Central moment

From WikiMD's Wellness Encyclopedia

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[edit | edit source]

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[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 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[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