Moments In Statistics (2012)

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 measure of dispersions 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 about Mean

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 $\sum^{n}_{i=1}f_i=n$

Moments about Arbitrary Value

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 $r=1,2,\cdots$.

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}

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

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

Moments about Zero

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

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}\\

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

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

Moments in Statistics

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.

Moments In Statistics (2012)

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