i - \bar{x})^4$ |
|---|---|---|---|---|
| 10 | -14 | 196 | -2,744 | 38,416 |
| 15 | -9 | 81 | -729 | 6,561 |
| 20 | -4 | 16 | -64 | 256 |
| 25 | 1 | 1 | 1 | 1 |
| 50 | 26 | 676 | 17,576 | 456,976 |
| 970 | 14,040 | 502,210 |
$$m_2 = \frac{970}{5} = 194$$
$$m_3 = \frac{14,040}{5} = 2,808$$
$$m_4 = \frac{502,210}{5} = 100,442$$
$$\text{Skewness} = \frac{2,808}{(13.93)^3} = 1.13$$ (Positively skewed)
Central Moments for Grouped Data
Formula
For grouped data:
$$m_r = \frac{\sum f_i(m_i - \bar{x})^r}{N}$$
where:
- $f_i$ = frequency of class i
- $m_i$ = midpoint of class i
- $\bar{x}$ = mean
- N = total frequency
Example: Grouped Data
| Class | Frequency | Midpoint |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 12 | 15 |
| 20-30 | 18 | 25 |
| 30-40 | 10 | 35 |
| 40-50 | 5 | 45 |
Calculate central moments (assume mean = 23.5).
Solution:
| Class | $f$ | $m_i$ | $(m_i - \bar{x})$ | $f(m_i - \bar{x})^2$ | $f(m_i - \bar{x})^3$ | $f(m_i - \bar{x})^4$ |
|---|---|---|---|---|---|---|
| 0-10 | 5 | 5 | -18.5 | 1,711.25 | -31,658.125 | 585,075.3 |
| 10-20 | 12 | 15 | -8.5 | 867 | -7,369.5 | 62,640.75 |
| 20-30 | 18 | 25 | 1.5 | 40.5 | 60.75 | 91.125 |
| 30-40 | 10 | 35 | 11.5 | 1,322.5 | 15,208.75 | 174,900.625 |
| 40-50 | 5 | 45 | 21.5 | 2,306.25 | 49,584.375 | 1,066,064.0625 |
| N=50 | 6,247.5 | 25,826.25 | 1,888,771.875 |
$$m_2 = \frac{6,247.5}{50} = 124.95$$
$$m_3 = \frac{25,826.25}{50} = 516.525$$
$$m_4 = \frac{1,888,771.875}{50} = 37,775.44$$
Relationship Between Central Moments and Distribution Shape
| Moment | Property | Interpretation |
|---|---|---|
| m₂ | Variance | Data spread around mean |
| m₃ | Related to skewness | Data asymmetry |
| m₄ | Related to kurtosis | Tail weight and peakedness |
Central Moments vs. Raw Moments
Raw Moment: $$\mu_r’ = \frac{\sum x_i^r}{n}$$
Conversion Formula: $$m_r = \sum \binom{r}{k} (-1)^k \bar{x}^k \mu_{r-k}’$$
Central moments are more useful for statistical analysis because they’re invariant to location (translation).
Applications of Central Moments
- Distribution Characterization: Understanding shape properties
- Skewness Calculation: Measuring asymmetry
- Kurtosis Calculation: Measuring tailedness
- Statistical Tests: Basis for normality tests
- Moment Matching: Estimating parameters
- Risk Assessment: Analyzing financial returns
Properties of Central Moments
- m₁ = 0 (always)
- m₂ > 0 (always positive)
- m₃ = 0 for symmetric distributions
- m₄ ≥ m₂² (Cauchy-Schwarz inequality)
- Scale Property: If $y = cx$, then $m_r(y) = c^r m_r(x)$
Comparison: Symmetric vs. Skewed Data
Symmetric Distribution
- $m_1 = 0$
- $m_3 = 0$
- Mean = Median = Mode
Right-Skewed Distribution
- $m_1 = 0$
- $m_3 > 0$ (positive)
- Mean > Median > Mode
- Longer tail on right
Left-Skewed Distribution
- $m_1 = 0$
- $m_3 < 0$ (negative)
- Mean < Median < Mode
- Longer tail on left
Practical Example: Quality Control
Manufacturing measurements (n = 100):
- $m_2 = 2.5$ → SD = 1.58 (process spread)
- $m_3 = 0.15$ → Slight positive skew (slight bias)
- $m_4 = 18.2$ → Relatively high kurtosis (more variability)
Conclusion: Process slightly biased high with more extreme variations than normal.
Computational Notes
- For large datasets, use computational formulas to reduce rounding errors
- Center data (subtract mean) before calculations when possible
- For grouped data, use class midpoints as representatives
- Higher central moments are more sensitive to outliers
Limitations
- Can be sensitive to extreme values
- Interpretation becomes complex for moments beyond the fourth
- Calculation can be computationally intensive for large datasets
- Different assumptions needed for different distribution types