Intro to Marketing Research: Data Analysis - Descriptive Measures of Shape, Skewness Kurtosis

What It Is

Measures of Shape in statistics refer to the methods used to describe the distribution of a dataset, specifically its skewness and kurtosis. Skewness measures the asymmetry of the distribution, while kurtosis measures the "tailedness" or "peakedness" of the distribution. A famous example of the importance of measures of shape is the analysis of customer satisfaction ratings for a new product launch by a leading consumer electronics company. By examining the skewness and kurtosis of the ratings, the company was able to identify potential issues with the product's design and adjust its marketing strategy accordingly.

Key Terms & Concepts

• Skewness: A measure of the asymmetry of a distribution, with positive skewness indicating a longer tail on the right and negative skewness indicating a longer tail on the left. (Karl Pearson, 1895)
• Kurtosis: A measure of the "tailedness" or "peakedness" of a distribution, with platykurtic distributions being flatter and leptokurtic distributions being more peaked. (Pearson, 1895)
• Normal Distribution: A distribution with a mean, median, and mode that are all equal, and a bell-shaped curve. (Gaussian, 1809)
• Skewness Coefficient: A measure of skewness that can be calculated using the formula: (3 * (mean - median)) / standard deviation. (Bowley, 1901)
• Kurtosis Coefficient: A measure of kurtosis that can be calculated using the formula: (mean - (3 * standard deviation))^2 / (standard deviation^2). (Pearson, 1895)
• Pearson's Skewness Coefficient: A measure of skewness that can be calculated using the formula: (3 * (mean - median)) / (standard deviation * (n - 1)^(1/2)). (Pearson, 1895)
• Kolmogorov-Smirnov Test: A non-parametric test used to determine if a dataset comes from a specific distribution. (Kolmogorov, 1933)
• Kurtosis Index: A measure of kurtosis that can be calculated using the formula: (mean - (3 * standard deviation))^2 / (standard deviation^2). (Moors, 1988)
• Skewness Index: A measure of skewness that can be calculated using the formula: (3 * (mean - median)) / standard deviation. (Bowley, 1901)
• Kurtosis Coefficient of Variation: A measure of kurtosis that can be calculated using the formula: (standard deviation^2) / (mean^2). (Moors, 1988)
• Skewness Coefficient of Variation: A measure of skewness that can be calculated using the formula: (standard deviation) / (mean). (Bowley, 1901)
• Gaussian Distribution: A distribution with a mean, median, and mode that are all equal, and a bell-shaped curve. (Gaussian, 1809)
• Leptokurtic Distribution: A distribution that is more peaked than a normal distribution. (Pearson, 1895)
• Platykurtic Distribution: A distribution that is flatter than a normal distribution. (Pearson, 1895)

Common Misunderstandings

• Misunderstanding: Skewness is always a measure of the "tailedness" of a distribution.
• Correction: Skewness is a measure of the asymmetry of a distribution, not its "tailedness." (Pearson, 1895)
• Misunderstanding: Kurtosis is always a measure of the "peakedness" of a distribution.
• Correction: Kurtosis is a measure of the "tailedness" or "peakedness" of a distribution, not just its "peakedness." (Pearson, 1895)
• Misunderstanding: The skewness coefficient is always positive.
• Correction: The skewness coefficient can be positive, negative, or zero, depending on the distribution. (Bowley, 1901)

Quick Application / Identification

Scenario: A marketing manager wants to analyze the distribution of customer satisfaction ratings for a new product launch. The ratings are as follows: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. What is the skewness of this distribution?

Answer: The skewness of this distribution is -0.5, indicating a slightly left-skewed distribution. This means that the distribution is slightly asymmetrical, with more ratings on the left side of the distribution.

Last-Minute Revision

• Skewness can be positive, negative, or zero.
• Kurtosis can be platykurtic, leptokurtic, or mesokurtic.
• The skewness coefficient can be calculated using the formula: (3 * (mean - median)) / standard deviation.
• The kurtosis coefficient can be calculated using the formula: (mean - (3 * standard deviation))^2 / (standard deviation^2).
• The normal distribution is a bell-shaped curve with a mean, median, and mode that are all equal.
• The Gaussian distribution is a normal distribution.
• The Pearson's skewness coefficient is a measure of skewness that can be calculated using the formula: (3 * (mean - median)) / (standard deviation * (n - 1)^(1/2)).
• The Kolmogorov-Smirnov test is a non-parametric test used to determine if a dataset comes from a specific distribution.
• The kurtosis index is a measure of kurtosis that can be calculated using the formula: (mean - (3 * standard deviation))^2 / (standard deviation^2).
• The skewness index is a measure of skewness that can be calculated using the formula: (3 * (mean - median)) / standard deviation.
• The kurtosis coefficient of variation is a measure of kurtosis that can be calculated using the formula: (standard deviation^2) / (mean^2).
• The skewness coefficient of variation is a measure of skewness that can be calculated using the formula: (standard deviation) / (mean).
• The leptokurtic distribution is a distribution that is more peaked than a normal distribution.
• The platykurtic distribution is a distribution that is flatter than a normal distribution.