Is The Mean Greater Than The Median In Right Skewed

Is the Mean Greater Than the Median in a Right-Skewed Distribution? A Deep Dive

Understanding the relationship between the mean and median is crucial in descriptive statistics, particularly when analyzing the shape of a data distribution. One key concept is skewness, which describes the asymmetry of a probability distribution. This article will delve deep into the relationship between mean, median, and right skewness, providing a comprehensive explanation with illustrative examples.

Understanding Skewness: A Visual Guide

Skewness refers to the extent to which a distribution is asymmetrical. We can broadly categorize distributions into three types:

• Symmetrical: The mean, median, and mode are all equal. The distribution is perfectly balanced around its center. A classic example is the normal distribution.

• Right-Skewed (Positively Skewed): The tail on the right side is longer than the tail on the left. The mean is greater than the median, which is greater than the mode. This indicates a concentration of data points on the lower end, with a few extremely high values pulling the mean upwards.

• Left-Skewed (Negatively Skewed): The tail on the left side is longer than the tail on the right. The mean is less than the median, which is less than the mode. This suggests a concentration of data points on the higher end, with a few extremely low values pulling the mean downwards.

Why is the Mean Greater Than the Median in Right-Skewed Distributions?

The fundamental reason why the mean is typically greater than the median in a right-skewed distribution lies in how these measures are calculated.

• The Median: The median is the middle value when the data is arranged in ascending order. It is resistant to extreme values; outliers have little effect on its value.

• The Mean: The mean is the average of all values. It's highly sensitive to extreme values. Outliers significantly influence the mean, pulling it in the direction of the outliers.

In a right-skewed distribution, the presence of a few extremely high values (outliers) significantly affects the mean. These high values are much farther from the median than the low values. Because the mean considers all values equally, these extreme high values inflate the average, causing the mean to be larger than the median. The median, being resistant to outliers, remains relatively unaffected by these extreme values and thus sits closer to the bulk of the data.

Illustrative Examples: Exploring Right Skewed Data

Let's consider several examples to solidify our understanding:

Example 1: Income Distribution

Income distribution often exhibits right skewness. Most people earn within a certain range, but a small percentage of high earners (CEOs, celebrities, etc.) significantly increase the average income. The median income provides a more representative measure of the "typical" income, while the mean income is skewed upwards by the high earners.

Example 2: House Prices in a City

In a city, you might find many houses priced within a reasonable range, but a few luxury mansions with extremely high prices. The mean house price will be inflated by these high-priced properties, while the median house price would give a more accurate reflection of the typical house price.

Example 3: Test Scores

Imagine a class where most students score around the average, but a few students achieve exceptionally high scores. The mean score would be pulled upwards by these high scores, making it greater than the median score. The median would be a better representation of the typical student performance.

Example 4: Company Sales Data

A company might have consistently moderate sales across most months but experience a few exceptionally high sales months due to promotions or seasonal factors. The mean monthly sales will be higher than the median monthly sales because of the influence of these high sales months.

Mathematical Explanation and Statistical Measures

The difference between the mean and the median can be quantified using statistical measures like skewness coefficients:

• Pearson's moment coefficient of skewness: This coefficient uses the mean, median, and standard deviation to estimate skewness. A positive value indicates right skewness, suggesting the mean is greater than the median.

• Bowley's skewness: This measure is based on quartiles and is less sensitive to extreme outliers than Pearson's coefficient. A positive value indicates right skewness.

• Quantile-based skewness: These measures use quantiles (e.g., percentiles) to characterize the distribution's skewness. Different quantile-based skewness measures exist, each with advantages and disadvantages depending on the specific data properties and goals.

While these coefficients provide a numerical measure of skewness, the visual inspection of a histogram or box plot is still invaluable for understanding the shape of the distribution.

When the Mean Isn't Greater Than the Median in a Right-Skewed Distribution (Exceptions)

While the general rule is that the mean is greater than the median in a right-skewed distribution, there are exceptions. These are rare but possible scenarios:

• Limited Data: With very small datasets, random variations can cause the mean to be less than the median even in a right-skewed distribution. The relationships become more reliable with larger sample sizes.

• Outliers Affecting the Median: In extremely rare cases, a powerful outlier at the lower end of the data could shift the median sufficiently to become greater than the mean in a nominally right-skewed distribution. This is particularly likely when the number of extremely high values is small.

• Complex Distributions: With multimodal or complex distributions, the relationship between mean and median might be less straightforward, leading to unexpected results.

Conclusion: The Importance of Understanding Mean, Median, and Skewness

The relationship between the mean and the median provides crucial information about the shape and characteristics of a data distribution. In a right-skewed distribution, the mean is generally greater than the median due to the influence of extreme values. However, it is essential to remember that while this is a common rule of thumb, exceptions can exist, especially with small datasets or unusually shaped distributions. Understanding skewness and the behavior of the mean and median is fundamental in various fields including finance, economics, healthcare, and engineering, enabling better data interpretation and informed decision-making. Visualizing the data through histograms, box plots, and using relevant statistical measures helps confirm the presence of skewness and its impact on the relationship between mean and median. Always consider the context of the data and the implications of choosing between these measures of central tendency.