Textbook Question
Finding the Mean of a Frequency Distribution In Exercises 49–52, approximate the mean of the frequency distribution.
Social Media The average daily amounts of time (in minutes) spent on Snapchat
Ch. 2 - Descriptive Statistics
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 2, Problem 2.3.52
Finding the Mean of a Frequency Distribution In Exercises 49–52, approximate the mean of the frequency distribution.
Populations The populations (in thousands) of the counties in Montana in 2019 (Source: U.S. Census Bureau)
Verified step by step guidance1
Step 1: Identify the midpoints of each population interval. The midpoint is calculated as the average of the lower and upper boundaries of each interval. For example, for the interval 0–19, the midpoint is (0 + 19) / 2 = 9.5.
Step 2: Multiply each midpoint by its corresponding frequency to find the 'weighted total' for each interval. For example, for the interval 0–19, the weighted total is 9.5 * 46.
Step 3: Sum all the weighted totals obtained in Step 2. This gives the total of all midpoints weighted by their frequencies.
Step 4: Sum all the frequencies from the table. This gives the total number of counties.
Step 5: Divide the sum of the weighted totals (from Step 3) by the sum of the frequencies (from Step 4). This quotient is the approximate mean of the frequency distribution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mean of a Frequency Distribution
The mean of a frequency distribution is calculated by multiplying each population midpoint by its corresponding frequency, summing these products, and then dividing by the total number of observations. This provides a weighted average that reflects the distribution of data across different population ranges.
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Midpoint Calculation
The midpoint of a class interval is found by averaging the lower and upper boundaries of the interval. For example, the midpoint for the population range 0-19 is (0 + 19) / 2 = 9.5. This value is essential for calculating the mean, as it represents a single value for each range that can be used in the mean formula.
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Total Frequency
Total frequency is the sum of all frequencies in a frequency distribution. It represents the total number of observations or data points being analyzed. In this case, it is crucial for determining the denominator when calculating the mean, ensuring that the mean reflects the average population across all counties.
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Related Practice
Textbook Question
Finding a Weighted Mean In Exercises 41– 46, find the weighted mean of the data.
Grades A student receives the grades shown below, with an A worth 4 points, a B worth 3 points, a C worth 2 points, and a D worth 1 point. What is the student’s grade point average?
Textbook Question
Constructing a Frequency Distribution and a Frequency Polygon In Exercises 35 and 36, construct a frequency distribution and a frequency polygon for the data set using the indicated number of classes. Describe any patterns.
Declaration of Independence
Number of classes: 5
Data set: Number of children of those who signed the Declaration of Independence (Source: The U.S. National Archives & Records Administration) 5 2 12 18 7 4 10 8 16 3 3 7 3 1 2 7 13 0 8 3 7 5 2 6 0 6 7 9 0 11 9 10 7 8 13 5 8 3 5 0 3 13 3 15 5 6 3 2 5 2 0 3 7 12 4 1
Textbook Question
Graphical Analysis In Exercises 19–22, use the box-and-whisker plot to determine whether the shape of the distribution represented is symmetric, skewed left, skewed right, or none of these. Justify your answer.
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.
Video Durations The lengths (in minutes) of seven educational videos from the Public Broadcasting Service (PBS) (Source: PBS)
83 67 90 55 56 119 52
Textbook Question
Comparing Variation in Different Data Sets In Exercises 45–50, find the coefficient of variation for each of the two data sets. Then compare the results.
Annual Salaries Sample annual salaries (in thousands of dollars) for entry level architects in Denver, CO, and Los Angeles, CA, are listed.