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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 2.3.6
Chapter 2, Problem 2.3.6

Constructing Data Sets In Exercises 5– 8, construct the described data set. The entries in the data set cannot all be the same.


Mean and median are the same and the data is bimodal.

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Step 1: Understand the requirements of the problem. The data set must satisfy three conditions: (1) the mean and median must be the same, (2) the data set must be bimodal (having two modes), and (3) the entries in the data set cannot all be the same.
Step 2: Begin by selecting a small set of numbers that meet the bimodal condition. For example, choose two numbers that appear most frequently in the data set (these will be the modes). Ensure that the frequencies of these modes are equal.
Step 3: Add additional numbers to the data set to balance the mean and median. The median is the middle value when the data is ordered, and the mean is the average of all values. Arrange the numbers so that the median equals the mean. For example, you can use symmetry around the median to balance the mean.
Step 4: Verify that the data set meets all conditions. Check that the mean and median are equal by calculating both. Confirm that the data set is bimodal by identifying the two modes. Ensure that not all entries are the same.
Step 5: Adjust the data set if necessary. If the mean and median are not equal or the data set is not bimodal, modify the values or frequencies of the numbers while maintaining the required conditions. Recheck all criteria after adjustments.

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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 data set, calculated by summing all the values and dividing by the number of entries. It provides a measure of central tendency, indicating where the center of the data lies. In the context of the question, the mean must be equal to the median, which implies a specific arrangement of values.
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04:52
Calculating the Mean

Median

The median is the middle value of a data set when arranged in ascending order. If there is an even number of entries, the median is the average of the two middle values. For the data set in the question, having the mean equal to the median suggests a symmetric distribution around the center, which is essential for constructing the required data set.
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03:26
Calculating the Median

Bimodal Distribution

A bimodal distribution is characterized by having two distinct modes or peaks in its frequency distribution. This means that there are two values that appear most frequently in the data set. In constructing the data set, it is crucial to ensure that these two modes are present while also maintaining the condition that the mean and median are equal.
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Intro to Frequency Distributions
Related Practice
Textbook Question

Construct a cumulative frequency distribution and an ogive for the data set using six classes. Then describe the location of the greatest increase in frequency.

Retirement Ages

Data set: Retirement ages of 35 English professors 72 62 55 61 53 62 65 66 69 55 66 63 67 69 55 65 67 57 67 68 73 75 65 54 71 57 52 58 58 71 72 67 63 65 61

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Textbook Question

Identifying the Shape of a Distribution In Exercises 53–56, construct a frequency distribution and a frequency histogram for the data set using the indicated number of classes. Describe the shape of the histogram as symmetric, uniform, negatively skewed, positively skewed, or none of these.


Heights of Males

Number of classes: 5

Data set: The heights (to the nearest inch) of 30 males

67 76 69 68 72 68 65 63 75 69

66 72 67 66 69 73 64 62 71 73

68 72 71 65 69 66 74 72 68 69

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Textbook Question

Construct a frequency distribution for the data set using the indicated number of classes. In the table, include the midpoints, relative frequencies, and cumulative frequencies. Which class has the greatest class frequency and which has the least class frequency.

Textbook Spending

Number of classes: 6

Data set: Amounts (in dollars) spent on textbooks for a semester 91 472 279 249 530 376 188 341 266 199 142 273 189 130 489 266 248 101 375 486 190 398 188 269 43 30 127 354 84 319

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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.


Weights (in pounds) of Packages on a Delivery Truck

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Textbook Question

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

Final Grade The scores and their percents of the final grade for a statistics student are shown below. What is the student’s mean score?

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Textbook Question

Interpreting Percentiles In Exercises 29–32, use the ogive, which represents the cumulative frequency distribution for quantitative reasoning scores on the Graduate Record Examination in a recent range of years. (Adapted from Educational Testing Service)

What percentile is a score of 170? How should you interpret this?

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