# 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

Kurtosis is a measure of peakedness of a distribution relative to the normal distribution. 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 is 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.

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.