style="font-size:15px;"> 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 would this affect the mean, median, and mode? (You may want to sketch the curve for help.)
Answers
1 They provide a single score to represent an entire set of numbers. They provide a quick “snapshot” of where the majority of the scores are located.
2 This indicates that the single score allows you to have a better understanding of where the majority of the scores from a data set are located.
3 The mode is the most commonly occurring score. It is not a very stable measure of central tendency because the addition of a few scores can drastically change the location of the mode. For example, a data set may have a mode of 3. However, after adding a few scores with the value of 15, the mode may become 15.
4 The median is the center score, or the score that occurs at the 50th percentile. It is not very sensitive to changes in a data set. In large data sets, a large number of scores would need to be added to heavily influence the median.
5 The mean differs from the median and mode in that it includes every score in its calculation. This makes it far more sensitive to changes in the data set than the other measures of central tendency.
6 This indicates that the total distance from the mean of scores above the mean is equal to the total distance from the mean of scores below the mean. This indicates that if you were to subtract the mean from each of the scores (X − M), the absolute value of the sum of these differences for the scores above the mean will be equal to the absolute value of the sum of these differences for the scores below the mean.
7 The mean includes all of the scores in the distribution in its calculation. If you were to add any score to a data set (other than a score that is the mean itself), the mean would change, whereas this may not be the case with the other measures of central tendency.
8 An outlier is an extreme score, or a score that differs greatly from all other scores. For example, when assessing the batting average of an entire baseball team, a very good player could drastically improve the average, whereas a very poor player could lower the average. The median would be a more appropriate measure of central tendency.
9 In a positively skewed distribution, you can expect the mode to be the smallest value, followed by the median, and the mean would be the largest value. In a negatively skewed distribution, the mean would be the smallest value, followed by the median, and finally the mode. In a symmetrical distribution, all three measures are equal.
10 The median would be the most appropriate because the mean is being pulled in the direction of the tail. The mode “ignores” the fact that the distribution is skewed by reporting the most commonly occurring score. The median, however, falls between these scores and so is the most appropriate.
11 You should report the mode because that will be the largest value.
12 You could report any measure because all three would be equal.
13 This should have no influence on the measures of central tendency because the data are normally distributed.
Multiple-Choice Questions
1 You are interested in using the most stable measure of central tendency. Which measure should you use?MeanMedianModeAll are equally stable.
2 How should you report central tendency when you have a bimodal distribution?Report the meanReport one modeReport the medianReport two modes
3 How many scores should you expect to fall above the mode?Fifty percent of the scoresThe mode should be the center, such that the numerical distance of the values above the mode is equal to the numerical distance of the scores below the mode.This would depend on the skew of the distribution.There is no set number of scores you should expect to fall above the mode.
4 In the following set of scores, what is the median: 9, 6, 6, 8, 4, 1, 10, 11?6.58.597
5 In the following set of scores, what is the median: 12, 4, 1, 2, 1, 6, 8?421.53
6 What is the difference between M and μ?M is for population means, and μ is for sample means.M is for sample means, and μ is for population means.M and μ have the same definition.M is for median, and μ is for mean.
7 A high school football team is having tryouts and asks five participants to throw a football. Here is how far each person threw the ball (in feet): 79, 84, 93, 166, 88. Which score would be considered an outlier?798493166
8 Using the information from Question 7, what would the mean of the distances with the outlier and the mean of the distances without the outlier be, respectively?102 and 8686 and 10286 and 8888 and 86
9 You are collecting data for a study on hospital service. You have obtained data from patients at 30 different hospitals who have completed a satisfaction survey. The measures of central tendency are as follows: mean = 50, median = 50, and mode = 50. What can you tell about all the scores you have collected?There are numerous outliers.The scale ranges from 1 to 50.The scores are distributed in a symmetrical shape.All the participants provided identical information.
The following information is the ice cream sales of an ice cream shop’s nine best-selling flavors in hundreds of scoops. Use this information to answer Questions 10 to 15.
6, 8, 15, 10, 23, 17, 23, 32, 19
10. What is the average number of ice cream scoops?16.5221718.9
11. What is the modal number of ice cream scoops?17271523
12. What is the median number of ice cream scoops?17193423
13. How would you expect this distribution to appear without sketching it?SymmetricalPositively skewedNegatively skewedBimodal
14.
