Central moment

Central moment in statistics is a measure of a probability distribution's shape, related to its moments about its mean. The nth central moment of a distribution is the expected value of the nth power of the deviations from the mean. Central moments provide insights into the properties of a distribution, such as its skewness and kurtosis.

Definition

The nth central moment of a random variable X is defined as:

\(\mu_n = E[(X - \mu)^n]\)

where:

• \(E\) denotes the expected value,
• \(X\) is the random variable,
• \(\mu\) is the mean of X (the first moment about the origin), and
• \(n\) is the order of the moment.

The second central moment, \(\mu_2\), is also known as the variance of the distribution, which measures the dispersion of the distribution around the mean. Higher-order central moments (when \(n > 2\)) describe other aspects of the distribution's shape.

Importance

Central moments are crucial in understanding the characteristics of a distribution beyond its mean and variance. For example:

• The third central moment, when normalized, gives the skewness of the distribution, indicating its asymmetry.
• The fourth central moment, when normalized, provides the kurtosis, which measures the "tailedness" of the distribution.

These moments help in describing the shape of the probability distribution more comprehensively.

Calculation

To calculate the nth central moment of a dataset, one can use the formula:

\(\mu_n = \frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})^n\)

where:

• \(N\) is the number of observations,
• \(x_i\) represents each observation, and
• \(\bar{x}\) is the sample mean.

Applications

Central moments are used in various fields, including statistics, probability theory, finance, and engineering, to analyze and interpret the behavior of random variables and model uncertainties.

See also

• Mean
• Variance
• Skewness
• Kurtosis
• Moment