Textbook Question
Use frequency distribution formulas to estimate the sample mean and the sample standard deviation of the data set in Exercise 2.
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.T.1a
The overall averages of 12 students in a statistics class prior to taking the final exam are listed.
67 72 88 73 99 85 81 87 63 94 68 87
a. Find the mean, median, and mode of the data set. Which best represents the center of the data?
Verified step by step guidance1
Step 1: To find the mean, calculate the sum of all the data points and divide by the total number of data points. Use the formula: , where is the sum of all values and is the number of values.
Step 2: To find the median, first arrange the data in ascending order: 63, 67, 68, 72, 73, 81, 85, 87, 87, 88, 94, 99. Since there are 12 data points (an even number), the median is the average of the 6th and 7th values in the ordered list. Use the formula: .
Step 3: To find the mode, identify the value(s) that appear most frequently in the data set. In this case, check the frequency of each value in the list: 63, 67, 68, 72, 73, 81, 85, 87, 87, 88, 94, 99.
Step 4: Compare the mean, median, and mode to determine which best represents the center of the data. Consider the distribution of the data (e.g., whether it is symmetric, skewed, or has outliers) to decide which measure of central tendency is most appropriate.
Step 5: Conclude which measure (mean, median, or mode) is the best representation of the center of the data based on the context and the characteristics of the data set.
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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 values. It provides a central value that represents the overall data but can be influenced by extreme values (outliers). In the context of the given student scores, calculating the mean will give a quick overview of their performance.
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Median
The median is the middle value of 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 the typical performance of the students.
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Mode
The mode is the value that appears most frequently in a data set. It can be particularly useful in identifying the most common score among the students. In cases where data is categorical or has repeated values, the mode can provide insights into trends that the mean and median may not reveal.
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Related Practice
Textbook Question
The overall averages of 12 students in a statistics class prior to taking the final exam are listed.
67 72 88 73 99 85 81 87 63 94 68 87
d. Display the data in a stem-and-leaf plot. Use one line per stem.
Textbook Question
"According to data from the city of Toronto, Ontario, Canada, there were nearly 112,000 parking infractions in the city for December 2020, with fines totaling over 5,500,000 Canadian dollars. The fines (in Canadian dollars) for a random sample of 105 parking infractions in Toronto, Ontario, Canada, for December 2020 are listed below. (Source: City of Toronto)
In Exercises 1–5, use technology. If possible, print your results.
Draw a histogram for the data. Does the distribution appear to be bell-shaped?"
Textbook Question
According to data from the city of Toronto, Ontario, Canada, there were nearly 112,000 parking infractions in the city for December 2020, with fines totaling over 5,500,000 Canadian dollars. The fines (in Canadian dollars) for a random sample of 105 parking infractions in Toronto, Ontario, Canada, for December 2020 are listed below. (Source: City of Toronto)
In Exercises 1–5, use technology. If possible, print your results.
Find the sample mean of the data.
Textbook Question
The data set represents the number of movies that a sample of 20 people watched in a year.
121 148 94 142 170 88 221 106 18 67
149 28 60 101 134 168 92 154 53 66
c. Display the data using a relative frequency histogram.
Textbook Question
In Exercises 1 and 2, use the data set, which represents the overall average class sizes for 20 national universities. (Adapted from Public University Honors)
37 34 42 44 39 40 41 51 49 31
52 26 31 40 30 27 36 43 48 35
Construct a relative frequency histogram using the frequency distribution in Exercise 1. Then determine which class has the greatest relative frequency and which has the least relative frequency.
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