Example A: Consider two populations in the same state, where both populations are the same size.
Population 1 consists of all students at the state university. Population 2 consists of all residents in a small
town. Consider the variable age. Which population would most likely have the larger standard deviation?
Explain.
If you are considering the variable age, population 2 would most likely have the larger standard
deviation. If you think about a small town, ages range from infants all the way up to those in
their 90’s and 100’s. Therefore, there is going to be a lot of variability present. However, if you
think about a college campus, all the students are roughly the same age. Therefore, there won’t
be as much variability in age as you would see in the small town.
Example B: A test is given to 100 students, and the median score is determined. After grading the test, the
instructor realizes that the 10 students with the highest scores did exceptionally well. The instructor
decides to award these 10 students a bonus of five additional points. How will the median of the new score
distribution change compared to that of the original distribution? Explain.
The median would not change. Once the test scores are ordered, the top 10 scores are just going
to get even bigger, but still remain the top 10 scores. The median is the middle value in the data
set, so the middle value won’t change if you’re making the top 10 scores even larger than they
already are. However, the mean would increase since the mean uses the test scores themselves
in the calculation. Therefore, if the top 10 scores are raised, the mean should also increase.
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Example C: The following histogram shows the distribution of the ages of male Oscar winners.
Which boxplot is graphing the same data as the histogram? Explain.
a.
c.
b.
d.
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Example D: Four histograms are presented below. Each histogram displays the quiz scores on a scale of 0
to 10 for one of four different STAT 110 classes.
Recall, the standard deviation measures the average distance from the mean. Therefore, we
want to look at how the observations are arranged in relationship to the mean. The mean for
each of the plots is roughly the same (or 5).
24. Which of the classes would you expect to have the smallest standard deviation? Explain.
Class 4 would have the smallest standard deviation because the tallest bar (i.e., the most
points) is pretty much at the mean and there are very few values which are furthest from
the mean (near 0 or near 10).
25. Which of the classes would you expect to have the largest standard deviation? Explain.
Class 3 would have the largest standard deviation because the tallest bars (i.e., the most
points) are furthest from the mean of 5 (near 0 or near 10) and the smallest bars (near 5)
are the closest to the mean.
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