Statistics for Data Science & Analytics

If the curve is symmetrical, a deviation below the mean exactly equals the corresponding deviation above the mean. This is called symmetry. Here, we will discuss Skewness and Measures of Skewness.

Skewness is the degree of asymmetry or departure from the symmetry of a distribution. Positive Skewness means when the tail on the right side of the distribution is longer or fatter. The mean and median will be greater than the mode. Negative Skewness is when the tail of the left side of the distribution is longer or fatter than the tail on the right side.

Measures of Skewness

Karl Pearson Measures of Relative Skewness

In a symmetrical distribution, the mean, median, and mode coincide. In skewed distributions, these values are pulled apart; the mean tends to be on the same side of the mode as the longer tail. Thus, a measure of the asymmetry is supplied by the difference ($mean – mode$). This can be made dimensionless by dividing by a measure of dispersion (such as SD).

The Karl Pearson measure of relative skewness is
$$\text{SK} = \frac{\text{Mean}-\text{mode}}{SD} =\frac{\overline{x}-\text{mode}}{s}$$
The value of skewness may be either positive or negative.

The empirical formula for skewness (called the second coefficient of skewness) is

$$\text{SK} = \frac{3(\text{mean}-\text{median})}{SD}=\frac{3(\tilde{X}-\text{median})}{s}$$

Bowley Measures of Skewness

In a symmetrical distribution, the quartiles are equidistant from the median ($Q_2-Q_1 = Q_3-Q_2$). If the distribution is not symmetrical, the quartiles will not be equidistant from the median (unless the entire asymmetry is located in the extreme quarters of the data). The Bowley suggested measure of skewness is

$$\text{Quartile Coefficient of SK} = \frac{Q_(2-Q_2)-(Q_2-Q_1)}{Q_3-Q_1}=\frac{Q_2-2Q_2+Q_1}{Q_3-Q_1}$$

This measure is always zero when the quartiles are equidistant from the median and is positive when the upper quartile is farther from the median than the lower quartile. This measure of skewness varies between $+1$ and $-1$.

Moment Coefficient of Skewness

In any symmetrical curve, the sum of odd powers of deviations from the mean will be equal to zero. That is, $m_3=m_5=m_7=\cdots=0$. However, it is not true for asymmetrical distributions. For this reason, a measure of skewness is devised based on $m_3$. That is

\begin{align}
\text{Moment of Coefficient of SK}&= a_3=\frac{m_3}{s^3}=\frac{m_3}{\sqrt{m_2^3}}\\
&=b_1=\frac{m_3^2}{m_2^3}
\end{align}

For perfectly symmetrical curves (normal curves), $a_3$ and $b_1$ are zero.

Real-Life Examples of Skewness

1. Income Distribution: Income distribution in most countries is right-skewed. A large number of people earn relatively low incomes, while a smaller number earn significantly higher incomes, creating a long tail on the right side of the distribution.
2. Insurance Claims: Insurance claim amounts are typically right-skewed. Most claims are for smaller amounts, but there are a few very large claims that create a long tail on the right.
3. Age at Retirement: The age at which people retire is often right-skewed. Most people retire around a certain age, but some continue to work much later in life, creating a long tail on the right.
4. Test Scores: In some educational settings, test scores can be left-skewed if the test is very easy, with most students scoring high and a few scoring much lower, creating a long tail on the left.
5. Hospital Stay Duration: The length of hospital stays is often right-skewed. Most patients stay for a short period, but some patients with severe conditions stay much longer, creating a long tail on the right.
6. House Prices: In many housing markets, the distribution of house prices is right-skewed. There are many houses priced within a certain range, but a few very expensive houses create a long tail on the right.
7. Web Traffic: The number of visitors to different websites can be highly right-skewed. A few popular sites get a huge number of visitors, while the majority of sites get much less traffic.
8. Customer Spending: In retail, customer spending can be right-skewed. Most customers spend a small amount, but a few spend a lot, creating a long tail on the right.
9. The lifespan of Products: The lifespan of certain products can be right-skewed. Most products last for a certain period, but a few last much longer, creating a long tail on the right.
10. Natural Disasters: The severity of natural disasters, such as earthquakes or hurricanes, can be right-skewed. Most events are of low to moderate severity, but a few are extremely severe, creating a long tail on the right.

FAQs about SKewness

1. What is skewness?
2. If a curve is symmetrical then what is the behavior of deviation below and above the mean?
3. What is Bowley’s Measure of Skewness?
4. What is Karl Person’s Measure of Relative Skewness?
5. What is the moment coefficient of skewness?
6. What is the positive and negative skewness?

Skewness

Online MCQs Test Preparation Website