Moments In Statistics (2012)

Introduction to Moments in Statistics

The measure of central tendency (location) and the measure of dispersion (variation) are useful for describing a data set. Both the measure of central tendencies and the measures of dispersion fail to tell anything about the shape of the distribution. We need some other certain measure called the moments. Moments in Statistics are used to identify the shape of the distribution known as skewness and kurtosis.

Moments are fundamental statistical tools for understanding the characteristics of any dataset. They provide quantitative measures that describe the data:

• Central tendency: The “center” of the data. It is the most common measure of central tendency, but other moments can also be used.
• Spread: Indicates how scattered the data is around the central tendency. Common measures of spread include variance and standard deviation.
• Shape: Describes the overall form of the data distribution. For instance, is it symmetrical? Does it have a long tail on one side? Higher-order moments like skewness and kurtosis help analyze the shape.

The moments about the mean are the mean of deviations from the mean after raising them to integer powers. The $r$th population moment about the mean is denoted by $\mu_r$ is

$\mu_r=\frac{\sum^{N}_{i=1}(y_i – \bar{y} )^r}{N}$

where $r=1,2,\cdots$

The corresponding sample moment denoted by $m_r$ 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 the 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 $r$th sample moment  about the 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 $\sum^{n}_{i=1}f_i=n$

The $r$th 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 $r=1,2,\cdots$.

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 $r$th 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 moments 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*}

If variable $y$ assumes $n$ values $y_1, y_2, \cdots, y_n$ then $r$th moment about zero can be obtained by taking $a=0$ so the moment about arbitrary value will be
$m’_r = \frac{\sum y^r}{n}$

where $r=1,2,3,\cdots$.

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 the 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

${\rm Kurtosis} = \frac{\sum^{i=1}_{n} (y_i -\bar{y})^4}{ns^4}$

In summary, moments are quantitative measures that describe the distribution of a dataset around its central tendency. Different types of moments, provide specific information about the shape and characteristics of data. By understanding and utilizing moments, one can get a deeper understanding of the data and make more informed decisions in statistical analysis.

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