Basic Statistics and Data Analysis

The Moments in Statistics

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.

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 denote by μr 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 $\sum^{n}_{i=1}f_i=n$

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*}

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

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

Skewness

Skewness is the degree of asymmetry or departure from symmetry of the distribution of a real valued random variable.

Positive Skewed
If the frequency curve of a distribution has a longer tail to the right of the central maximum than to the left, the distribution is said to be skewed to the right or to have positive skewed. In a positive skewed distribution, the mean is greater than the media and median is greater than the mode i.e. Mean > Median > Mode.

Negative Skewed
If the frequency curve has a longer tail to the left of the central maximum than to the right, the distribution is said to be skewed to the left or to have negative skewed. In a negatively skewed distribution, mode is greater than median and median is greater than mean i.e. Mode > Median > Mean.

In a symmetrical distribution the mean, median and mode coincide. In skewed distribution these values are pulled apart.

Pearson’s Coefficient of Skewness
Karl Pearson, (1857-1936) introduced a coefficient of skewness to measure the degree of skewness of a distribution or curve, which is denote by $S_k$ and define by

\begin{eqnarray*}
S_k &=& \frac{Mean – Mode}{Standard Deviation}\\
S_k &=& \frac{3(Mean – Median)}{Standard Deviation}\\
\end{eqnarray*}
Usually this coefficient varies between –3 (for negative) to +3 (for positive) and the sign indicates the direction of skewness.

Bowley’s Coefficient of Skewness or Quartile Coefficient of Skewness
Arthur Lyon Bowley (1869-1957) proposed a measure of skewness based on the median and the two quartiles.

$S_k=\frac{Q_1+Q_3-2Median}{Q_3 – Q_1}$
Its values lie between 0 and ±1.

Moment Coefficient of Skewness
This measure of skewness is the third moment expressed in standard units (or the moment ratio) thus given by

$S_k=\frac{\mu_3}{\sigma^3}$
Its values lie between -2 and +2.

If $S_k$ is greater than zero, the distribution or curve is said to be positive skewed. If $S_k$ is less than zero the distribution or curve is said to be negative skewed. If $S_k$ is zero the distribution or curve is said to be symmetrical.

The skewness of the distribution of a real valued random variable can easily be seen by drawing histogram or frequency curve.

The skewness may be very extreme and in such a case these are called J-shaped distributions.

Measure of Kurtosis

In statistics, measure of kurtosis is a measure of the “tailedness” of the probability distribution of a real-valued random variable. The standard measure of kurtosis, is based on a scaled version of the fourth moment of the data or population. Therefore, measure of kurtosis is related to the tails of the distribution, not its peak.

Sometimes, Measure of Kurtosis is characterized as a measure of peakedness is mistaken. A distribution having a relatively high peak is called leptokurtic. A distribution which is flat-topped is called platykurtic. The normal distribution which is neither very peaked nor very flat-topped is also called mesokurtic.  The histogram in some cases can be used as an effective graphical technique for showing both the skewness and kurtosis of data set.

Data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Moment ratio and Percentile Coefficient of kurtosis are used to measure the kurtosis

Moment Coefficient of Kurtosis= $b_2 = \frac{m_4}{S^2} = \frac{m_4}{m^{2}_{2}}$

Percentile Coefficient of Kurtosis = $k=\frac{Q.D}{P_{90}-P_{10}}$
where Q.D = $\frac{1}{2}(Q_3 – Q_1)$ is the semi-interquartile range. For normal distribution this has the value 0.263.

Dr Wheeler defines kurtosis as:

The kurtosis parameter is a measure of the combined weight of the tails relative to the rest of the distribution.

So, kurtosis is all about the tails of the distribution – not the peakedness or flatness.

A normal random variable has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3, it is said to be Leptokurtic. If its kurtosis is less than 3, it is said to be Platykurtic.

A large value of kurtosis indicates a more serious outlier issue and hence may lead the researcher to choose alternative statistical methods.

Some Examples of Kurtosis

• In finance, risk and insurance are examples of needing to focus on the tail of distribution and not assuming normality.
• Kurtosis helps in determining whether resource used within an ecological guild is truly neutral or which it differs among species.
• The accuracy of the variance as an estimate of the population $\sigma^2$ depends heavily on kurtosis.