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Chapter 3: Problem 6

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.

Short Answer

Expert verified
In most quantitative data sets, we can calculate the measures of mean, median and mode. An exception to this is a data set with all unique values, where we cannot determine a mode since no value repeats. Also, an empty data set doesn't have a mean, median, or mode.

Step by step solution

01

Understanding Mean, Median, Mode

In statistical language, the mean is the average, calculated by adding all numbers in a data set and dividing by the amount of numbers in the set. The median is the mid-point value in a data set when the values are arranged in order, while Mode is the number which occurs most frequently in a data set.
02

Example for No Mode

Consider the data set: \[ \] {1, 2, 3, 4, 5}. Every number in this set appears once, so there isn't a mode here since no number appears more than once. This set, however, clearly has a mean (3) and median (3).
03

Example for No Mean and No Median

If you have an empty set, there will be no mean or median, as there are no values to calculate these summary measures. Obviously, there will be no mode for an empty set as well.

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Most popular questions from this chapter

Briefly explain Chebyshev's theorem and its applications.

The following data give the odometer mileage (rounded to the nearest thousand miles) for all 20 cars that are for sale at a dealership. \(\begin{array}{llllllllll}62 & 86 & 58 & 84 & 72 & 40 & 27 & 38 & 50 & 43 \\\ 27 & 40 & 90 & 43 & 94 & 36 & 28 & 48 & 86 & 77\end{array}\) a. Calculate the mean and median. Do these data have a mode? Why or why not? b. Calculate the \(10 \%\) trimmed mean for these data. c. Compute the range, variance, standard deviation, and coefficient of variation for these data.

The prices of all college textbooks follow a bell-shaped distribution with a mean of \(\$ 180\) and a standard deviation of \(\$ 30\). a. Using the empirical rule, find the (approximate) percentage of all college textbooks with their prices between i. \(\$ 150\) and \(\$ 210\) ii. \(\$ 120\) and \(\$ 240\) b. Using the empirical rule, find the interval that contains the prices of (approximate) \(99.7 \%\) of college textbooks.

The following data give the odometer mileage (rounded to the nearest thousand miles) for all 20 cars that are for sale at a dealership. \(\begin{array}{llllllllll}62 & 86 & 58 & 84 & 72 & 40 & 27 & 38 & 50 & 43 \\\ 27 & 40 & 90 & 43 & 94 & 36 & 28 & 48 & 86 & 77\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the number 77 fall in relation to these quartiles? b. Find the approximate value of the 18 th percentile. Give a brief interpretation of this percentile. c. Calculate the percentile rank of 72 . Give a brief interpretation of this percentile rank.

The one-way commuting times from home to work for all employees working at a large company have a bell-shaped curve with a mean of 34 minutes and a standard deviation of 8 minutes. Using the empirical rule, find the approximate percentages of the employees at this company who have one-way commuting times in the following intervals. a. 10 to 58 minutes b. 26 to 42 minutes c. 18 to 50 minutes
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