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Chapter 3: Problem 28

The range, as a measure of spread, has the disadvantage of being influenced by outliers. Illustrate this with an example.

Short Answer

Expert verified
The range of the initial data set \( \{2, 4, 6, 8, 10\} \) is 8. By introducing an outlier (100), the range increases drastically to 98, illustrating that the range as a measure of spread is greatly influenced by outliers.

Step by step solution

01

Define the Initial Data Set

The initial data set can consist of multiple numbers which are close to each other. Let's define the initial set as \( \{2, 4, 6, 8, 10\} \). The range is calculated as the difference between the highest and lowest values. You get the result by subtracting the lowest value (2) from the highest (10), giving us a range of 8.
02

Add an Outlier to The Data Set

Introduce an outlier into the dataset. An outlier is a value that is substantially different from the other values in a data set. Let's introduce 100 as an outlier, therefore the new set will be \( \{2, 4, 6, 8, 10, 100\} \).
03

Calculate the new Range

Now calculate the range of the new data set, again by subtracting the lowest value (2) from the highest (100), thus resulting in a new range of 98.
04

Highlight the Impact of the Outlier

It is clear that the introduction of the outlier considerably increased the range (from 8 to 98), therefore, illustrating how sensitive the range is to outliers in a data set.

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