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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

The following data give the total food expenditures (in dollars) for the past one month for a sample of 20 families. \(\begin{array}{rrrrrrrrrr}1125 & 530 & 1234 & 595 & 427 & 872 & 1480 & 699 & 1274 & 1187 \\ 933 & 1127 & 716 & 1065 & 934 & 930 & 1046 & 1199 & 1353 & 441\end{array}\) a. Calculate the mean and median for these data. b. Calculate the \(20 \%\) trimmed mean for these data.

The following data give the number of new cars sold at a dealership during a 20-day period. $$ \begin{array}{rrrrrrrrrr} 8 & 5 & 12 & 3 & 9 & 10 & 6 & 12 & 8 & 8 \\ 4 & 16 & 10 & 11 & 7 & 7 & 3 & 5 & 9 & 11 \end{array} $$ a. Calculate the values of the three quartiles and the interquartile range. Where does the value of 4 lie in relation to these quartiles? b. Find the (approximate) value of the 25 th percentile. Give a brief interpretation of this percentile. c. Find the percentile rank of 10 . Give a brief interpretation of this percentile rank.

The following data give the number of driving citations received during the last three years by 12 drivers. \(\begin{array}{llllllllllll}4 & 8 & 0 & 3 & 11 & 7 & 4 & 14 & 8 & 13 & 7 & 9\end{array}\) a. Find the mean, median, and mode for these data. b. Calculate the range, variance, and standard deviation. c. Are the values of the summary measures in parts a and \(\mathrm{b}\) population parameters or sample statistics?

Melissa's grade in her math class is determined by three 100 -point tests and a 200-point final exam. To determine the grade for a student in this class, the instructor will add the four scores together and divide this sum by 5 to obtain a percentage. This percentage must be at least 80 for a grade of \(\mathrm{B}\). If Melissa's three test scores are 75,69 , and 87 , what is the minimum score she needs on the final exam to obtain a B grade?

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.
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