owner of the ice cream shop does not like the shape of the distribution of recent sales. He would like a stronger negative skew. To adjust the distribution so that it is more negatively skewed, where must the owner add scores?Above the meanBelow the meanPrecisely at the meanA few extreme scores above the mean
15. Here is the commute time (in hours) per day of seven people who work at the same company: 4, 1, 0.5, 0.75, 2, 0.25, 1.5 hr. What is the average amount of time it takes these people to commute?2.11.671.431.54
16. Using the information from Question 15, the person with the 4-hr commute begins to work for a different company and is no longer included in the calculation. What is the new mean amount of time it takes for the remaining people to commute?1.451.071.471.0
17. To replace the person with the 4-hr commute, the company hires a new person, who commutes only 0.1 hr to work each day (because he lives very close). What is the new mean amount of time it takes for the workers to commute? (Use information from Questions 15 and 16.)1.171.00.640.87
18. Using the original set of scores (from Question 15), which measure of central tendency would best describe the distribution?MeanMedianModeNone
19. A farmer is interested in finding out which vegetable is preferred by elementary school children. He asks them to select from the following categories: peas, broccoli, carrots, onions, and celery. Which measure of central tendency should he use to summarize his data?MeanMedianModeNone
Multiple-Choice Answers
1 A
2 D
3 D
4 D
5 A
6 B
7 D
8 A
9 C
10 C
11 D
12 A
13 C
14 A
15 C
16 D
17 D
18 B
19 D
Module Quiz
1 Here are the results of a pop quiz given in an American history course: 2, 1, 9, 3, 2, 4, 5, 6, 7, 8, 2, 3, 7, 7, 2, 4, 7, 8, 5. The quiz was graded on a scale of 1 to 10. What are the mean, median, and mode of these scores?
2 Which measure of central tendency would best describe the distribution of the scores given in Question 1?
3 You have just completed teaching a computer science course for the first time. Many of your students did not do well, and the distribution of final grades was positively skewed. You would rather not terrify your students at the start of your next class, so you decide to be a little deceptive with how you report the performance of your previous students. Which measure of central tendency would be the most appropriate?
4 Following the example from Question 3, you only have 10 students enroll in your course the following semester. After the first test, you obtain the following grades (on a 0–70 scale): 45, 64, 57, 34, 65, 49, 44, 58, 58, 62. What is the mean of your class grades?
5 If you were to add one outlier to a symmetrical distribution, what would you expect to happen with regard to the measures of central tendency?
Quiz Answers
1 Mean = 4.84; median = 4.75—that is, ; mode = 2 and 7 (both four occurrences).
2 The distribution is bimodal, so there is no appropriate measure of central tendency, although both 2 and 7 may be reported.
3 You should report the mean because it is the highest value. (And you should make your tests easier!)
4 The mean would be 53.6.
5 The mean would change toward the direction of the outlier, but the mode and median would be relatively unaffected.
Module 6 Range, Variance, and Standard Deviation
Learning Objectives
Understand what dispersion means as it pertains to a set of data
Calculate the range of a set of data
Calculate the variance of a set of data
Understand how a standard deviation is obtained from the variance
Determine when the mean absolute deviation is used
Distinguish between descriptive (N) and inferential (n − 1) formulas for the variance and standard deviation
Module Summary
Dispersion is the extent to which the scores in a distribution are spread out or clustered together. Similar to measures of central tendency, measures of dispersion, or variability, are expressed as single values.
The range is the difference between the highest score and the lowest score in a data set. The range is the simplest measure of dispersion and is very insensitive to changes in the distribution; adding scores to the center of the distribution will not affect the range. The only way the range can be affected is by changing the most extreme scores. Due to these properties, the range is not used often.
Variance (s2 for samples; σ2 for populations) is the average squared distance of each score from the mean. The formula for variance is (depending on your instructor’s preference):
The definition of variance becomes more understandable by reviewing the formula. The numerator states that for every score in the distribution, you should obtain the deviation score—the distance of each score from the mean. This is found by subtracting each score from the mean. All the deviation scores sum to 0 (this is a good way to check your work when finding variances). To proceed with obtaining the variance, you must square each deviation score. This will remove any signs (+ or −) from the deviation scores and allow them to sum to a value other than 0. You then divide the sum of the squared deviation scores by the number of scores in the sample (or 1 fewer) to obtain the variance, or the average squared distance from the mean.
Unfortunately, the average squared distance from the mean is difficult to interpret. The variance tells you the average distance from the mean in area units (which we are unable to interpret) as opposed to linear units (which we normally use). Linear distance is the distance in original units, or regular score points (e.g., the linear distance between 3 and 5 is 2). To revert the variance (in area units) back to linear units, you take the square root of the variance. This result is referred to as the standard deviation. The standard deviation (s for samples, σ for populations) is the average amount a score differs (or deviates) from the mean. The formula for standard deviation is (depending
