In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the moments of a
function are certain quantitative measures related to the shape of the function's
graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
, and the second moment is the
moment of inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular accele ...
. If the function is a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
, then the first moment is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
, the second
central moment is the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
, the third
standardized moment is the
skewness, and the fourth standardized moment is the
kurtosis. The mathematical concept is closely related to the concept of
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
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* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
in physics.
For a distribution of mass or probability on a
bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (
Hausdorff moment problem). The same is not true on unbounded intervals (
Hamburger moment problem).
In the mid-nineteenth century,
Pafnuty Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebysh ...
became the first person to think systematically in terms of the moments of
random variables.
Significance of the moments
The '-th raw moment (i.e., moment about zero) of a distribution is defined by
where
The -th moment of a
real-valued
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
''f''(''x'') of a real variable about a value ''c'' is the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
It is possible to define moments for
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s in a more general fashion than moments for real-valued functions — see
moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with ''c'' = 0.
For the second and higher moments, the
central moment (moments about the mean, with ''c'' being the mean) are usually used rather than the moments about zero, because they provide clearer information about the distribution's shape.
Other moments may also be defined. For example, the th inverse moment about zero is