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Chapter 3: Problem 37

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 a dataset without outliers, [2, 4, 6, 8, 10], is 8. When an outlier (100) is added to the dataset, [2, 4, 6, 8, 10, 100], the range dramatically increases to 98. This illustrates that the range, as a measure of spread, is significantly influenced by outliers and thus may not always provide a realistic representation of a dataset.

Step by step solution

01

Understanding the Range and Outliers

The range of a dataset is the difference between the highest and the smallest values. An outlier is an extreme value that is significantly higher or lower than the other values in the dataset.
02

Providing an Example Without Outliers

To illustrate the influence of outliers, let's start by considering a simple dataset without outliers: [2, 4, 6, 8, 10]. The range for this dataset is calculated as 'highest value - smallest value', or \(10 - 2 = 8\).
03

Adding an Outlier to the Dataset

Now, introduce an outlier into the dataset. Let's add the number 100: [2, 4, 6, 8, 10, 100]. Calculating the range again: 'highest value - smallest value', or \(100 - 2 = 98\). The range has now dramatically increased due to the outlier.
04

Comparison

Comparing the calculated ranges with and without the outlier shows the influence that an extreme value can have on the range. The range has increased from 8 to 98 due to the introduction of an outlier. This example illustrates that the range can be significantly affected by outliers, which is a considerable disadvantage when using it as a measure of spread of a dataset.

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