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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 2.5.57a
Chapter 2, Problem 2.5.57a

Song Lengths Side-by-side box-and-whisker plots can be used to compare two or more different data sets. Each box-and-whisker plot is drawn on the same number line to compare the data sets more easily. The lengths (in seconds) of songs played at two different concerts are shown.

a. Describe the shape of each distribution. Which concert has less variation in song lengths?

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Step 1: Observe the box-and-whisker plots for both concerts. The Punk Rock Concert has a box that is more compact, indicating less variation in song lengths. The Jazz Concert has a wider box and longer whiskers, suggesting greater variation in song lengths.
Step 2: Describe the shape of the distribution for the Punk Rock Concert. The box is relatively small, and the whiskers are short, indicating that the data is tightly clustered around the median. This suggests a more consistent song length.
Step 3: Describe the shape of the distribution for the Jazz Concert. The box is wider, and the whiskers are longer, indicating that the data is more spread out. This suggests a greater range of song lengths.
Step 4: Compare the interquartile ranges (IQR) of both concerts. The IQR for the Punk Rock Concert is from 200 to 220 seconds, which is 20 seconds. The IQR for the Jazz Concert is from 224 to 288 seconds, which is 64 seconds. This confirms that the Jazz Concert has more variation in song lengths.
Step 5: Compare the overall range of song lengths. The range for the Punk Rock Concert is from 177 to 240 seconds, which is 63 seconds. The range for the Jazz Concert is from 200 to 390 seconds, which is 190 seconds. This further supports that the Jazz Concert has greater variation in song lengths.

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Key Concepts

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

Box-and-Whisker Plot

A box-and-whisker plot is a graphical representation of a data set that displays its minimum, first quartile, median, third quartile, and maximum. The 'box' shows the interquartile range (IQR), which contains the middle 50% of the data, while the 'whiskers' extend to the smallest and largest values within 1.5 times the IQR from the quartiles. This visualization helps in comparing distributions and identifying outliers.
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Variation in Data

Variation refers to how spread out or clustered the data points are within a data set. It can be assessed using measures such as range, variance, and standard deviation. In the context of box-and-whisker plots, a smaller interquartile range indicates less variation, meaning the data points are closer to the median, while a larger IQR suggests greater variability among the data points.
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Distribution Shape

The shape of a distribution describes how data points are arranged across the range of values. Common shapes include normal (bell-shaped), skewed (asymmetrical), and uniform (evenly spread). Analyzing the shape helps in understanding the underlying characteristics of the data, such as whether it has outliers or if it is concentrated around certain values, which is crucial for interpreting the results of the concerts' song lengths.
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Related Practice
Textbook Question

Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.


Use the sample statistics from Exercise 29 and assume the number of vehicles in the sample is 75.


a. Estimate the number of vehicles whose speeds are between 63 miles per hour and 71 miles per hour.

Textbook Question

Extending Concepts


Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.


a. Find the 10% trimmed mean for the data in Exercise 65.

Textbook Question

Using and Interpreting Concepts


Using and Interpreting Concepts Finding Quartiles, Interquartile Range, and Outliers In Exercises 11 and 12,

(a) find the quartiles


56 63 51 60 57 60 60 54 63 59 80 63 60 62 65

Textbook Question

use the ogive to approximate

the number in the sample.

1
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Textbook Question

Yoga Classes The data sets at the left show the ages of all participants in two yoga classes.

a. Make a back-to-back stem-and-leaf plot as described in Exercise 41 to display the data.

Textbook Question

U.S. Trade Deficits The table at the left shows the U.S. trade deficits (in billions of dollars) with 18 countries in 2020. (Source: U.S. Department of Commerce)


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a. Find the mean and the median of the trade deficits.