75  Moments

75.1 Why Moments Matter

Moments summarize different aspects of a distribution:

  • first non-centered moment: mean (location)
  • second centered moment: variance (spread)
  • third centered moment: skewness-related behavior (asymmetry)
  • fourth centered moment: kurtosis-related behavior (tail weight)

For this reason, moments serve as building blocks for many descriptive and inferential statistics.

75.2 Definition of General Moments

\[ m_j = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - c \right)^j \]

where \(j\) stands for the order of the Moment.

If \(c=0\) then \(m_j\) represents a Non-Centered Moment.

If \(c\neq0\) then \(m_j\) represents a Centered Moment around \(c\).

75.3 Definition of Non-Centered Moments

\[ m_j = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - c \right)^j \]

where \(c = 0\).

Non Centered Moments are also written as

\[ m_j' = \frac{1}{n} \sum_{i=1}^{n} x_i^j \]

where \(j\) stands for the order of the Non-Centered Moment. A special case is obtained for \(j=1\) because \(m_1' = \bar{x}\) (i.e. the Arithmetic Mean is the Non-Centered Moment of order 1).

75.4 Definition of Centered Moments around \(\bar{x}\)

\[ m_j = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - c \right)^j \]

where \(c = \bar{x}\).

The Centered Moment around \(\bar{x}\) can also be written as

\[ m_j = \frac{1}{n} \sum_{i=1}^{n} \left( x_i - \bar{x} \right)^j \]

where \(j\) stands for the order of the Centered Moment around \(\bar{x}\). A special case is obtained for \(j=2\) because \(m_2 = \text{V}(x) = \sigma_x^2\) (i.e. the Centered Moment of order 2 is equal to the Variance).

75.5 Property 1

There is a mathematical relationship between Centered and Non-Centered Moments:

\[ m_k = \sum_{i=0}^{k} \binom{k}{i} m_{k-i}' \left( -m_1' \right)^i \]

where

\[ \begin{cases}\binom{k}{i} = \frac{k!}{i!(k-i)!} \\m_0 = m_0' = 1 \\m_1 = 0\end{cases} \]

75.5.1 Special case: \(k = 2\)

\[ \begin{align*}&m_2 = \sum_{i=0}^{2} \binom{2}{i} m_{2-i}' \left( -m_1' \right)^i \\&m_2 = \binom{2}{0} m_{2-0}' \left( -m_1' \right)^0 + \binom{2}{1} m_{2-1}' \left( -m_1' \right)^1 + \binom{2}{2} m_{2-2}' \left( -m_1' \right)^2 \\&m_2 = m_2' - 2m_1' m_1' + {m'}_1^2 \\&m_2 = m_2' - {m'}_1^2\end{align*} \]

75.5.2 Special cases: \(k = 1, 2, 3, 4\)

\[ \begin{align*}&m_1 = 0 \\&m_2 = m_2' - {m'}_1^2 \\&m_3 = m_3' - 3m_1' m_2' + 2{m'}_1^3 \\&m_4 = m_4' - 4m_1'm_3' + 6{m'}_1^2 m_2' - 3{m'}_1^4\end{align*} \]

75.6 Property 2

Non-Centered Moments can also be expressed in terms of Centered Moments

\[ m_k' = \sum_{i=0}^{k} \binom{k}{i} m_{k-i} \left( m_1' \right)^i \]

where

\[ \begin{cases}\binom{k}{i} = \frac{k!}{i!(k-i)!} \\m_0 = m_0' = 1 \\m_1 = 0\end{cases} \]

75.7 Property 2 (for \(k = 2\))

\[ \begin{align*}&m_2' = \sum_{i=0}^{2} m_{2-i} \left( m_1' \right)^i \\&m_2' = \binom{2}{0} m_{2-0} \left( m_1' \right)^0 + \binom{2}{1} m_{2-1} \left( m_1' \right)^1 + \binom{2}{2} m_{2-2} \left( m_1' \right)^2 \\&m_2' = m_2 + {m'}_1^2\end{align*} \]

75.8 R Module

75.8.0.0.1 Public website

The General Moments module is available on the public website:

75.8.1 RFC

The General Moments module can be found in RFC under the “Descriptive / Moments”.

To compute General Moments on your local machine, the following script can be used in the R console:

library(e1071)
x <- runif(200)
par1 <- 6 #Maximum order
for (i in 1:par1) {
  print(paste('Uncentered moment of order', i, moment(x,order=i,center=F,absolute=F,na.rm=T)))
  print(paste('Centered moment of order', i, moment(x,order=i,center=T, absolute=F,na.rm=T)))
}
[1] "Uncentered moment of order 1 0.50840566675528"
[1] "Centered moment of order 1 -4.71844785465692e-17"
[1] "Uncentered moment of order 2 0.34248288992888"
[1] "Centered moment of order 2 0.0840065679399998"
[1] "Uncentered moment of order 3 0.2582521478915"
[1] "Centered moment of order 3 -0.0012869244857838"
[1] "Uncentered moment of order 4 0.206921586803157"
[1] "Centered moment of order 4 0.012446444355051"
[1] "Uncentered moment of order 5 0.17218541440635"
[1] "Centered moment of order 5 -0.000487717421671315"
[1] "Uncentered moment of order 6 0.147038541222848"
[1] "Centered moment of order 6 0.00219594101803263"

To compute General Moments, the R code uses the moment function from the e1071 library. The parameter par1 determines the maximum order to be used in the computations.

75.9 Purpose

Moments are used to derive various types of descriptive and inferential statistics.