Moment

In statistics, the term moment has been referred to as quantitative-measure, measuring set-of-point’s shape. The 2^nd moment is used widely & the width of a particular set-of-points is measured by it in 1 attribute or in a higher attribute it measures cloud or group of-point’s shape as an-ellipsoid could fit the clouds. Other distribution aspects, like how distributions are skewed or peaked from the mean, are described by other types of moments. The statistical concepts are related closely to the moment’s concept in the study of physics. However, moments in the physics are represented in a different manner. Characterization of distribution of any type can be done through numerous features, like skewness, variance, mean etc, & an operation’s moment describes the distribution's nature.

By the symbol μ_1, we denote the 1^st moment. The distribution’s 1^st moment of given random-variable – X has been considered as expectation manipulator, that is, population-mean (condition is that if there is existence of the 1^st mean).

In advanced orders, central-moments are very much interesting in comparison to those moments which are nearly Zero. The equation which shows the actual valued-random-variable “X” i.e. the probability distribution’s k^th central-moment, with μ i.e. the expected-value is as follows:

μ_{k =}  E ( (X – μ) ^k )

Therefore Zero (“0”) is the 1^st central-moment. The zero ^th central-moment i.e.μ₀ is 1.

Likewise, other types of moments can be also defined. Say, for e.g. the n ^th inverse-moment near Zero (“0”) is as follows:

E (X ^-n)

And n ^th inverse-moment near Zero (“0”) is as follows:

E (ln ⁿ (x))

Variance:

The 2^nd central-moments near mean are called as variance. Standard deviation (σ) is variance's positive-square root.

Normalized moments:

The standardized-moment or normalized-n ^th central-moment is n ^th central-moment which is divided-by- σⁿ ; the formula is as follows:

X’s normalized-n ^th central-moment = E ((x − μ) ⁿ) / σⁿ

The mentioned above normalized-central moments, are quantities without any dimension representing the statistical distribution severally of the scale's any one dimensional change.

Skewness:

The 3^rd central moment has been considered as distribution’s lopsidedness’ measure. Zero’s symmetrical distribution has a 3^rd central moment. The normalized 3^rd central moment has been defined as skewness, and often as γ. Any distribution which has been skewed towards the left, (i.e. distribution’s tail is heavier towards the left side) has negative-skewness and any distribution which has been skewed towards the right, (i.e. distribution’s tail is heavier towards the right side) has positive-skewness.

Kurtosis

The 4^th central moment has been considered as a measure, measuring the features of distribution i.e. whether it is skinny and tall or squat and short when compared with normal distribution having similar variance. Kurtosis has been defined by following equation:

Kurtosis (k) = normalized 4^th central-moment – 3.

Mixed moments:

These are the moments which involve many variables like cokurtosis, covariance and coskewness.