Understanding Pearson's 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. In simpler terms, it tells us about the shape of the data distribution – specifically, if it leans to one side or is symmetrical.

A distribution can be:

• Symmetrical: Data is evenly distributed around the mean (like a normal distribution). The skewness is zero.
• Positively Skewed (Right-Skewed): The tail of the distribution is longer on the right side. The mass of the distribution is concentrated on the left. The skewness is positive.
• Negatively Skewed (Left-Skewed): The tail of the distribution is longer on the left side. The mass of the distribution is concentrated on the right. The skewness is negative.

Pearson's Coefficients of Skewness

Pearson developed two coefficients to measure skewness:

1. Pearson's First Coefficient of Skewness (Mode Skewness): This is based on the difference between the mean and the mode.

   Formula: \( Sk_1 = \frac{\text{Mean} - \text{Mode}}{\text{Standard Deviation}} \)

2. Pearson's Second Coefficient of Skewness (Median Skewness): This is based on the difference between the mean and the median, and is often used when the mode is ill-defined or the distribution is moderately skewed.

   Formula: \( Sk_2 = \frac{3(\text{Mean} - \text{Median})}{\text{Standard Deviation}} \)

Both coefficients use the standard deviation to standardize the measure, making it unitless and comparable across different datasets.

Range of Pearson's Coefficient of Skewness

While theoretically, for certain extreme distributions, Pearson's coefficient could fall outside this range, for most practical datasets encountered in statistics, the value of Pearson's coefficient of skewness (both \( Sk_1 \) and \( Sk_2 \)) typically lies between -3 and +3. This range is widely accepted and used in applied statistics to interpret the degree of skewness.

Therefore, the generally accepted range for Pearson's coefficient of skewness is from -3 to 3.

Revision Table: Key Concepts on Skewness

Additional Information on Data Distribution Measures

Skewness is one of several measures used to describe the shape and characteristics of a data distribution. Other important measures include:

• Measures of Central Tendency: These describe the center of the data, such as the mean, median, and mode.
• Measures of Dispersion (Variability): These describe the spread of the data, such as range, variance, and standard deviation.
• Kurtosis: This measures the "tailedness" of the probability distribution, describing how heavily the tails differ from the tails of a normal distribution.

Analyzing skewness along with these other measures provides a comprehensive understanding of the data's characteristics and distribution shape. Pearson's coefficient helps quantify the degree and direction of skewness.