Understanding Pearson's Skewness Coefficients

Skewness is a measure that describes the asymmetry of a probability distribution in statistics. It tells us whether the data is skewed to the left (negative skewness) or to the right (positive skewness) compared to a symmetric distribution like the normal distribution.

Pearson developed two methods to measure the skewness of a data set, known as Pearson's skewness coefficients. These coefficients use common statistical measures like the mean, median, mode, and standard deviation.

Pearson's First Skewness Coefficient

Pearson's first skewness coefficient is based on the difference between the mean and the mode, scaled by the standard deviation. The formula is:

\(\text{Pearson's First Coefficient} = \frac{{Mean - Mode}}{\sigma }\)

Here, \(\sigma\) represents the standard deviation.

Pearson's Second Skewness Coefficient

Pearson's second skewness coefficient is used when the mode is not well-defined or the distribution is moderately skewed. It is based on the relationship between the mean and the median, scaled by the standard deviation. The formula is:

\(\text{Pearson's Second Coefficient} = \frac{{3(Mean - Median)}}{\sigma }\)

Again, \(\sigma\) represents the standard deviation.

Analyzing the Given Options

Let's look at the provided options, which propose different formulas for Pearson's first skewness coefficient:

1. \(\frac{{3(Mean - Median)}}{\sigma }\)
2. \(\frac{{3(Mean - Mode)}}{\sigma }\)
3. \(\frac{{3(Mean + Medium)}}{{\sigma}}\)
4. \(\frac{{3(Mean + S\tan darddeviation)}}{{Mean}}\)

Identifying the Correct Formula

Comparing the options with the standard formulas for Pearson's skewness coefficients:

• Option 1: \(\frac{{3(Mean - Median)}}{\sigma }\). This matches Pearson's second skewness coefficient formula.
• Option 2: \(\frac{{3(Mean - Mode)}}{\sigma }\). This is not a standard Pearson coefficient formula. Pearson's first coefficient uses \((Mean - Mode) / \sigma\).
• Option 3: \(\frac{{3(Mean + Medium)}}{{\sigma}}\). This includes a sum of mean and median, which is not part of Pearson's formulas.
• Option 4: \(\frac{{3(Mean + S\tan darddeviation)}}{{Mean}}\). This includes standard deviation in the numerator and mean in the denominator, which is not a Pearson coefficient formula.

Although the question asks for Pearson's *first* skewness coefficient, the formula presented in option 1, \(\frac{{3(Mean - Median)}}{\sigma }\), is the standard formula for Pearson's *second* skewness coefficient. This formula is commonly used as an alternative measure of skewness based on the median when the mode is less representative. The provided correct option uses this formula.

Revision Table: Skewness Measures

Additional Information: Types of Distributions and Skewness

• Symmetric Distribution: In a perfectly symmetric distribution (like the normal distribution), the mean, median, and mode are all equal. The skewness coefficient is 0.
• Positively Skewed (Right-skewed): The tail of the distribution is longer on the right side. The mean is typically greater than the median, which is greater than the mode (Mean > Median > Mode). Pearson's skewness coefficient is positive.
• Negatively Skewed (Left-skewed): The tail of the distribution is longer on the left side. The mean is typically less than the median, which is less than the mode (Mean < Median < Mode). Pearson's skewness coefficient is negative.

Understanding the relationship between mean, median, and mode helps interpret the direction of skewness in a data set, complementing the calculation of Pearson's skewness coefficients.