Moments

Posted on by MDP

The concept of “moment” has various interpretations depending on context, but generally refers to a unit of time, a specific occasion, or a particular state. In mechanics, a “moment” is the tendency of a force to rotate a body around a point or axis. In other contexts, it can refer to a precise instant in time, an emotionally significant period in psychology, or a notable event or period in history and culture.

Our focus, however, is on statistical “moments,” which are used to describe the distribution of data.

Absolute moments

Absolute moments are calculated around the origin. They consider the values of a variable to the zero point, rather than the mean value. In essence, absolute moments measure how much our values deviate from zero.

\mu'_k = E(X^k) = \int_{-\infty}^{\infty} x^k f(x) dx

For example, if our variable measures heights, absolute moments evaluate how much each individual value deviates from zero, without considering the mean.

In the formula, the expected value of the first moment (k=1) represents the mean.

Central moments

Central moments are more significant as they are calculated relative to the mean.

The most important central moments are:

  • First central moment: always equals zero, representing the average difference between values and their mean.
  • Second central moment: the variance (σ2) of the distribution, measuring dispersion around the mean.
  • Third central moment: this measures skewness, describing the distribution’s symmetry relative to the mean.
  • Fourth central moment: measures kurtosis, indicating the “heaviness” of the distribution’s tails—how pronounced or thin they are compared to a normal distribution.

The general formula for central moments is:

\mu_k = \frac{1}{N} \sum_{i=1}^{N} (x_i - \overline{x})^k

Where: N is the number of data points, xi is each individual data point, x̄ is the mean of the data, k is the order of the moment.

For k = 1 (first-order moment), we obtain the mean—the expected value of a distribution.

For k = 2 (second-order moment), we obtain the variance, which measures how much the data deviates from the mean.

For k = 3 (third-order moment), we obtain the skewness, indicating how much the distribution is tilted to the right or left.

For k = 4 (fourth-order moment), we obtain the kurtosis, representing the “heaviness” of the distribution’s tails.

These concepts form the foundation of data analysis, statistical inference, and hypotheses about random phenomena.

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