The sample central moment is defined by $$ m_j = \frac{1}{n} \sum_{i = 1}^{n} \left( x_i - \bar{x} \right)^j %(\#eq:dist-moments-sample-central) $$ where
\(n\) is the sample size,
\(x = \{x_1 \dots x_n\}\) is a set of values of a random variable \(X\),
\(\bar{x}\) is the sample mean of \(x\), and
\(j\) is the \(j\)th central moment.
moment(x, j)
| x | Numeric vector. Sample data. |
|---|---|
| j | Integer.
|
$$ m_0 = \frac{1}{n} \sum_{i = 1}^{n} \left( x_i - \bar{x} \right)^0 = \frac{1}{n} \sum_{i = 1}^{n} \left( 1 \right) = \frac{1}{n} n = \frac{n}{n} = 1 %(\#eq:dist-moments-sample-central-zero) $$
$$ m_1 = \frac{1}{n} \sum_{i = 1}^{n} \left( x_i - \bar{x} \right)^1 = \frac{1}{n} \sum_{i = 1}^{n} \left( 0 \right) = \frac{1}{n} 0 = \frac{0}{n} = 0 %(\#eq:dist-moments-sample-central-first) $$
$$ m_2 = \frac{1}{n} \sum_{i = 1}^{n} \left( x_i - \bar{x} \right)^2 %(\#eq:dist-moments-sample-central-second) $$
$$ m_3 = \frac{1}{n} \sum_{i = 1}^{n} \left( x_i - \bar{x} \right)^3 %(\#eq:dist-moments-sample-central-third) $$
$$ m_4 = \frac{1}{n} \sum_{i = 1}^{n} \left( x_i - \bar{x} \right)^4 %(\#eq:dist-moments-sample-central-fourth) $$
The "zeroth" central moment is 1.
The first central moment is 0.
The second cental moment is the variance.
The third central moment is used to define skewness.
The fourth central moment is used to define kurtosis.
Wikipedia: Standardized Moment