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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 data set, the median is the middlemost number when listed in ascending or descending order. For an even number data set, the median is the arithmetic mean of the two middlemost numbers in the ordered set.

Step by step solution

01

Understanding the Median for odd amount of observations

To determine the median for a data set with an odd amount of observations, this set needs to be organized in ascending or descending order. The median is then the value that is exactly in the middle of this ordered data set. This middle value will separate the data set in two halves, having an equal amount of values on each side.
02

Calculating the Median for odd amount of observations

Once the data set is ordered, find out the total number \( n \) of the observations. The position (index) of the median will be \( (n+1) / 2 \). Hence the median is the number at this position in the ordered data set.
03

Understanding the Median for even amount of observations

For a data set with an even amount of observations, there won't be a single middle number. Instead, there will be two middle numbers. According to this, once the data set is ordered, the two middlemost numbers need to be identified.
04

Calculating the Median for even amount of observations

After identifying the two middle numbers at positions \( n / 2 \) and \( (n/2) + 1 \) in the ordered set, the median is calculated as the arithmetic mean of these two numbers, which means it is their sum divided by 2.

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Most popular questions from this chapter

One property of the mean is that if we know the means and sample sizes of two (or more) data sets, we can calculate the combined mean of both (or all) data sets. The combined mean for two data sets is calculated by using the formula $$ \text { Combined mean }=\bar{x}=\frac{n_{1} \bar{x}_{1}+n_{2} \bar{x}_{2}}{n_{1}+n_{2}} $$ where \(n_{1}\) and \(n_{2}\) are the sample sizes of the two data sets and \(\bar{x}_{1}\) and \(\bar{x}_{2}\) are the means of the two data sets, respectively. Suppose a sample of 10 statistics books gave a mean price of \(\$ 140\) and a sample of 8 mathematics books gave a mean price of \(\$ 160\). Find the combined mean. (Hint: For this example: \(n_{1}=10, n_{2}=8, \bar{x}_{1}=\$ 140, \bar{x}_{2}=\$ 160 .\) )

The following data give the annual salaries (in thousand dollars) of 20 randomly selected health care workers. \(\begin{array}{llllllllll}50 & 71 & 57 & 39 & 45 & 64 & 38 & 53 & 35 & 62 \\\ 74 & 40 & 67 & 44 & 77 & 61 & 58 & 55 & 64 & 59\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 57 fall in relation to these quartiles? b. Find the approximate value of the 30 th percentile. Give a brief interpretation of this percentile. c. Calculate the percentile rank of 61 . Give a brief interpretation of this percentile rank.

Is it possible for a (quantitative) data set to have no mean, no median, or no mode? Give an example of a data set for which this summary measure does not exist.

For any data, the sum of all values is equal to the product of the sample size and mean; that is, \(\Sigma x=n \bar{x}\). Suppose the average amount of money spent on shopping by 10 persons during a given week is \(\$ 105.50 .\) Find the total amount of money spent on shopping by these 10 persons.

Briefly describe how the three quartiles are calculated for a data set. Illustrate by calculating the three quartiles for two examples, the first with an odd number of observations and the second with an even number of observations.
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