Understanding Skewness in Quartiles and Coefficient of Skewness Calculation

Statistical analysis plays a vital role in understanding and interpreting data. One of the key aspects of statistical analysis is understanding the distribution of data, which includes measures like the median, quartiles, and the coefficient of skewness. This article discusses how to calculate the coefficient of skewness given the relationship between the median and the quartiles, specifically when the distance of the median from the first quartile is five times the distance from the third quartile to the median.

Defining Key Terms

In statistical analysis, several important parameters are used to describe a data set. Let's define these terms before moving to the solution:

Median (M): The middle value of the distribution. First Quartile (Q1): The value below which 25% of the data falls. Third Quartile (Q3): The value below which 75% of the data falls.

Problem Statement

The problem statement provides a specific relationship between the median and the quartiles:

The distance of the median from the first quartile is 5 times the distance of the third quartile from the median. Mathematically, this can be represented as:

M - Q1 5(Q3 - M)

Deriving the Coefficient of Skewness

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The coefficient of skewness is one way of measuring skewness. The formula for the coefficient of skewness using quartiles is:

Sk (Q3 - Q1 - 2M) / (Q3 - Q1)

Given the relationship M - Q1 5(Q3 - M), we first need to find an expression for the median (M) in terms of Q1 and Q3:

Expand the right side: M - Q1 5Q3 - 5M Rearrange the equation: M 5M Q1 5Q3 Simplify: 6M Q1 5Q3 Solve for M: M (Q1 5Q3) / 6

Now, substitute M (Q1 5Q3) / 6 into the skewness formula:

Calculate Q3 - Q1 - 2M: Q3 - Q1 - 2(M) Q3 - Q1 - 2((Q1 5Q3) / 6) Q3 - Q1 - (2Q1 10Q3) / 6 (3Q3 - 3Q1 - 2Q1 - 10Q3) / 6 -7Q3 - 5Q1 / 6 (3Q1 - 2Q3 - 7Q3) / 3 Substitute into the skewness formula:

Sk (3Q1 - 2Q3) / (3Q3 - Q1)

Interpreting the Results

The coefficient of skewness, Sk (3Q1 - 2Q3) / (3Q3 - Q1), represents the skewness of the distribution. Since M - Q1 5(Q3 - M) implies M is closer to Q1 than Q3, this indicates a positive skewness. Positive skewness means that the data is skewed to the right, with a longer tail on the right side of the distribution.

Conclusion

The coefficient of skewness is a powerful tool for understanding the shape of a distribution. Given the specific relationship between the median and the quartiles in this problem, we have derived the formula for the coefficient of skewness as Sk (3Q1 - 2Q3) / (3Q3 - Q1). This formula helps in identifying the direction and strength of the skewness in the data.