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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 2 - Descriptive StatisticsProblem 2.3.28
Chapter 2, Problem 2.3.28

Using and Interpreting Concepts


Finding and Discussing the Mean, Median, and Mode In Exercises 17–34, find the mean, the median, and the mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.


Judicial System The responses of a sample of 34 young adult United Kingdom males in custodial sentences who were asked what is affected by such sentences (Adapted from User Voice)
Mental health: 8
Trust: 3
Education: 8
Personal development: 5
Family: 3
Future opportunities: 3
Other: 4

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1
Step 1: Organize the data into a frequency distribution table. The categories and their respective frequencies are: Mental health (8), Trust (3), Education (8), Personal development (5), Family (3), Future opportunities (3), Other (4).
Step 2: Calculate the mean. To find the mean, sum the products of each category's frequency and its corresponding value, then divide by the total number of responses (34). Use the formula: Mean=∑xfN, where x is the category value, f is the frequency, and N is the total number of responses.
Step 3: Determine the median. Arrange the data in ascending order based on frequency. Since there are 34 responses (an even number), the median is the average of the 17th and 18th values in the ordered dataset. Identify these positions and calculate their average.
Step 4: Identify the mode. The mode is the category with the highest frequency. Look at the frequency distribution table and determine which category appears most frequently.
Step 5: Interpret the results. Discuss whether the mean, median, and mode effectively represent the center of the data. Consider factors such as the distribution of frequencies and whether any measure might be skewed or less representative due to the nature of the data.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mean

The mean, or average, is calculated by summing all the values in a dataset and dividing by the number of values. It provides a central value that represents the dataset as a whole. However, the mean can be sensitive to extreme values (outliers), which may skew the result and misrepresent the data's center.
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Median

The median is the middle value of a dataset when it is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle numbers. The median is a robust measure of central tendency, as it is not affected by outliers, making it useful for skewed distributions.
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Mode

The mode is the value that appears most frequently in a dataset. A dataset may have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values occur with the same frequency. The mode is particularly useful for categorical data where we wish to know which is the most common category.
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Related Practice
Textbook Question

Using Technology to Find Quartiles and Draw Graphs In Exercises 23–26, use technology to draw a box-and-whisker plot that represents the data set.


Hourly Earnings The hourly earnings (in dollars) of a sample of 21 employees at a consulting firm

25.89 27.09 31.76 28.28 26.19 27.43 24.06

25.61 22.56 29.76 18.01 23.66 38.24 37.27

32.70 31.12 25.87 15.06 23.12 30.62 19.85

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Textbook Question

Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.

Credit Card Balance For the month of October, a credit card has a balance of \$115.63 for 12 days, \$637.19 for 6 days, \$1225.06 for 7 days, \$0 for 2 days, and \$34.88 for 4 days. What is the account’s mean daily balance for October?

267
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Textbook Question

Using and Interpreting Concepts


Finding and Discussing the Mean, Median, and Mode In Exercises 17–34, find the mean, the median, and the mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.


College Credits The number of credits being taken by a sample of 14 full-time college students for a semester

12 14 16 15 13 14 15

18 16 16 12 16 15 17

120
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Textbook Question

Building Basic Skills and Vocabulary


True or False? In Exercises 1–4, determine whether the statement is true or false. If it is false, rewrite it as a true statement.


A data set can have the same mean, median, and mode.

103
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Textbook Question

Using Chebychev’s Theorem Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 100, the mean interval between Old Faithful’s eruptions is 101.56 minutes and the standard deviation is 42.69 minutes. Using Chebychev’s Theorem, determine at least how many of the intervals lasted between 16.18 minutes and 186.94 minutes. (Adapted from Geyser Times)

278
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Textbook Question

Using and Interpreting Concepts


Finding and Discussing the Mean, Median, and Mode In Exercises 17–34, find the mean, the median, and the mode of the data, if possible. If any measure cannot be found or does not represent the center of the data, explain why.


Cholesterol The cholesterol levels of a sample of 10 female employees

154 240 171 188 235 203 184 173 181 275

129
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