Understanding and Calculating Coefficient of Skewness with Mean and Mode

The coefficient of skewness is a measure of the degree of asymmetry in a distribution. This article discusses how to calculate the coefficient of skewness when given the mean and mode, but not the standard deviation. We'll also cover a scenario where the standard deviation is provided, as well as the importance of understanding the significance of skewness in data analysis.

Introduction to Coefficient of Skewness

Skewness is a statistical measure that describes the asymmetry of the probability distribution of a real-valued random variable about its mean. A distribution is considered:

Positively skewed (right-skewed): The tail on the right side of the distribution is longer. Negatively skewed (left-skewed): The tail on the left side of the distribution is longer.

Calculating Skewness with Mean and Mode

The coefficient of skewness can be calculated using Pearson's first coefficient of skewness formula, which is:

Skewness frac{(3*Mean - Mode)}{Standard Deviation}

Given that we only have the mean and mode and not the standard deviation, we can express the skewness in terms of these variables. Let's consider an example where the mean (μ) is 40 and the mode (Mo) is 60.

Example Without Standard Deviation

Using the formula:

Skewness frac{(3*40 - 60)}{Standard Deviation}

Simplifying this expression:

Skewness frac{120 - 60}{Standard Deviation} frac{60}{Standard Deviation}

In this case, the coefficient of skewness is:

Skewness -60 / Standard Deviation

Negative skewness indicates that the distribution is left-skewed or negatively skewed, meaning the left tail is longer than the right tail. If you have the standard deviation, you can substitute it into the equation to find the exact value of skewness.

Calculating Skewness with Standard Deviation

Let's consider a scenario where the standard deviation (σ) is given. If σ is 10, we can calculate the coefficient of skewness as follows:

Example with Standard Deviation

Using the given values:

Mean (μ) 40 Mode (Mo) 60 Standard Deviation (σ) 10

Substituting these values into the formula:

Skewness frac{(40 - 60)}{10} -20 / 10 -2

This indicates a negative skewness, confirming that the distribution is negatively skewed.

Conclusion and Significance

Understanding and calculating the coefficient of skewness is crucial for interpreting the distribution of data. It helps in analyzing the asymmetry and determining the shape of the distribution. Knowing whether a distribution is positively or negatively skewed can be vital in various fields such as finance, economics, and social sciences.

For further reading and detailed information, you can refer to advanced statistics textbooks or online resources. Understanding these concepts can help you make more informed decisions in data analysis and interpretation.