An Introduction To Central Moment

Introduction

In terms of probability theory and statistics, a central moment can be defined as a moment of the probability distribution of a random variable about the variable’s own mean. In other words, it is the projected value of a particular integer power of the deviation of the random variable from the mean. The different moments create a set of values which can be used to successfully portray the properties of a probability distribution. Ordinary moments are calculated from the zero; central moments are preferred over them as they are calculated using deviations from the mean. The reason for this is that central moments are only related to the spread and shape of the distribution and not to its location.

Univariate Moments

Univariate moments are related to the probability distribution of just one random variable also known as univariate distribution. There are different interpretations of the first few central moments of a univariate distribution. The zeroth moment of the distribution is one. The first moment of the probability distribution is zero. The second is known as the variance. The third and fourth of the probability distribution are used to ascertain the standardized moments which are used to define skewness and kurtosis, respectively. Univariate central moments possess a number of different properties. The first one is that they are translation invariant. The second property is that univariate are homogeneous. In some cases where independent random variables are involved, the central moments also have an additivity property. They are closely related to moments about the origin. At times, it can be relatively easy to change moments about the origin to moments about the mean which are the central moments. A symmetric univariate distribution is generally unaffected by being moved about its mean. Therefore in such a distribution, the value of all odd moments will equal to zero. The reason for this is that in the formula for calculating the central moment, each term involving a value of a random variable less than the mean by a certain amount exactly cancels out the term involving a value of the random variable greater than the mean by the same amount.

Multivariate moments

Multivariate distribution refers to the probability distribution that takes into account two or more random variables. It is also known as joint probability distribution. The formula for calculating the central moment in case of a multivariate distribution is an extension of the formula for univariate distribution.

Conclusion

Defining the central moment of a probability distribution has a number of applications in the field of statistics.

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