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Chapter 3: Q3 (page 91)

Measures of Center In what sense are the mean, median, mode, and midrange measures of 鈥渃enter鈥�?

Short Answer

Expert verified
  • Mean is the central measure based on all observations.
  • Median is the observation at the center as per the measurements.
  • Mode is the value with the most frequency.
  • Midrange is the central measure based on two extreme observations only.

Step by step solution

01

Mean

Mean is the measure for the central location of the data set. It is obtained using the measure of each observation along with the count of observations.

Thus, mean finds the center of all observations through the measures.

02

Median 

Median determines the observation that divides the entire data set into two halves. Thus, 50% of observations are, respectively, lesser or greater than the median value.

Thus, median finds the central observation of the data set by arranging the observations as per the measurement.

03

Mode

Mode forms the center of all observations as per the corresponding frequencies or number of repetitions in the entire dataset.

Thus, mode is/are the observation(s) with the highest frequency.

04

Midrange

Midrange is mathematically the average of two values; the highest and lowest measuring observations.

Thus, midrange provides a tentative measure of central observation based on two extreme values of the data.

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Most popular questions from this chapter

Critical Thinking. For Exercises 5鈥�20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question

Sales of LP Vinyl Record Albums Listed below are annual U.S. sales of vinyl record albums (millions of units). The numbers of albums sold are listed in chronological order, and the last entry represents the most recent year. Do the measures of center give us any information about a changing trend over time?

0.3 0.6 0.8 1.1 1.1 1.4 1.4 1.5 1.2 1.3 1.4 1.2 0.9 0.9 1 1.9 2.5 2.8 3.9 4.6 6.1

Critical Thinking. For Exercises 5鈥�20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question

Most Expensive Colleges Listed below are the annual costs (dollars) of tuition and fees at the 10 most expensive colleges in the United States for a recent year (based on data from U.S. News & World Report). The colleges listed in order are Columbia, Vassar, Trinity, George Washington, Carnegie Mellon, Wesleyan, Tulane, Bucknell, Oberlin, and Union. What does this 鈥渢op 10鈥� list tell us about those costs for the population of all U.S. college tuitions?

49,138 47,890 47,510 47,343 46,962 46,944 46,930 46,902 46,870 46,785

Mean Absolute Deviation Use the same population of {9 cigarettes, 10 cigarettes, 20 cigarettes} from Exercise 45. Show that when samples of size 2 are randomly selected with replacement, the samples have mean absolute deviations that do not centre about the value of the mean absolute deviation of the population. What does this indicate about a sample mean absolute deviation being used as an estimator of the mean absolute deviation of a population?

Symbols Identify the symbols used for each of the following: (a) sample standard deviation;(b) population standard deviation;(c) sample variance;(d) population variance.

Resistant Measures Here are four of the Verizon data speeds (Mbps) from Figure 3-1: 13.5, 10.2, 21.1, 15.1. Find the mean and median of these four values. Then find the mean and median after including a fifth value of 142, which is an outlier. (One of the Verizon data speeds is 14.2 Mbps, but 142 is used here as an error resulting from an entry with a missing decimal point.) Compare the two sets of results. How much was the mean affected by the inclusion of the outlier? How much is the median affected by the inclusion of the outlier?
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