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Chapter 3: Problem 1

Explain how the value of the median is determined for a data set that contains an odd number of observations and for a data set that contains an even number of observations.

Short Answer

Expert verified
For an odd number of data points, the median is the value at position \((n+1)/2\), where n is the number of observations, in the ordered dataset. For an even number of data points, the median is the average of the two middle numbers, at positions \(n/2\) and \(n/2 + 1\) in the ordered dataset.

Step by step solution

01

Determine the Median for Odd Number of Observations

For a dataset with an odd number of observations, first, arrange the data in ascending or descending order. The median is then the value exactly in the middle of this arrangement. If the number of observations is \(n\), the position of the median can be determined using the formula \(\frac{n+1}{2}\). This will tell you the position of the median in the ordered dataset.
02

Determine the Median for Even Number of Observations

If the number of data points is even, after arranging the data in ascending or descending order, there will be two values in the middle. The median is calculated as the average of these two. If the total number of observations is \(n\), the two central positions are \(\frac{n}{2}\) and \(\frac{n}{2} + 1\). To find the median, take these two values from the ordered dataset and calculate their average: \(\frac{value1 + value2}{2}\).

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Most popular questions from this chapter

According to an estimate, Americans were expected to spend an average of 1669 hours watching television in 2004 . Assume that the average time spent watching television by Americans in 2009 has a distribution that is skewed to the right with a mean of 1750 hours and a standard deviation of 450 hours. a. Using Chebyshev's theorem, find at least what percentage of Americans watched television in 2009 for i. 850 to 2650 hours ii. 400 to 3100 hours "b. Using Chebyshev's theorem, find the interval that will contain the television viewing times of at least \(84 \%\) of all Americans.

The following data give the prices of seven textbooks randomly selected from a university bookstore. \(\begin{array}{lllllll}\$ 89 & \$ 170 & \$ 104 & \$ 113 & \$ 56 & \$ 161 & \$ 147\end{array}\) a. Find the mean for these data. Calculate the deviations of the data values from the mean. Is the sum of these deviations zero? b. Calculate the range, variance, and standard deviation.

Nixon Corporation manufactures computer monitors. The following are the numbers of computer monitors produced at the company for a sample of 30 days: \(\begin{array}{llllllllll}24 & 32 & 27 & 23 & 33 & 33 & 29 & 25 & 23 & 28 \\\ 21 & 26 & 31 & 20 & 27 & 33 & 27 & 23 & 28 & 29 \\ 31 & 35 & 34 & 22 & 26 & 28 & 23 & 35 & 31 & 27\end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

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Can the standard deviation have a negative value? Explain.
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