Student Study Guide to Accompany Statistics Alive!. Wendy J. Steinberg

skewed distributions: mode < median < mean; negatively skewed distributions: mode > median > mean.

       It is not appropriate to report any single measure of central tendency when you have multimodal data. If reporting the mode, report multiple modes. A graph is the best method for displaying this distribution.

      Computational Exercises

      The following are the ratings of 23 restaurant customers on a measure of satisfaction in a store. The scale ranges from 1 to 10. Use these data for Questions 1 to 7.

       6, 2, 3, 4, 5, 2, 6, 3, 4, 5, 6, 8, 7, 7, 5, 6, 8, 7, 5, 3, 4, 4, 5

      1 What is the mean amount of customer satisfaction? What is the mode amount of customer satisfaction?

      2 What is the median amount of customer satisfaction?

      3 What is the numerical distance of the scores above the mean? What is the numerical distance of the scores below the mean? Which is greater?

      4 Based on the measures of central tendency (use a ballpark estimate for the median), how would you expect the distribution of these scores to appear? (Do not sketch your answer.)

      5 If we were to add a score of 45 to this distribution, what would the new measures of central tendency be? Which measure would be the most affected?

      6 If we were to add another score, 5, to the original distribution, instead of the score of 45, what would the new measures of central tendency be? Which measure would be the most affected?

      7 In Question 4, we added a score of 45 to the distribution. How would we expect the shape of the distribution to change if this score were to be added? How would we expect the shape to look if we were to add another score of 5?The following are data obtained from a measure that assesses positivity. The scale that was used ranged from 1 to 50. M = 29. Use the following data to answer Questions 8 to 11.

      8 What is the modal response?

      9 What is the median response?

      10 How would you expect the distribution to appear, based on the measures of central tendency? (Do not sketch the distribution.)

      11 Which measure of central tendency would best describe the distribution?

      12 In a study looking at the distance people commute to work, the average distance in an area was 25 miles. If the sum of all the distances in the sample was 100, what was n for the study?

      Computational Answers

      1 Mean = 5, mode = 5.

      2 

      3 Sum above = 16. Sum below = −16. The difference is 0. Neither is greater. The mean is the balance point of the distribution.

      4 It would appear normally distributed because the mean, median, and mode are equal.

      5 Mean = 6.67, median = 6.1, mode = 5. The mean has changed the most.

      6 Mean = 5, median = 5, mode = 5. None of them would have changed.

      7 If we were to add a score of 45, the distribution would have a strong positive skew. If we were to add another score of 5, the distribution would still appear normal (symmetrical).

      8 40

      9 

      10 Positively skewed

      11 The median, because it is a skewed distribution.

      12 n = 4

      True/False Questions

      1 A measure of central tendency provides a single value that is considered highly representative of a distribution.

      2 The mean is always the optimal measure of central tendency.

      3 Each measure of central tendency (mean, median, and mode) can be found in the exact center of all distributions.

      4 The mode is a highly stable measure of central tendency.

      5 The numbers of scores falling above and below the median are always equal.

      6 The median is the score that occurs at the 50th percentile rank.

      7 One of the reasons why the mean is considered a strong measure of central tendency is that it includes every score in its computation.

      8 The total distance of the scores below the mean will always be equal to the total distance of the scores above the mean.

      9 In a positively skewed distribution, you can expect all measures of central tendency to be equal.

      10 There is no best single measure of central tendency to describe a bimodal distribution.

      11 Outliers are scores close to the mean.

      12 Multiple outliers of different values will heavily influence the mode.

      13 The median is considered to be a better measure of central tendency in a skewed distribution.

      14 When describing a distribution with bimodal data, you should report both modes.

      15 In a negatively skewed distribution, you can expect that mean > median > mode.

      True/False Answers

      1 True

      2 False

      3 False

      4 False

      5 True

      6 True

      7 True

      8 True

      9 False

      10 True

      11 False

      12 False

      13 True

      14 True

      15 False

      Short-Answer Questions

      1 Why are measures of central tendency useful?

      2 What does it mean that a score is “representative of an entire data set”?

      3 What is the mode? Is the mode a very stable measure of central tendency? Why or why not?

      4 Define the median. How sensitive is the median to changes in the data set?

      5 How is the mean different from the median and mode as a measure of central tendency?

      6 Sometimes the mean is said to be the center of a seesaw. What does this mean?

      7 What about the mean makes it the measure of central tendency that is the most sensitive to change?

      8 What is an outlier? Describe a situation in which an outlier would have a heavy influence on the mean. Which measure of central tendency is most appropriate to use in this situation?

      9 Compare and contrast the locations of the mean, median, and mode in the following distributions: positively skewed, negatively skewed, and symmetrical.

      10 You have just completed a study and discover that you have a skewed distribution. Explain which measure of central tendency is the most appropriate to use with skewed data.

      11 You are concerned that children are watching an excessive amount of violence on TV. You collect a sample of viewing times and notice that the distribution is negatively skewed. If you want to report the largest number as a measure of central tendency so as to make the biggest impact in the news, which measure of central tendency should you report?

      12 A cupcake company has asked you to summarize its customers’ opinions on a new type of cupcake filling. The company has provided an opinion scale with ratings of 1 to 5. If the data are normally distributed, which measure of central tendency should you report?

      13 In looking over the data from the previous study, you notice that the data are symmetrical but leptokurtic. How