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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.RE.14
Chapter 2, Problem 2.RE.14

In Exercises 13 and 14, 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.


The responses of 1019 adults who were asked how much money they think they will spend on Christmas gifts in a recent year (Adapted from Gallup)


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Step 1: Understand the data categories and their frequencies. The table provides the following categories: '\$1000 or more' (306), '\$250–999' (336), 'Less than \$250' (234), 'Not sure' (51), and 'None/do not celebrate Christmas' (92). These represent the responses of 1019 adults.
Step 2: Calculate the mean. To find the mean, assign a representative value to each category. For example, use \$1000 for '\$1000 or more', the midpoint of \$250 and \$999 for '\$250–999', and a value less than \$250 for 'Less than \$250'. Multiply each representative value by its frequency, sum these products, and divide by the total number of responses (1019).
Step 3: Determine the median. To find the median, arrange the data in ascending order based on the categories. Then, identify the middle value. Since there are 1019 responses, the median corresponds to the 510th response. Use cumulative frequencies to locate the category containing the median.
Step 4: Identify the mode. The mode is the category with the highest frequency. From the table, '\$250–999' has the highest frequency (336), so it is the mode.
Step 5: Interpret the results. Discuss whether the mean, median, and mode are appropriate measures of central tendency for this data. Consider the distribution of the data and whether any categories (e.g., 'Not sure' or 'None/do not celebrate Christmas') might affect the interpretation of the center.

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Key Concepts

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

Mean

The mean is the average of a set of values, calculated by summing all the values and dividing by the number of values. In this context, it would involve determining the total estimated spending on Christmas gifts and dividing by the number of respondents. The mean provides a measure of central tendency but can be influenced by extreme values.
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Median

The median is the middle value in a data set when the values are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers. This measure is particularly useful in skewed distributions, as it is less affected by outliers compared to the mean.
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Mode

The mode is the value that appears most frequently in a data set. In this case, it would indicate the most common spending category among the respondents. The mode is useful for understanding the most popular choice but may not always represent the overall trend if the data is multimodal (having multiple modes).
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Related Practice
Textbook Question

In Exercises 37– 40, use the data set, which represents the model 2020 vehicles with the highest fuel economies (in miles per gallon) in the most popular classes. (Source: U.S. Environmental Protection Agency)

36 30 30 45 31 113 113 33 33 33 52 141 56 117 58

118 50 26 23 23 27 48 22 22 22 121 41 105 35 35


Find the five-number summary of the data set.

Textbook Question

In Exercises 5 and 6, use the data set, which represents the number of rooms reserved during one night’s business at a sample of hotels.


153 104 118 166 89 104 100 79 93 96 116

94 140 84 81 96 108 111 87 126 101 111

122 108 126 93 108 87 103 95 129 93 124


Construct a frequency distribution for the data set with six classes and draw a frequency polygon.

Textbook Question

For the four test scores 96, 85, 91, and 86, the first 3 test scores are 20% of the final grade, and the last test score is 40% of the final grade. Find the weighted mean of the test scores.

Textbook Question

In Exercises 37– 40, use the data set, which represents the model 2020 vehicles with the highest fuel economies (in miles per gallon) in the most popular classes. (Source: U.S. Environmental Protection Agency)

36 30 30 45 31 113 113 33 33 33 52 141 56 117 58

118 50 26 23 23 27 48 22 22 22 121 41 105 35 35


About how many vehicles fall on or below the third quartile?

Textbook Question

In Exercises 25 and 26, find the range, mean, variance, and standard deviation of the population data set.


The mileages (in thousands of miles) for a rental car company’s fleet.

4 2 9 12 15 3 6 8 1 4 14 12 3 3

Textbook Question

In Exercises 1 and 2, use the data set, which represents the overall average class sizes for 20 national universities. (Adapted from Public University Honors)

37 34 42 44 39 40 41 51 49 31

52 26 31 40 30 27 36 43 48 35


Construct a relative frequency histogram using the frequency distribution in Exercise 1. Then determine which class has the greatest relative frequency and which has the least relative frequency.

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