4.2 Tests of Skewness

Skewness quantifies the asymmetry of a dataset's probability distribution. It's a crucial measure that indicates whether the data is concentrated on one side or evenly distributed around the mean. Understanding skewness helps in selecting appropriate statistical methods and interpreting results accurately.

A distribution can be:

• Symmetric: The left and right sides of the distribution are mirror images.
• Positively Skewed (Right-Skewed): The right tail of the distribution is longer or fatter than the left tail. The mean is typically greater than the median, which is greater than the mode.
• Negatively Skewed (Left-Skewed): The left tail of the distribution is longer or fatter than the right tail. The mean is typically less than the median, which is less than the mode.

Methods to Test and Measure Skewness

Several methods are available to assess skewness, ranging from visual inspections to numerical calculations.

1. Visual Inspection

Visual analysis using graphical representations is a quick and intuitive way to assess skewness. Common tools include:

• Histograms: Bar charts that show the frequency distribution of data.
• Box Plots (Box-and-Whisker Plots): Displays the distribution of data through its quartiles, highlighting the median and potential outliers.
• Density Plots: Smoothed versions of histograms, showing the probability density function of the data.

Interpretation:

• Positive Skewness: The tail on the right side of the distribution is longer or fatter.
• Negative Skewness: The tail on the left side of the distribution is longer or fatter.
• Symmetric Distribution: Both tails are roughly equal in length, and the data is centered.

Example:

Imagine plotting a histogram of monthly incomes for a group of people. If most people earn around ₹30,000 but a few individuals earn significantly more (e.g., over ₹1,00,000), this will create a long tail extending to the right, indicating positive skewness.

2. Pearson's First Coefficient of Skewness (Moment Coefficient of Skewness)

This is a widely used numerical method that compares the mean and the mode of the distribution.

Formula:

Skewness = (Mean - Mode) / Standard Deviation

Note: Sometimes, the denominator (Standard Deviation) is omitted, and the skewness is simply represented by (Mean - Mode). The interpretation remains the same regarding the direction of skewness.

Interpretation:

• Mean > Mode: The distribution is positively skewed. A larger positive value indicates greater positive skewness.
• Mean < Mode: The distribution is negatively skewed. A larger negative value indicates greater negative skewness.
• Mean = Mode: The distribution is symmetric. A value close to zero indicates a nearly symmetric distribution.

Examples:

• Scenario 1: Mean = 70 Mode = 60 Standard Deviation = 10 Skewness = (70 - 60) / 10 = 10 / 10 = +1 Interpretation: Positive skewness.

• Scenario 2: Mean = 50 Mode = 60 Standard Deviation = 12 Skewness = (50 - 60) / 12 = -10 / 12 ≈ -0.83 Interpretation: Negative skewness.

3. Quartile-Based Skewness (Bowley's Method)

This method is particularly useful when the mode is difficult to determine or when dealing with distributions where the mode might not be clearly defined (e.g., bimodal or multimodal distributions). It uses quartiles and the median.

Formula:

Skewness = (Q3 + Q1 - 2 * Median) / (Q3 - Q1)

Where:

• Q1 is the first quartile (25th percentile).
• Q3 is the third quartile (75th percentile).
• Median (Q2) is the second quartile (50th percentile).

Key Insight:

The formula essentially compares the distance of the median from the first quartile versus the distance of the median from the third quartile.

• Positive Skewness: The distance between the median and Q3 is greater than the distance between Q1 and the median (Q3 - Median > Median - Q1). This leads to a positive skewness value.
• Negative Skewness: The distance between Q1 and the median is greater than the distance between the median and Q3 (Median - Q1 > Q3 - Median). This leads to a negative skewness value.
• Symmetric Distribution: The distances are equal (Q3 - Median = Median - Q1), resulting in a skewness value of zero.

Example:

Given the following quartiles for a dataset:

• Q1 = 20
• Median = 30
• Q3 = 50

Calculate the distances:

• Distance from Median to Q3: Q3 - Median = 50 - 30 = 20
• Distance from Q1 to Median: Median - Q1 = 30 - 20 = 10

Now, apply the formula:

Skewness = (50 + 20 - 2 * 30) / (50 - 20) Skewness = (70 - 60) / 30 Skewness = 10 / 30 ≈ 0.33

Interpretation: Since Q3 - Median (20) is greater than Median - Q1 (10), the distribution is positively skewed. The calculated value of 0.33 confirms this.

Summary

Assessing skewness is a fundamental step in data analysis. The methods discussed—visual inspection, Pearson's coefficients, and quartile-based measures—provide different lenses through which to understand the asymmetry of a data distribution. Recognizing and quantifying skewness helps analysts choose appropriate statistical models, validate assumptions, and derive more accurate conclusions from their data.

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Interview Questions

• What are the common methods to test skewness in a dataset?
• How can you visually identify skewness in data?
• Explain Pearson’s first coefficient of skewness and when it is appropriate to use.
• What is quartile-based skewness (Bowley's Method), and how is it calculated?
• Why might the mode be difficult to use in skewness calculation?
• How do you interpret the skewness value derived from Pearson's coefficient?
• When would you prefer using quartile-based skewness over Pearson's method?
• How does skewness affect the choice of statistical tests?
• Can skewness be zero in real-world data? What does it imply?
• How do outliers influence skewness measurements?