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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.CR.15a
Chapter 2, Problem 2.CR.15a

Tail lengths (in feet) for a sample of American alligators are listed.
6.5 3.4 4.2 7.1 5.4 6.8 7.5 3.9 4.6


a. Find the mean, median, and mode of the tail lengths. Which best describes a typical American alligator tail length? Explain your reasoning.

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Step 1: To find the mean, calculate the sum of all the tail lengths and divide by the total number of data points. Use the formula: Mean = ∑xn, where ∑x is the sum of the data points and n is the number of data points.
Step 2: To find the median, first arrange the tail lengths in ascending order: 3.4, 3.9, 4.2, 4.6, 5.4, 6.5, 6.8, 7.1, 7.5. Since there are 9 data points (an odd number), the median is the middle value in the ordered list.
Step 3: To find the mode, identify the value(s) that appear most frequently in the dataset. If no value repeats, then the dataset has no mode.
Step 4: Compare the mean, median, and mode to determine which best describes a typical American alligator tail length. Consider whether the data is symmetric or skewed, and whether outliers might affect the mean.
Step 5: Explain your reasoning by discussing the distribution of the data and how each measure of central tendency (mean, median, mode) reflects the dataset. For example, if the data is skewed, the median might be a better representation of a typical tail length than the mean.

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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 numbers, calculated by summing all values and dividing by the count of values. It provides a central value that represents the data set, but can be influenced by extreme values (outliers). In the context of tail lengths, the mean gives a general idea of the average size of alligator tails.
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Median

The median is the middle value in a data set when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers. It is a robust measure of central tendency that is less affected by outliers, making it useful for understanding typical values in skewed distributions.
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Mode

The mode is the value that appears most frequently in a data set. It is particularly useful for categorical data or when identifying the most common value in a numerical set. In the case of alligator tail lengths, the mode can indicate the most typical tail length, providing insight into common characteristics within the sample.
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Related Practice
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

Extending Concepts


Midquartile Another measure of position is called the midquartile. You can find the midquartile of a data set by using the formula below.

Midquartile = (Q₁ + Q₃) / 2

In Exercises 55 and 56, find the midquartile of the data set.


5 7 1 2 3 10 8 7 5 3

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Textbook Question

Graphing Data Sets In Exercises 17–32, organize the data using the indicated type of graph. Describe any patterns.


Engineering Degrees Use a time series chart to display the data shown in the table. The data represent the number of bachelor’s degrees in engineering (in thousands) conferred in the U.S. (Source: U.S. Deapartment of Education)


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Textbook Question

The mean annual salary for a sample of electrical engineers is \$86,500, with a standard deviation of \(1500. The data set has a bell-shaped distribution.


b. The salaries of three randomly selected electrical engineers are \)93,500, \$85,600, and \$82,750. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.

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Textbook Question

You are a member of your local apartment association. The association represents rental housing owners and managers who operate residential rental property throughout the greater metropolitan area. Recently, the association has received several complaints from tenants in a particular area of the city who feel that their monthly rental fees are much higher compared to other parts of the city.

You want to investigate the rental fees. You gather the data shown in the table at the right. Area A represents the area of the city where tenants are unhappy about their monthly rents. The data represent the monthly rents paid by a random sample of tenants in Area A and three other areas of similar size. Assume all the apartments represented are approximately the same size with the same amenities.

a. What type of graph would you choose to display the data? Explain your reasoning.

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Textbook Question

You are a member of your local apartment association. The association represents rental housing owners and managers who operate residential rental property throughout the greater metropolitan area. Recently, the association has received several complaints from tenants in a particular area of the city who feel that their monthly rental fees are much higher compared to other parts of the city.

You want to investigate the rental fees. You gather the data shown in the table at the right. Area A represents the area of the city where tenants are unhappy about their monthly rents. The data represent the monthly rents paid by a random sample of tenants in Area A and three other areas of similar size. Assume all the apartments represented are approximately the same size with the same amenities.

c. Based on your data displays, does it appear that the monthly rents in Area A are higher than the rents in the other areas of the city? Explain.

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