The Moments in Statistics
The measure of central tendency (location) and measure of dispersion (variation) both are useful to describe a data set but both of them fail to tell anything about the shape of the distribution. We need some other certain measure called the moments to identify the shape of the distribution known as skewness and kurtosis.
Moments about Mean
The moments about mean are the mean of deviations from the mean after raising them to integer powers. The rth population moment about mean is denoted by $\mu_r$ is
\[\mu_r=\frac{\sum^{N}_{i=1}(y_i – \bar{y} )^r}{N}\]
where r=1, 2, …
Corresponding sample moment denoted by mr is
\[\mu_r=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^r}{n}\]
Note that if r=1 i.e. the first moment is zero as $\mu_1=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^1}{n}=0$. So first moment is always zero.
If r=2 then the second moment is variance i.e. \[\mu_2=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^2}{n}\]
Similarly the 3rd and 4th moments are
\[\mu_3=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^3}{n}\]
\[\mu_4=\frac{\sum^{n}_{i=1}(y_i – \bar{y} )^4}{n}\]
For grouped data the rth sample moment about sample mean $\bar{y}$ is
\[\mu_r=\frac{\sum^{n}_{i=1}f_i(y_i – \bar{y} )^r}{\sum^{n}_{i=1}f_i}\]
where $\sum^{n}_{i=1}f_i=n$
Moments about Arbitrary Value
The rth sample sample moment about any arbitrary origin “a” denoted by $m’_r$ is
\[m’_r = \frac{\sum^{n}_{i=1}(y_i – a)^2}{n} = \frac{\sum^{n}_{i=1}D^r_i}{n}\]
where $D_i=(y_i -a)$ and r = 1, 2, ….
therefore
\begin{eqnarray*}
m’_1&=&\frac{\sum^{n}_{i=1}(y_i – a)}{n}=\frac{\sum^{n}_{i=1}D_i}{n}\\
m’_2&=&\frac{\sum^{n}_{i=1}(y_i – a)^2}{n}=\frac{\sum^{n}_{i=1}D_i ^2}{n}\\
m’_3&=&\frac{\sum^{n}_{i=1}(y_i – a)^3}{n}=\frac{\sum^{n}_{i=1}D_i ^3}{n}\\
m’_4&=&\frac{\sum^{n}_{i=1}(y_i – a)^4}{n}=\frac{\sum^{n}_{i=1}D_i ^4}{n}
\end{eqnarray*}
The rth sample moment for grouped data about any arbitrary origin “a” is
\[m’_r=\frac{\sum^{n}_{i=1}f_i(y_i – a)^r}{\sum^{n}_{i=1}f} = \frac{\sum f_i D_i ^r}{\sum f}\]
The moment about the mean are usually called central moments and the moments about any arbitrary origin “a” are called non-central moments or raw moments.
One can calculate the moments about mean from the following relations by calculating the moments about arbitrary value
\begin{eqnarray*}
m_1&=& m’_1 – (m’_1) = 0 \\
m_2 &=& m’_2 – (m’_1)^2\\
m_3 &=& m’_3 – 3m’_2m’_1 +2(m’_1)^3\\
m_4 &=& m’_4 -4 m’_3m’_1 +6m’_2(m’_1)^2 -3(m’_1)^4
\end{eqnarray*}
Moments about Zero
If variable y assumes n values $y_1, y_2, \cdots, y_n$ then rth moment about zero can be obtained by taking a=0 so moment about arbitrary value will be
\[m’_r = \frac{\sum y^r}{n}\]
where r = 1, 2, 3, ….
therefore
\begin{eqnarray*}
m’_1&=&\frac{\sum y^1}{n}\\
m’_2 &=&\frac{\sum y^2}{n}\\
m’_3 &=&\frac{\sum y^3}{n}\\
m’_4 &=&\frac{\sum y^4}{n}\\
\end{eqnarray*}
The third moment is used to define the skewness of a distribution
\[{\rm Skewness} = \frac{\sum^{i=1}_{n} (y_i – \bar{y})^3}{ns^3}\]
If distribution is symmetric then the skewness will be zero. Skewness will be positive if there is a long tail in the positive direction and skewness will be negative if there is a long tail in the negative direction.
The fourth moment is used to define the kurtosis of a distribution
\[{\rm Kurtosis} = \frac{\sum^{i=1}_{n} (y_i -\bar{y})^4}{ns^4}\]
