In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" (in a particular sense) of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by an ellipsoid. Other moments describe other aspects of a distribution such as how the distribution is skewed from its mean, or peaked. The mathematical concept is closely related to the concept of moment in physics, although moment in physics is often represented somewhat differently. Any distribution can be characterized by a number of features (such as the mean, the variance, the skewness, etc.), and the moments of a function describe the nature of its distribution.
The 1st moment is denoted by μ1. The first moment of the distribution of the random variable X is the expectation operator, i.e., the population mean (if the first moment exists).
In higher orders, the central moments (moments about the mean) are more interesting than the moments about zero. The kth central moment, of a real-valued random variable probability distribution X, with the expected value μ is
The first central moment is thus 0. The zero-th central moment, μ0 is one. See also central moment.
Other moments may also be defined. For example, the n th inverse moment about zero is E(X − n) and the n th logarithmic moment about zero is E(lnn(x))
The fourth central moment is a measure of whether the distribution is tall and skinny or short and squat, compared to the normal distribution of the same variance. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always non-negative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is 3σ4.
The kurtosis κ is defined to be the normalized fourth central moment minus 3. (Equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance.) Some authorities do not subtract three, but it is usually more convenient to have the normal distribution at the origin of coordinates. If a distribution has a peak at the mean and long tails, the fourth moment will be high and the kurtosis positive (leptokurtic); and conversely; thus, bounded distributions tend to have low kurtosis (platykurtic).
The kurtosis can be positive without limit, but κ must be greater than or equal to γ2 − 2; equality only holds for binary distributions. For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of γ2 and 2γ2.
The inequality can be proven by considering
Mixed moments are moments involving multiple variables.
Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.
High-order moments are moments beyond 4th-order moments. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality