# Central moment

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

## Definition[edit | edit source]

The *n*th 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[edit | edit source]

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[edit | edit source]

To calculate the *n*th 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[edit | edit source]

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[edit | edit source]

## References[edit | edit source]

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Contributors: Prab R. Tumpati, MD