Moments

Moments is a term used in several disciplines to denote a quantitative measure that describes the influence or distribution of a quantity around a reference point or axis. While the specifics vary by field, the common idea is to capture how a quantity is spread, rotated, or aggregated relative to location, time, or another variable.

In physics and engineering, the moment of a force about a point, or torque, measures the turning effect produced when a force acts at a distance from that point. It is defined as the cross product M = r × F and is measured in newton-meters. Moments about a line or axis are central to rotational equilibrium: if the sum of moments about every point on an axis is zero, the system is balanced. The moment of inertia is a related concept that describes an object’s resistance to angular acceleration; for a collection of point masses, I = Σ m_i r_i^2, where r_i is the distance to the rotation axis.

In statistics and probability, moments describe the shape of a distribution. Raw moments m_k = E[X^k] summarize the distribution’s powers, while central moments μ_k = E[(X − μ)^k] describe deviations from the mean μ. The first raw moment equals the mean, the second central moment equals the variance, the third central moment relates to skewness, and the fourth to kurtosis. Moment generating functions, M_X(t) = E[e^{tX}], encode all moments.

In geometry and image analysis, geometric or spatial moments summarize the distribution of coordinates in a shape, aiding tasks such as finding the centroid, orientation, and various shape descriptors. Across applications, moments provide concise, interpretable summaries of how quantities are arranged around a reference.