4.9 Karl Pearson’s Measure of Skewness

Karl Pearson’s Coefficient of Skewness is a widely used statistical method to quantify the asymmetry of a dataset. It leverages measures of central tendency (mean, median, mode) and the standard deviation to provide a numerical indicator of a distribution's shape. This measure helps determine if the data is symmetrically distributed, positively skewed (tail extends to the right), or negatively skewed (tail extends to the left).

Definition

Karl Pearson's coefficient of skewness is defined as the ratio of the difference between the mean and the mode to the standard deviation. Alternatively, it can be calculated using the mean and median when the mode is not readily available or is ill-defined.

Formulas

There are two primary formulas for calculating Karl Pearson’s Coefficient of Skewness:

1. Basic Skewness (using Mean and Mode)

This formula is directly derived from Pearson's initial concept and is most effective when the mode is clearly identifiable.

Skewness = Mean - Mode

2. Karl Pearson’s Coefficient of Skewness (Sk)

This is the standardized version of the skewness measure, allowing for comparisons across different datasets.

Using Mean and Median:

This is the most common form when the mode is not easily determined or is less representative of the distribution's center.

Sk = (3 * (Mean - Median)) / Standard Deviation

Using Mean and Mode:

This formula is used when the mode is well-defined.

Sk = (Mean - Mode) / Standard Deviation

Where:

• Sk: Coefficient of Skewness
• Mean: The average of the dataset.
• Median: The middle value of the dataset when ordered.
• Mode: The most frequent value in the dataset.
• Standard Deviation: A measure of the dispersion or spread of the data points around the mean.

Interpretation of Skewness Values

The value of the coefficient of skewness provides insight into the shape of the data distribution:

• Sk = 0: Indicates a perfectly symmetrical distribution. The mean, median, and mode are all equal.
• Sk > 0: Indicates a positively skewed distribution (right-skewed). The tail of the distribution extends further to the right. The mean is typically greater than the median, which is greater than the mode.
• Sk < 0: Indicates a negatively skewed distribution (left-skewed). The tail of the distribution extends further to the left. The mean is typically less than the median, which is less than the mode.

General Guidelines for Interpretation:

Note: These are general guidelines and can vary depending on the context and field of study.

Example: Step-by-Step Calculation

Let's analyze the following dataset:

Dataset: 85, 88, 92, 94, 96, 98, 100, 100, 100, 100

Step 1: Calculate the Mean

The mean is the sum of all values divided by the number of values.

Mean = (85 + 88 + 92 + 94 + 96 + 98 + 100 + 100 + 100 + 100) / 10
     = 953 / 10
     = 95.3

Step 2: Calculate the Median

Since there are 10 values (an even number), the median is the average of the two middle values (the 5th and 6th values when ordered).

The ordered dataset is: 85, 88, 92, 94, 96, 98, 100, 100, 100, 100

Median = (96 + 98) / 2
       = 194 / 2
       = 97

Step 3: Calculate the Standard Deviation

First, calculate the variance (σ²), which is the average of the squared differences from the mean.

• Calculate the squared difference for each data point: (85-95.3)² = 106.09 (88-95.3)² = 53.29 (92-95.3)² = 10.89 (94-95.3)² = 1.69 (96-95.3)² = 0.49 (98-95.3)² = 7.29 (100-95.3)² = 22.09 (100-95.3)² = 22.09 (100-95.3)² = 22.09 (100-95.3)² = 22.09

• Sum of squared differences: 106.09 + 53.29 + 10.89 + 1.69 + 0.49 + 7.29 + 22.09 + 22.09 + 22.09 + 22.09 = 268.1

• Calculate the variance (using N for population variance or n-1 for sample variance; for simplicity, we'll use N here, but sample standard deviation is more common in practice):

  Variance (σ²) = Σ(xi - mean)² / N
                = 268.1 / 10
                = 26.81

• Calculate the standard deviation (the square root of the variance):

  Standard Deviation (σ) = √26.81
                         ≈ 5.18

Step 4: Find the Mode

The mode is the value that appears most frequently in the dataset.

Mode = 100  (appears 4 times)

Step 5: Apply the Formulas to Calculate Skewness

A. Using Mean and Median:

Sk = (3 * (Mean - Median)) / Standard Deviation
   = (3 * (95.3 - 97)) / 5.18
   = (3 * -1.7) / 5.18
   = -5.1 / 5.18
   ≈ -0.98

B. Using Mean and Mode:

Sk = (Mean - Mode) / Standard Deviation
   = (95.3 - 100) / 5.18
   = -4.7 / 5.18
   ≈ -0.91

Conclusion

Using the provided dataset, we found the following coefficients of skewness:

• Using Mean and Median: Sk ≈ -0.98
• Using Mean and Mode: Sk ≈ -0.91

Both values indicate that the data is negatively skewed. This suggests that the tail of the distribution is longer on the left side. The presence of a few lower values (compared to the bulk of the data) is pulling the distribution towards the left.

Why Use Karl Pearson’s Skewness?

Karl Pearson’s method of skewness is valuable in data analysis for several reasons:

• Simplicity of Calculation: It is relatively easy to compute, especially with readily available statistical software.
• Versatility: It works well for interval and ratio data where measures of central tendency are meaningful.
• Quick Symmetry Check: Provides a rapid assessment of whether a dataset is symmetrically distributed or exhibits bias.
• Data Preprocessing: Understanding skewness is crucial for data preprocessing in statistical modeling, as many algorithms assume normally distributed data. Correcting for skewness can improve model performance.

Related Concepts and Interview Questions

Interview Questions:

1. What is Karl Pearson’s Coefficient of Skewness? It's a measure that quantifies the degree and direction of asymmetry in a probability distribution of a real-valued random variable about its mean.

2. How do you calculate skewness using mean and median? You calculate it by taking three times the difference between the mean and the median, and then dividing that result by the standard deviation: Sk = (3 * (Mean - Median)) / Standard Deviation.

3. When should you use mode vs. median in Pearson’s formula? The formula using the mode (Sk = (Mean - Mode) / Standard Deviation) is preferred when the mode is clearly defined and unimodal. The formula using the median (Sk = (3 * (Mean - Median)) / Standard Deviation) is generally more robust and preferred when the distribution might be bimodal, multimodal, or the mode is difficult to ascertain.

4. How do you interpret a skewness value of -0.98? A value of -0.98 indicates a moderately to highly negatively skewed distribution. The tail of the data distribution is longer on the left side, meaning there are more extreme low values relative to the mean and median.

5. What does a negative Pearson’s skewness indicate? A negative Pearson's skewness indicates that the tail on the left side of the probability density function is longer or fatter than the tail on the right side. In practical terms, this means the distribution has more low values than high values relative to the mean.

6. Why is Karl Pearson’s method of skewness useful in data analysis? It provides a single, interpretable number to describe the asymmetry of a dataset, helping analysts understand the data's shape, identify potential biases, and inform decisions about data transformation or model selection.

7. What are the limitations of using Pearson’s skewness coefficient?

   • It can be sensitive to outliers, especially when using the mean.
   • The formula using the mode is not reliable for multimodal distributions.
   • The interpretation of "moderate" or "high" skewness is subjective and context-dependent.

8. How do outliers affect Karl Pearson’s skewness? Outliers, particularly extreme ones, can significantly influence the mean, and thus the skewness calculation. A few very low outliers can cause negative skewness, while a few very high outliers can cause positive skewness.

9. Give an example of a real-world dataset where Pearson’s skewness is useful.

   • Income Distribution: Income data is often positively skewed, with a few high earners stretching the tail to the right. Pearson's skewness helps quantify this.
   • Reaction Times: In psychology or performance studies, reaction times can sometimes be negatively skewed if there are occasional very slow responses.
   • Test Scores: If a test is too difficult, the scores might be negatively skewed.

10. What’s the difference between Karl Pearson’s and Bowley’s skewness measures? Karl Pearson's measure uses the mean, median (or mode), and standard deviation. Bowley's skewness (also known as the quartile coefficient of skewness) uses quartiles, specifically: BQ = (Q3 + Q1 - 2*Q2) / (Q3 - Q1), where Q1, Q2 (median), and Q3 are the first, second, and third quartiles, respectively. Bowley's measure is less sensitive to extreme values (outliers) because it relies on quartiles rather than the mean or standard deviation.