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Chapter 2: Q37E (page 94)

Explain how the relationship between the mean and median provides information about the symmetry or skewness of the data鈥檚 distribution.

Short Answer

Expert verified

If the mean is to the right, the distribution is skewed to the right.

If the mean is to the left, the distribution is skewed to the left.

Step by step solution

01

Definitions

A mean is a way to find the central tendency value that gives the average around which most of the observations lie.

A median represents that value of observation which lies in the middle of all the observations. 50% of observations lie above the median, and 50% lie below the median.

02

Relationship between the mean and the median

When the mean and the median are the same, it means that the distribution is symmetric.

The mean is influenced by the outliers in the distribution, whereas the median is not. Therefore, where the mean lies around the median determines the skewness of the data.

If the mean is greater than the median, the distribution is skewed to the right, and if the mean is lesser than the median, the distribution is skewed to the left.

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Most popular questions from this chapter

Is honey a cough remedy?Refer to the Archives of Pediatrics and Adolescent Medicine(December 2007) study of honeyas a remedy for coughing, Exercise 2.31 (p. 86). The coughingimprovement scores (as determined by the children鈥檚 parents)for the patients in the over-the-counter cough medicinedosage (DM) group, honey dosage group, and control groupare reproduced in the accompanying table.

Honey Dosage

12, 11, 15, 11, 10, 13, 10, 4 ,15, 16, 9, 14, 10, 6, 10, 8, 11, 12, 12, 8, 12, 9, 11, 15, 10, 15, 9, 13, 8, 12, 10, 8, 9, 5, 12

DM Dosage

4, 6, 9, 4, 7, 7, 7, 9, 12 ,10, 11, 6, 3, 4, 9, 12, 7, 6, 8, 12, 12, 4, 12, 13, 7, 10

No Dosage
(Control)

13, 9, 4, 4, 10, 15, 9, 5, 8, 6, 1, 0, 8, 12, 8, 7, 7, 1, 6, 7, 7, 12, 7, 9, 7, 9, 5, 11, 9, 5, 6, 8, 8, 6, 7, 10, 9, 4, 8, 7, 3, 1, 4, 3

a.Find the standard deviation of the improvement scores for the honey dosage group.

b.Find the standard deviation of the improvement scores for the DM dosage group.

c.Find the standard deviation of the improvement scores for the control group.

d.Based on the results, parts a鈥揷, which group appears to have the most variability in coughing improvement scores? The least variability?

Ranking driving performance of professional golfers.A group of researchers developed a new method forranking the total driving performance of golfers on theProfessional Golf Association (PGA) tour (The Sport Journal,Winter 2007). The method requires knowinga golfer鈥檚 average driving distance (yards) and drivingaccuracy (percent of drives that land in the fairway).The values of these two variables are used to compute adriving performance index. Data for 40 PGA golfers (asranked by the new method for a recent tour year) aresaved in the accompanying file. The first five and last fiveobservations are listed in the table below.

a.Find the mean, median, and mode for the 40 driving performance index values.

b.Interpret each of the measures of central tendency, part a.

c.Use the results, part a,to make a statement about the type of skewness in the distribution of driving performance indexes. Support your statement with a graph.

Answer the following questions about the variability of data sets:

a.What is the primary disadvantage of using the range to compare the variability of data sets?

b.Describe the sample variance using words rather than a formula. Do the same with the population variance.

c.Can the variance of a data set ever be negative? Explain. Can the variance ever be smaller than the standard deviation? Explain.

Voltage sags and swells. Refer to the Electrical Engineering (Vol. 95, 2013) study of transformer voltage sags and swells, Exercise 2.76 (p. 110). Recall that for a sample of 103 transformers built for heavy industry, the mean number of sags per week was 353 and the mean number of swells per week was 184. Assume the standard deviation of the sag distribution is 30 sags per week and the standard deviation of the swell distribution is 25 swells per week. Suppose one of the transformers is randomly selected and found to have 400 sags and 100 swells in a week.

a. Find the z-score for the number of sags for this transformer. Interpret this value.

b. Find the z-score for the number of swells for this transformer. Interpret this value.

Question: For any set of data, what can be said about the percentage of the measurements contained in each of the following intervals?

a.xbar- sto xbar+ s

b.xbar- 2sto xbar+ 2s

c.xbar- 3sto xbar+ 3s
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