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[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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