Coefficient of Skewness

The coefficient of skewness is a measure used to determine the direction of the skewness in a dataset using the mean, median, or mode. If the data is skewed to the left, it has negative skewness. If the data is symmetric or in two equal halves, it has zero skewness. If the data is skewed to the right, then it has positive skewness. The coefficient of skewness indicates whether the data points are more spread out on one side of the distribution’s mean than the other.

The coefficient of skewness can be interpreted based on its sign:

• If the coefficient of skewness is positive, the data distribution is skewed to the right, indicating a longer tail on the right side.

• If the coefficient of skewness is negative, then the data distribution is skewed to the left, indicating a longer tail on the left side.

• If the coefficient of skewness is zero, then the data distribution is symmetric.
   

There are various types of skewness that are used to show whether the data points in a dataset are spread out or not. We can classify skewness into two main types:

Positive Skewness:

In a positively skewed distribution, the mean will be greater than the median, which is greater than the mode. This implies that the distribution has a longer tail towards the right side, where the extreme values pull the mean towards the right.

Negative Skewness:

In a negative skewed distribution, the mean will be less than the median, which is less than the mode. This means that the distribution has a longer tail towards the left side, with a few extreme values pulling the mean towards the left.

There are several measures that we use to quantify the skewness in a distribution. Some of the most commonly used measures are:

Pearson’s First Coefficient:

Also known as the moment coefficient of skewness. It is a measure of skewness used to compare the mean and mode of a data distribution. It determines the direction and the extent of the skewness in the data. The formula we use for Pearson’s First Coefficient is: Pearson’s First Coefficient Formula = (Mean - Mode) / Standard Deviation

Where:

Mean is the average of the values in the dataset

Mode is the most frequently occurring value in the dataset

Standard Deviation is a measure of the amount of variation in the dataset.

If mean > mode, the skewness is positive (right-skewed)

If mean < mode, the skewness is negative (left-skewed)

If mean ≈ mode, the skewness is symmetric

Pearson’s Second Coefficient of Skewness:

Compared to Pearson’s first coefficient, it is less influenced by outliers or any extreme values in the distributions. We use Pearson’s second coefficient if the mode is not well-defined. The formula we use is:
Pearson’s Second Coefficient Formula = 3(Mean - Median) / Standard Deviation

Where:

Mean is the average of the values in the dataset

Median is the central value in the dataset

Standard Deviation is a measure of the amount of variation in the dataset.

If mean > median, the skewness is positive (right-skewed)

If mean < median, the skewness is negative (left-skewed)

If mean ≈ median, the skewness is symmetric

These are the two formulas used to calculate Pearson’s coefficient of skewness.

The coefficient of skewness is used to determine how data is distributed. Here are some real-world applications of the coefficient of skewness:

Finance and investing:

Skewness helps investors analyze stock returns and understand whether there is a high gain or any risks of extreme losses.

Healthcare:

Hospitals use skewness to analyze the patient's test results or disease spread. It can be used to determine whether a particular disease affects younger people or older people depending on the direction of the skewness.

Education analysis:

To grade and standardize the tests, schools use skewness to determine how many students score high or low.