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

## Table of Contents

**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 the 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 be negatively skewed. In a negatively skewed distribution, the mode is greater than the median and the median is greater than the 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 **Formula**

Karl Pearson, (1857-1936) introduced a coefficient to measure the degree of skewness of distribution or curve, which is denoted by $S_k$ and defined 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 Formula (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 Formula

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.

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