Coefficient of Skewness

The coefficient of skewness is a statistical measure that indicates both the direction and extent of skewness in a dataset. It uses values like the mean, median, or mode to show how much a sample distribution deviates from a perfectly normal, symmetric distribution. A larger skewness value means the sample distribution is more uneven and differs more strongly from a normal distribution.


For example,
A dataset of monthly incomes with a skewness of +1.8 indicates a strongly right-skewed distribution. This means a small number of people earn very high incomes, making the distribution very different from a standard, symmetric shape.
 

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

• If the coefficient of skewness is positive, the distribution is right-skewed, indicating a longer right tail.

• If the coefficient of skewness is negative, the distribution is left-skewed, indicating a longer left tail.

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

Skewness indicates how data points are spread in a dataset. 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: Pearson’s First Coefficient, also known as the moment coefficient of skewness, measures the skewness of a distribution. 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.

Understanding the coefficient of skewness can be challenging, so a few helpful tips can make the concept easier to master. Below are some practical guidelines.

• A positive skew indicates a long right tail, while a negative skew shows a long left tail. Try visualizing the shape to avoid confusion.
  
   
• Create histograms or frequency curves to see whether the data is symmetric or skewed quickly.
  
   
• Working with examples like income levels or student scores helps you see how skewness appears in everyday situations.
  
   
• Since it is dimensionless, you can easily compare skewness values across datasets.
  
   
• Tools such as Excel, SPSS, or Python can calculate skewness for you. Focus on understanding how to interpret the values rather than just computing them.
  
   
• Parents can ask children to sort small sets of numbers and check whether each set is symmetric or skewed.
  
   
• Teachers can conduct quick activities where students guess whether a graph is left-skewed or right-skewed.
  
   
• Children can use colorful charts or stickers to understand how values tend to cluster toward one side of the distribution.

Karl Pearson proposed two formulas for calculating the coefficient of skewness, one is based on the mode and the other is based on the median. The formulas are.

Using the mode:

\(\text{sk}_1 = \frac{\bar{x} - \text{Mode}}{s} \)
 

By using the median:

\(\text{sk}_2 = \frac{3(\bar{x} - \text{Median})}{s} \)

Where,
x = mean
s = standard deviation

The first formula uses the mode, but since the mode can be unreliable in small or multimodal datasets, researchers often choose the second formula, which uses the median, because it provides a more stable and accurate measure of skewness.

Depending on the available data, either of the two formulas can be used to calculate the coefficient of skewness. Suppose the mean of a data set is 48, the mode is 60, the median is 55, and the standard deviation is 12. The steps to calculate the coefficient of skewness are as follows:

By using the mode

Step 1: Subtract the mode from the mean.
 

\(48 – 60 = -12\)
 

Step 2: Divide this value by the standard deviation to get the coefficient of skewness.
 

Thus, 
 

\(sk_1=-12/12=-1\)
 

By using the median
 

Step 1: Subtract the median from the mean.
 

\(48 – 55 = - 7\)
 

Step 2: Multiply this value by 3.
 

This gives -21.
 

Step 3: Divide the value from step 2 by the standard deviation to obtain the coefficient of skewness.
 

Thus,
 

\(sk_2=-21/12=-1.75\)

Understanding the coefficient of skewness can be challenging, so a few helpful tips can make the concept easier to master. Below are some practical guidelines.

• A positive skew indicates a long right tail, while a negative skew shows a long left tail. Try visualizing the shape to avoid confusion.
  
   
• Create histograms or frequency curves to see whether the data is symmetric or skewed quickly.
  
   
• Working with examples like income levels or student scores helps you see how skewness appears in everyday situations.
  
   
• Since it is dimensionless, you can easily compare skewness values across datasets.
  
   
• Tools such as Excel, SPSS, or Python can calculate skewness for you. Focus on understanding how to interpret the values rather than just computing them.
  
   
• Parents can ask children to sort small sets of numbers and check whether each set is symmetric or skewed.
  
   
• Teachers can conduct quick activities where students guess whether a graph is left-skewed or right-skewed.
  
   
• Children can use colorful charts or stickers to understand how values tend to cluster toward one side of the distribution.

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. 
  
   
• Weather forecasting: Meteorologists use skewness to study rainfall data or temperature variations. A positively skewed distribution might indicate more days with extremely high temperatures or heavy rainfall.
  
   
• Real estate: Property analysts use skewness to understand housing prices in an area. If the data is positively skewed, it shows that while most houses are moderately priced, a few luxury homes raise the average price.