Description
Chamberlain University
MATH225N: Statistical Reasoning for Health Sciences
Week 3: Understanding Measures of Central Tendency
Question
The following data set represents the math test scores for a class of 20 students.
90, 60, 85, 100, 100, 90, 100, 75, 100, 95, 95, 85, 30, 100, 40, 15, 100, 90, 70, 80
Identify the best measure of central tendency for this data set.
- the mode, 100
- the mean, 80
- the median, 95
- the median, 90
- the mean, 85
Answer Explanation
Correct answer:
the median, 90
The ordered data set is as follows: 15, 30, 40, 60, 70, 75, 80, 85, 85, 90, 90, 90, 95, 95, 100, 100, 100, 100, 100, 100. The mean is 80, the median is 90, and the mode is 100. The data set is skewed because the values in the lower half of the data are much more spread out than the values in the upper half. The mean is generally not a good measure of central tendency when there are outliers or the data set is skewed, as is the case here. Here there are 13 data values above the mean compared to 6 data values below the mean. The median, which is the middle value of the ordered data set (or, as in this case, the average of the middle two values) is the best measure of central tendency in these circumstances. The median is 90.
Question
The following data set represents the dollar amounts of donations collected at the entrance to a free museum during one hour.
| Donation Amount ($) | Frequency |
| 1 | 1 |
| 5 | 5 |
| 10 | 3 |
| 15 | 1 |
| 600 | 1 |
Is the median a reasonably good measure of central tendency for this data set? What if the outlier were removed from consideration?
- The median is a good measure regardless of whether the outlier is included.
- The median is a very poor measure regardless of whether the outlier is included.
- The median is a good measure when the outlier is included, but it would be a very poor measure if the outlier were removed from consideration.
- The median is a very poor measure when the outlier is included, but it would be a good measure if the outlier were removed from consideration.
Answer Explanation
Correct answer:
The median is a good measure regardless of whether the outlier is included.
The data value 600 is significantly greater than the other data values in the data set, so 600 is the outlier. The median is the middle value in the ordered data set (or, for an odd number of data values, the average of the middle two data values). When the outlier is included, the middle value is the sixth data value in the ordered data set, which is 5. When the outlier is removed, there are ten data values and the middle two data values are both 5, so the median is 5. In both cases, the number of data values greater than or equal to the median is close to the number of data values less than or equal to the median, so the median is a reasonably good measure of central tendency for the data set.
Question
The following histogram shows the monthly rents reported in a survey of university students. Which of the following would be a reasonable measure of central tendency for this data set? Select all that apply.
A histogram has a horizontal x-axis labeled Rent in dollars from 300 to 1000 in increments of 100 and a vertical y-axis labeled Frequency from 0 to 20 in increments of 2. Vertical bars of width 50 start at the horizontal axis value 400. The heights of the bars are as follows, where the left horizontal axis label is listed first and the height is listed second: 400, 2; 450, 4; 500, 7; 550, 12; 600, 19; 650, 13; 700, 8; 750, 3; 800, 2; 850, 1.
- the mean
- the median
- the mode
Answer Explanation
Correct answer:
the mean
the median
the mode
This data set has no outliers and is roughly symmetric (and is not skewed). As a result, the mean would be a reasonable measure of central tendency. The mode is the range from $600 to $650. 25 of the data values are below this range and 27 of the data values are above this range, so the mode is a reasonable measure of central tendency in this case. The median is also in the range from $600 to $650 and for similar reasons is a reasonable measure of central tendency.
Question
The following data set represents the math test scores for a class of 20 students.
90, 85, 95, 100, 100, 90, 100, 65, 100, 85, 80, 95, 80, 100, 85, 75, 100, 90, 90, 75
Would the mode be a good measure of central tendency for this data set?
- Yes, since this data set has a well-defined, unique mode.
- Yes, since this data set contains no outliers.
- No, since there are many more data values below the mode than above.
- No, since there are many more data values above the mode than below.
- No, since this data set does not have a well-defined, unique mode.
- No, since this data set contains no outliers.
Answer Explanation
Correct answer:
No, since there are many more data values below the mode than above.
The mode is the data value that appears most often. In this case, the mode is 100. Since 100 appears six times in the data set and all other values appear fewer than six times. There are 14 data values below the mode and 0 data values above the mode. Since there are many more data values below the mode than above, the mode would not be a good measure of central tendency.
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