Determine The Coefficient Of Skewness Using Pearson's Method

Determining the Coefficient of Skewness Using Pearson's Method: A Comprehensive Guide

Understanding the distribution of your data is crucial in statistical analysis. While measures of central tendency (mean, median, mode) describe the center of a dataset, they don't reveal the shape of the distribution. This is where measures of skewness come into play. Skewness quantifies the asymmetry of a probability distribution. A perfectly symmetrical distribution has a skewness of zero. A distribution with a long tail to the right (positive skewness) indicates a concentration of data points at the lower end, while a distribution with a long tail to the left (negative skewness) indicates a concentration at the higher end.

This article provides a comprehensive guide on how to determine the coefficient of skewness using Pearson's method, one of the most commonly used techniques. We'll explore the concept of skewness, the different types of Pearson's skewness coefficients, and delve into practical examples and interpretations.

What is Skewness?

Skewness is a measure of the asymmetry of a probability distribution. It describes the extent to which a distribution deviates from perfect symmetry. A symmetrical distribution has equal mean, median, and mode, and its skewness is zero. However, real-world data is rarely perfectly symmetrical.

Types of Skewness:

• Positive Skewness (right-skewed): The tail on the right side of the distribution is longer than the left. The mean is typically greater than the median, which is greater than the mode. This often indicates that there are a few extremely high values influencing the mean.

• Negative Skewness (left-skewed): The tail on the left side of the distribution is longer than the right. The mean is typically less than the median, which is less than the mode. This suggests the presence of a few extremely low values impacting the mean.

• Zero Skewness (symmetrical): The distribution is perfectly symmetrical, with the mean, median, and mode being equal.

Pearson's Method for Calculating Skewness

Karl Pearson developed several methods for calculating skewness. The most common are:

• Pearson's first coefficient of skewness: This method uses the mean, median, and standard deviation. It's simple to calculate but sensitive to outliers.

• Pearson's second coefficient of skewness (mode coefficient): This utilizes the mean, mode, and standard deviation. It's effective when the mode is clearly defined but may be less reliable if the mode is difficult to determine accurately.

This article primarily focuses on Pearson's first coefficient of skewness due to its wider applicability and ease of calculation, even in situations with less clearly defined modes.

Formula for Pearson's First Coefficient of Skewness

The formula for Pearson's first coefficient of skewness (Sk) is:

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

Where:

• Mean: The average of the dataset.
• Median: The middle value of the dataset when arranged in ascending order.
• Standard Deviation: A measure of the dispersion or spread of the data around the mean.

Step-by-Step Calculation of Pearson's First Coefficient of Skewness

Let's walk through a step-by-step example to illustrate the calculation. Consider the following dataset representing the test scores of 10 students:

65, 70, 75, 80, 80, 85, 90, 90, 95, 100

Step 1: Calculate the Mean:

Sum of scores = 65 + 70 + 75 + 80 + 80 + 85 + 90 + 90 + 95 + 100 = 830

Mean = Sum of scores / Number of students = 830 / 10 = 83

Step 2: Calculate the Median:

Arrange the scores in ascending order: 65, 70, 75, 80, 80, 85, 90, 90, 95, 100

Since there are 10 scores (an even number), the median is the average of the two middle values: (80 + 85) / 2 = 82.5

Step 3: Calculate the Standard Deviation:

1. Find the deviation of each score from the mean: (65-83)=-18, (70-83)=-13, (75-83)=-8, (80-83)=-3, (80-83)=-3, (85-83)=2, (90-83)=7, (90-83)=7, (95-83)=12, (100-83)=17

2. Square each deviation: 324, 169, 64, 9, 9, 4, 49, 49, 144, 289

3. Sum of squared deviations: 1110

4. Variance = Sum of squared deviations / (Number of students - 1) = 1110 / 9 = 123.33

5. Standard Deviation = √Variance = √123.33 ≈ 11.10

Step 4: Calculate Pearson's First Coefficient of Skewness:

Sk = 3 * (Mean - Median) / Standard Deviation = 3 * (83 - 82.5) / 11.10 ≈ 0.135

Step 5: Interpret the Result:

The coefficient of skewness is approximately 0.135. This indicates a slightly positive skewness. The distribution is not perfectly symmetrical, with a slightly longer tail on the right. This could be because of a few higher scores influencing the mean.

Interpreting the Coefficient of Skewness

The interpretation of Pearson's coefficient of skewness depends on its value:

• Sk = 0: The distribution is perfectly symmetrical.
• Sk > 0: The distribution is positively skewed (right-skewed). The larger the value, the more pronounced the positive skewness.
• Sk < 0: The distribution is negatively skewed (left-skewed). The smaller the value (more negative), the more pronounced the negative skewness.

Note: The interpretation of the magnitude of skewness (how much is "slightly skewed" versus "highly skewed") is often subjective and context-dependent. Generally, values between -0.5 and 0.5 are considered to indicate a relatively symmetrical distribution. Values beyond this range suggest increasing levels of asymmetry.

Advantages and Disadvantages of Pearson's Method

Advantages:

• Simplicity: The formula is straightforward and easy to calculate.
• Wide applicability: It can be used for various types of data.
• Intuitive interpretation: The sign of the coefficient clearly indicates the direction of skewness.

Disadvantages:

• Sensitivity to outliers: Extreme values can significantly influence the mean and standard deviation, potentially distorting the skewness coefficient.
• Dependence on the mean and standard deviation: The method assumes that the data follows a roughly normal distribution or at least has a defined mean and standard deviation.
• Not as robust as other methods: Some alternative methods, like the method based on moments, might offer more robust estimations of skewness in the presence of extreme values or non-normal distributions.

Comparison with Other Skewness Measures

While Pearson's method is widely used, it's not the only approach for calculating skewness. Other methods exist, including:

• Method of Moments: This involves calculating higher-order moments of the distribution to quantify skewness. It is often considered more robust than Pearson's method, particularly for non-normal distributions.

• Quartile-based methods: These utilize quartiles to measure skewness, providing an alternative that is less sensitive to outliers than methods reliant on the mean.

Choosing the most appropriate method depends on the specific characteristics of the data and the research goals. For datasets with potential outliers or non-normal distributions, exploring alternative methods might be beneficial.

Conclusion

Pearson's method provides a simple yet effective way to determine the coefficient of skewness. Understanding the formula, the step-by-step calculation, and the interpretation of the results is essential for correctly analyzing data distribution. While it offers advantages in simplicity and ease of use, remembering its limitations regarding outliers and robustness is crucial. By combining Pearson's method with other analytical tools and potentially exploring alternative skewness measures, data analysts can gain a more comprehensive understanding of their datasets. Always consider the context of your data and the limitations of any statistical measure when interpreting your results. Remember to always visually inspect your data using histograms or box plots to confirm your findings from the skewness calculations. This combination of numerical analysis and visual exploration provides the most robust understanding of your data's distribution.