4.3 Skewness: Pearson's First Coefficient

Pearson's First Coefficient of Skewness, often referred to as the mode skewness, provides a straightforward method to assess the asymmetry of a dataset's distribution. It helps in determining whether a distribution is skewed to the left (negatively skewed), skewed to the right (positively skewed), or is symmetrical.

This measure is particularly valuable in various fields, including business analytics, economics, psychology, education, and natural sciences, for understanding the balance and potential distortion within data distributions.

Formula

Pearson's First Coefficient of Skewness is calculated using the following simple formula:

Skewness = Mean - Mode

This formula quantifies the difference between the mean and the mode, which directly reflects the overall shape and the extent of skewness in the dataset.

Interpretation

The value obtained from the skewness calculation provides direct insight into the distribution's asymmetry:

• Positive Value: Indicates right skewness (positive skew). The distribution has a longer tail extending towards the higher values (to the right).
• Negative Value: Indicates left skewness (negative skew). The distribution has a longer tail extending towards the lower values (to the left).
• Zero: Indicates a perfectly symmetrical distribution. The mean, median, and mode are typically coincident in such cases.

Example

Let's consider a dataset with the following summary statistics:

• Mean: 80
• Mode: 70

Applying Pearson's First Coefficient of Skewness formula:

Skewness = 80 - 70 = +10

Conclusion:

A skewness value of +10 signifies positive skewness. This suggests that the dataset has a tail that is longer on the right side, implying the presence of a few unusually high values that are pulling the mean upwards relative to the mode.

When to Use Pearson's First Coefficient of Skewness

This measure is particularly suitable in the following scenarios:

• Clearly Defined Mode: When the mode of the dataset is distinct and easily identifiable.
• Exploratory Data Analysis (EDA): For quick and intuitive assessments of skewness during the initial stages of data exploration.
• Normality Checks: As a preliminary step to evaluate data normality before applying statistical tests that assume symmetrical distributions.

Summary

Pearson's First Coefficient of Skewness offers an easily understandable and rapid method for detecting asymmetry in data. By examining the difference between the mean and the mode, analysts can effectively gauge the direction of skewness, which is crucial for making informed decisions about appropriate statistical methods and data interpretations.

Key Concepts & Related Terms

• Skewness: A measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
• Mean: The average of a dataset.
• Mode: The value that appears most frequently in a dataset.
• Symmetrical Distribution: A distribution where the left and right sides are mirror images of each other.
• Left-Skewed Distribution (Negative Skew): The tail on the left side of the distribution is longer or fatter than the tail on the right.
• Right-Skewed Distribution (Positive Skew): The tail on the right side of the distribution is longer or fatter than the tail on the left.

Interview Questions

1. What is Pearson's First Coefficient of Skewness and how is it calculated? Pearson's First Coefficient of Skewness, also known as mode skewness, is a measure of asymmetry calculated as the difference between the mean and the mode of a dataset (Skewness = Mean - Mode).

2. How does Pearson's skewness formula indicate the direction of skewness? A positive result indicates right skewness (mean > mode), a negative result indicates left skewness (mean < mode), and a zero result indicates a symmetrical distribution (mean ≈ mode).

3. When would you use Pearson's First Coefficient of Skewness over other skewness measures? It's ideal for quick assessments, especially when the mode is clearly defined and easily identifiable. It's a good starting point for exploratory data analysis.

4. What does a positive skewness value imply about a dataset’s distribution? A positive skewness value implies that the distribution has a longer tail on the right side, meaning there are a few unusually high values that are pulling the mean towards the higher end.

5. Can Pearson's skewness be zero? What does that signify? Yes, a skewness of zero signifies a perfectly symmetrical distribution, where the mean and mode are equal.

6. Why is the mode important in Pearson's skewness calculation? The mode represents the most frequent value in a dataset. Its position relative to the mean indicates the direction in which the data is skewed.

7. What are the limitations of using Pearson's First Coefficient of Skewness? Its primary limitation is its reliance on the mode, which may not be unique or well-defined in all datasets (e.g., bimodal or multimodal distributions, or continuous data with no repeated values). The median-based skewness measure (Pearson's Second Coefficient) is often more robust.

8. How would you interpret a negative skewness value using Pearson’s method? A negative skewness value (Mean < Mode) indicates a left-skewed distribution, where the tail extends towards the lower values, and there are likely a few unusually low values pulling the mean down.

9. Give an example of calculating Pearson’s skewness with sample data. If a dataset has a mean of 50 and a mode of 55, the skewness is 50 - 55 = -5. This indicates a left skew.

10. How does skewness affect assumptions in statistical testing? Many statistical tests (e.g., t-tests, ANOVA, linear regression) assume that the data is normally distributed or that the residuals are symmetrically distributed. Significant skewness can violate these assumptions, potentially leading to inaccurate results or a need for data transformation or non-parametric tests.