Understanding Pearson's Coefficient of Skewness
Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. Pearson's coefficient of skewness ($S_k$) is a widely used metric to quantify the degree of this asymmetry.
The Formula
Pearson’s first coefficient of skewness is defined as:
$$S_k = \frac{\bar{x} - M_0}{\sigma}$$
Where:
- $\bar{x}$ is the arithmetic mean.
- $M_0$ is the mode.
- $\sigma$ is the standard deviation.
Step-by-Step Solution
Given data:
- Number of items ($N$) = 50
- Sum of squares ($\sum x^2$) = 600
- Sum of values ($\sum x$) = 150
- Mode ($M_0$) = 1.75
1. Find the Arithmetic Mean ($\bar{x}$): $$\bar{x} = \frac{\sum x}{N} = \frac{150}{50} = 3$$
2. Find the Standard Deviation ($\sigma$): First, calculate variance ($\sigma^2$): $$\sigma^2 = \frac{\sum x^2}{N} - \left(\frac{\sum x}{N}\right)^2$$ $$\sigma^2 = \frac{600}{50} - (3)^2 = 12 - 9 = 3$$ $$\sigma = \sqrt{3} \approx 1.732$$
3. Calculate Skewness ($S_k$): $$S_k = \frac{3 - 1.75}{1.732} = \frac{1.25}{1.732} \approx 0.722$$
Intuition
A positive skewness ($S_k > 0$) indicates that the tail on the right side of the distribution is longer or fatter than the left side, and the mean is typically greater than the mode.