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Chapter 1: Problem 107

If a distribution is skewed to the right with no outliers, (a) mean < median. (d) mean > median. (b) mean ? median. (e) We can鈥檛 tell without xamining the data. (c) mean = median.

Short Answer

Expert verified
(d) mean > median.

Step by step solution

01

Understanding Skewness

A distribution is skewed to the right, which means that the tail of the distribution is longer on the right side. In such distributions, the mean is typically greater than the median because the mean is affected by the larger values that form the tail.
02

Identifying the Relationship

In any right-skewed distribution, due to the presence of larger values on the right, the mean is pulled in that direction more so than the median. Thus, the relationship mean > median should hold true for right-skewed distributions.
03

Confirming the Scenario

The problem states there's no presence of outliers, which might affect our decision. Since it confirms the skewed nature and no additional anomalies, we can rely on the typical behavior of a right-skewed distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean vs Median
In statistics, the mean and median are both measures of central tendency, which provide insight into the center point of a data set. However, they respond differently to changes in the data.

  • Mean: Calculated by adding all the data points together and dividing by the number of points. It gives the average value, but can be greatly affected by extreme values or outliers.
  • Median: The middle value when all points are ordered from smallest to largest. If the number of data points is even, the median is the average of the two middle numbers. The median is more robust against outliers and skewness than the mean.
In right-skewed distributions, where data is stretched out more on the higher end, the mean tends to be pulled in the direction of the longer tail, causing it to be higher than the median. This characteristic is underlined by the formulaic nature of the mean, which is more sensitive to the larger values present on the tail of a skewed distribution.
Skewness in Statistics
Skewness refers to the asymmetry in the distribution of data points. This can give us insight into the shape and direction of a data set.

  • Right-Skewed Distribution: If a distribution is right-skewed, it has a longer tail on the right side. This means there are more high-value data points at one end.
  • Left-Skewed Distribution: Conversely, left-skewed distributions have a longer tail on the left side.
  • Symmetrical Distribution: When a distribution is symmetrical, like a bell curve, both sides are mirror images and the mean equals the median.
Understanding skewness helps predict how different measures of central tendency, like the mean and median, will behave. In our exercise, since the distribution is right-skewed, larger data values on the right side pull the mean higher, making it greater than the median.
Impact of Skewness on Measures of Central Tendency
Skewness can significantly affect the measures of central tendency - particularly the mean and median. Let's explore how:

- In skewed distributions, whether to the right or left, the mean is pulled towards the direction of the skew. This is due to its sensitivity to extreme values. Thus, in right-skewed distributions, the mean is higher than the median.
- The median remains a more stable measure as it divides the distribution into two equal halves, rendering it less susceptible to the skewness caused by extremes at either end.

Why does this matter? The measure of central tendency chosen affects how we interpret data. In skewed data, relying solely on the mean could misrepresent the center, as it doesn't reflect the typical value. For interpretations regarding the "typical" data point, the median is often preferred.

Understanding the impact of skewness assists statisticians in data interpretation and choosing the appropriate measure of central tendency for accurate analysis.

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