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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 2.3.31
Chapter 2, Problem 2.3.31

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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Step 1: Interpret the stem-and-leaf plot. The 'stem' represents the tens place, and the 'leaf' represents the ones place. For example, a stem of 3 and a leaf of 0 corresponds to the value 30. List all the data points from the plot: 5, 8, 10, 11, 13, 16, 21, 23, 23, 23, 26, 27, 27, 30, 31, 32, 34, 34, 34, 34, 35, 37, 38, 43, 44, 45, 46, 49, 52.
Step 2: Calculate the mean. Add all the data points together to find the total sum, then divide by the number of data points. Use the formula: \( \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \).
Step 3: Find the median. Arrange the data points in ascending order (already done in Step 1). The median is the middle value if the number of data points is odd, or the average of the two middle values if the number of data points is even. Identify the middle value(s) based on the total number of data points.
Step 4: Determine the mode. The mode is the value(s) that appear most frequently in the data set. Count the frequency of each value and identify the one(s) with the highest frequency.
Step 5: Interpret the results. Discuss whether the mean, median, and mode represent the center of the data effectively. Consider the distribution of the data (e.g., symmetric, skewed) and whether any outliers might affect the measures of central tendency.

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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 measure of central tendency that is sensitive to extreme values, which can skew the result. For example, in a dataset of weights, if one package is significantly heavier than the others, it will raise the mean, potentially misrepresenting the typical weight.
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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. This measure is less affected by outliers and provides a better representation of the center for skewed distributions. For instance, in a dataset of package weights, the median would indicate the weight at which half the packages are lighter and half are heavier.
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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. In the context of package weights, the mode would indicate the weight that occurs most often among the packages.
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Related Practice
Textbook Question

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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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

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?

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 score represents the 65th percentile? How should you interpret this?

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Textbook Question

Determine whether the statement is true or false. If it is false, rewrite it as a true statement.

Class boundaries ensure that consecutive bars of a histogram touch.

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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