Skewness Introduction, formula, Interpretation

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

Positive Skewed
If the frequency curve of 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 positively skewed. In a positively skewed distribution, the mean is greater than the median 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 negatively skewed. In a negatively skewed distribution, the 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 a skewed distribution, these values are pulled apart.

Pearson’s Coefficient of Skewness
Karl Pearson, (1857-1936) introduced a coefficient to measure the degree of skewness of distribution or curve, which is denoted 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
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 positively skewed. If $S_k$ is less than zero the distribution or curve is said to be negatively 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 a histogram or frequency curve.

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