Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

3. Describing Data Numerically

Median

Statistics

3. Describing Data Numerically

Median

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2:55 minutes

Problem 2.T.1a

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

a. Find the mean, median, and mode of the data set. Which best represents the center of the data?

Verified step by step guidance

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:

Mean = \sum x

_in, where

\sum x

_i is the sum of all values and

n

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:

Median = x

₆+x₇2.

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.

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.

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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Textbook Question

In Exercises 37– 40, without performing any calculations, determine which measure of central tendency best represents the graphed data. Explain your reasoning.

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Textbook Question

Protein Powder During a quality assurance check, the actual contents (in grams) of six containers of protein powder were recorded as 1525, 1526, 1502, 1516, 1529, and 1511.

a. Find the mean and the median of the contents.

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.

Textbook Question

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.

Textbook Question

Weekly salaries (in dollars) for a sample of construction workers are listed.

1100 720 1384 1124 1255 976 718 1316

749 1062 1248 891 969 790 860 1100

a. Find the mean, median, and mode of the salaries. Which best describes a typical salary?

Textbook Question

Extending Concepts

Golf The distances (in yards) for nine holes of a golf course are listed.

336 393 408 522 147 504 177 375 360

a. Find the mean and the median of the data.

Textbook Question

Extending Concepts

Golf The distances (in yards) for nine holes of a golf course are listed.

336 393 408 522 147 504 177 375 360

d. Use your results from part (c) to explain how to quickly find the mean and the median of the original data set when the distances are converted to inches.

Textbook Question

Extending Concepts

Data Analysis Students in an experimental psychology class did research on depression as a sign of stress. A test was administered to a sample of 30 students. The scores are shown in the table at the left.

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a. Find the mean and the median of the data.

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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 87a. Find the mean, median, and mode of the data set. Which best represents the center of the data?

Key Concepts

Mean

Median

Mode

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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?