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Chapter 2: Problem 32

Resistance to an outlier Consider the following three sets of observations: Set 1: 8,9,10,11,12 Set 2: 8,9,10,11,100 Set 3: 8,9,10,11,1000 a. Find the median for each data set. b. Find the mean for each data set. c. What do these data sets illustrate about the resistance of the median and mean?

Short Answer

Expert verified
The median is resistant to outliers, while the mean is not.

Step by step solution

01

Find the Median of Set 1

To find the median, first order the numbers from smallest to largest. Set 1 is already ordered as 8, 9, 10, 11, and 12. Since there are 5 numbers (an odd amount), the median is the middle number. Thus, the median for Set 1 is 10.
02

Find the Median of Set 2

Set 2 is also ordered: 8, 9, 10, 11, and 100. Since there are 5 numbers, the median is again the middle number. Therefore, the median for Set 2 is 10.
03

Find the Median of Set 3

Set 3 is ordered as 8, 9, 10, 11, and 1000. The middle number of these 5 is 10, so the median of Set 3 is 10.
04

Find the Mean of Set 1

To find the mean, sum the numbers and divide by the count of numbers. For Set 1, calculate: \[\frac{8 + 9 + 10 + 11 + 12}{5} = \frac{50}{5} = 10.\] So, the mean of Set 1 is 10.
05

Find the Mean of Set 2

Sum the numbers in Set 2 and then divide by the count: \[\frac{8 + 9 + 10 + 11 + 100}{5} = \frac{138}{5} = 27.6.\] Thus, the mean of Set 2 is 27.6.
06

Find the Mean of Set 3

Sum the numbers in Set 3 and divide by the count: \[\frac{8 + 9 + 10 + 11 + 1000}{5} = \frac{1038}{5} = 207.6.\] Therefore, the mean of Set 3 is 207.6.
07

Compare the Resistance of Median and Mean

The results show that the median remains at 10 across all sets, indicating it is resistant to changes from outliers (100 and 1000). In contrast, the mean increases significantly from 10 to 27.6 to 207.6 when the outliers are present, showing it is affected greatly by outliers.

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Key Concepts

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

Median
The median is a measure of central tendency that identifies the middle value in an ordered dataset. It's useful because it divides the data into two equal halves. To find the median, list the numbers in the dataset from smallest to largest.
For example, in Set 1, which contains 5 numbers (8, 9, 10, 11, 12), the median is the third number: 10.
  • This is straightforward when there is an odd number of observations.
  • When there is an even number of observations, the median is the average of the two middle numbers.
The beauty of the median is its robustness when faced with outliers. No matter how large an outlier may be, as long as it doesn't change the ordering, the median remains the same, as illustrated in Sets 2 and 3, which both also have a median of 10.
Mean
The mean, or average, is another type of central tendency measurement. It's calculated by summing all numbers in a set and dividing by the number of observations.
For Set 1 (numbers: 8, 9, 10, 11, 12), the mean is calculated as follows:\[\frac{8 + 9 + 10 + 11 + 12}{5} = 10\]
  • The mean provides a quick idea of the data's center point.
  • It is more sensitive to extreme values, or outliers, than the median.
In Sets 2 and 3, the introduction of outliers (100 in Set 2 and 1000 in Set 3) drastically changes the mean, highlighting its lack of resistance to outliers, unlike the median.
Outliers
Outliers are data points that differ significantly from other observations. They can occur due to variability or errors and can heavily influence statistical measures.
Consider an ordinary dataset with numbers evenly distributed, for instance, like in Set 1. When large numbers like 100 and 1000 replace one observation in Sets 2 and 3, these become outliers.
  • Outliers can skew the results of an analysis, particularly affecting the mean.
  • They can occur naturally, or they may indicate an error or a different sub-group of data.
It's important to check for outliers in statistical data because, as illustrated here, just one can shift data interpretations significantly.
Resistance to Outliers
Resistance to outliers refers to how sensitive a measure is to extreme values. Measures that are resistant do not change much with the presence of outliers.
In the context of the exercise, the median is a resistant measure. Even with the introduction of extreme values such as 100 and 1000, the median remained at 10 in all data sets.
  • This highlights the robustness of the median as a measure of central tendency when dealing with skewed datasets.
  • In contrast, the mean is not resistant. It changed from 10, to 27.6, to 207.6 when outliers were introduced.
Understanding the resistance of different measures helps in choosing the right statistical tools for data analysis, especially in datasets with potential outliers.

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Most popular questions from this chapter

Vacation days \(\quad\) National Geographic Traveler magazine recently presented data on the annual number of vacation days averaged by residents of eight different countries. They reported 42 days for Italy, 37 for France, 35 for Germany, 34 for Brazil, 28 for Britain, 26 for Canada, 25 for Japan, and 13 for the United States. a. Report the median. b. By finding the median of the four values below the median, report the first quartile. c. Find the third quartile. d. Interpret the values found in parts \(a-c\) in the context of these data.

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