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Chapter 3: Problem 4

Consider a sample with data values of \(53,55,70,58,64,57,53,69,57,68,\) and \(53 .\) Compute the mean, median, and mode.

Short Answer

Expert verified
The mean is 59.727, the median is 57, and the mode is 53.

Step by step solution

01

Arrange Data in Ascending Order

First, arrange the data values from smallest to largest to make it easier to calculate the median and mode. The ordered data set is: 53, 53, 53, 55, 57, 57, 58, 64, 68, 69, 70.
02

Calculate the Mean

The mean is calculated by adding all the data values together and then dividing by the total number of values. The sum of the data values is:\(53 + 55 + 70 + 58 + 64 + 57 + 53 + 69 + 57 + 68 + 53 = 657\).There are 11 data points. So, the mean is:\(\frac{657}{11} = 59.727\).
03

Find the Median

The median is the middle value of the ordered data set. Since there are 11 data points, the median is the 6th value in the ordered list. In this list, the median is 57.
04

Determine the Mode

The mode is the most frequently occurring value in the data set. In the ordered list, the value 53 appears 3 times, which is more frequent than any other value. Thus, the mode is 53.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
The mean is a measure of central tendency that gives us the average value of a dataset. It is calculated by adding up all data points and dividing the total by the number of observations. This measure helps us understand the overall trend of the data. Here's how it works for our dataset:

To find the mean:
  • Add all the data values: \[53 + 55 + 70 + 58 + 64 + 57 + 53 + 69 + 57 + 68 + 53 = 657\]
  • Count the number of data points: 11
  • Divide the sum of the data by the number of data points: \[\text{Mean} = \frac{657}{11} = 59.727\]
Thus, the mean for this dataset is approximately 59.73.
The mean gives you a sense of what each value would be if all data were evenly distributed.
Median Calculation
The median is another measure of central tendency. It represents the middle value of a dataset when it is ordered from smallest to largest. Finding the median is particularly useful because it is not affected by outliers in the data.

Here's how to find the median in our example:
  • First, arrange the data in ascending order: 53, 53, 53, 55, 57, 57, 58, 64, 68, 69, 70.
  • Count the number of data points: 11
  • Determine the middle position: For an odd number of data points, the median is the value at position \[\frac{11 + 1}{2} = 6.\]
  • The 6th data point in our ordered list is 57
The median, therefore, is 57. It divides the dataset into two equal halves.
Mode Calculation
The mode is the measure of central tendency that identifies the most frequently occurring value in a dataset. This can be particularly helpful in understanding which values are more dominant or repetitive.

In our dataset, determine the mode by following these steps:
  • Look at the ordered dataset: 53, 53, 53, 55, 57, 57, 58, 64, 68, 69, 70
  • Count the frequency of each value
  • The value 53 appears 3 times, more than any other value in the dataset
The mode is 53. It signifies the most common value in our list, giving insight into the data's tendency to repeat certain values.

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Most popular questions from this chapter

Dividend yield is the annual dividend per share a company pays divided by the current market price per share expressed as a percentage. A sample of 10 large companies provided the following dividend yield data (The Wall Street Journal, January 16,2004 ). $$\begin{array}{lclc} \text { Company } & \text { Yield % } & \text { Company } & \text { Yield % } \\\ \text { Altria Group } & 5.0 & \text { General Motors } & 3.7 \\ \text { American Express } & 0.8 & \text { JPMorgan Chase } & 3.5 \\ \text { Caterpillar } & 1.8 & \text { McDonald's } & 1.6 \\ \text { Eastman Kodak } & 1.9 & \text { United Technology } & 1.5 \\ \text { ExxonMobil } & 2.5 & \text { Wal-Mart Stores } & 0.7 \end{array}$$ a. What are the mean and median dividend yields? b. What are the variance and standard deviation? c. Which company provides the highest dividend yield? d. What is the \(z\) -score for McDonald's? Interpret this z-score. e. What is the \(z\) -score for General Motors? Interpret this z-score. f. Based on z-scores, do the data contain any outliers?

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The following frequency distribution shows the price per share of the 30 companies in the Dow Jones Industrial Average (Barron s, February 2,2009 ). $$\begin{array}{cc} \text { Price per } & \text { Number of } \\ \text { Share } & \text { Companies } \\ \$ 0-9 & 4 \\ \$ 10-19 & 5 \\ \$ 20-29 & 7 \\ \$ 30-39 & 3 \\ \$ 40-49 & 4 \\ \$ 50-59 & 4 \\ \$ 60-69 & 0 \\ \$ 70-79 & 2 \\ \$ 80-89 & 0 \\ \$ 90-99 & 1 \end{array}$$ a. Compute the mean price per share and the standard deviation of the price per share for the Dow Jones Industrial Average companies. b. On January \(16,2006,\) the mean price per share was \(\$ 45.83\) and the standard deviation was \(\$ 18.14 .\) Comment on the changes in the price per share over the three-year period.

During the \(2007-2008\) NCAA college basketball season, men's basketball teams attempted an all-time high number of 3 -point shots, averaging 19.07 shots per game (Associated Press Sports, January 24,2009 ). In an attempt to discourage so many 3 -point shots and encourage more inside play, the NCAA rules committee moved the 3 -point line back from 19 feet, 9 inches to 20 feet, 9 inches at the beginning of the \(2008-2009\) basketball season. Shown in the following table are the 3 -point shots taken and the 3 -point shots made for a sample of 19 NCAA basketball games during the \(2008-2009\) season. $$\begin{array}{cccc} \text { 3-Point Shots } & \text { Shots Made } & \text { 3-Point Shots } & \text { Shots Made } \\ 23 & 4 & 17 & 7 \\ 20 & 6 & 19 & 10 \\ 17 & 5 & 22 & 7 \\ 18 & 8 & 25 & 11 \\ 13 & 4 & 15 & 6 \\ 16 & 4 & 10 & 5 \\ 8 & 5 & 11 & 3 \\ 19 & 8 & 25 & 8 \\ 28 & 5 & 23 & 7 \\ 21 & 7 & & \end{array}$$ a. What is the mean number of 3 -point shots taken per game? b. What is the mean number of 3 -point shots made per game? c. Using the closer 3 -point line, players were making \(35.2 \%\) of their shots. What percentage of shots were players making from the new 3 -point line? d. What was the impact of the NCAA rules change that moved the 3 -point line back to 20 feet, 9 inches for the \(2008-2009\) season? Would you agree with the Associated Press Sports article that stated," Moving back the 3 -point line hasn't changed the game dramatically"? Explain.
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