there are many outliers in a data set.
The skew of a distribution will affect the location of the measures of central tendency. In a symmetrical distribution, the mean, median, and mode are all equal. In a skewed distribution, however, the values of the measures of central tendency are as follows: positively 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
