Mathematical Design: Median, Mean, Quartiles, and Outliers
1.
Name:________________________
The Statistical Abstract of the United States gives the median size of a home garden as 663 square
feet.
a. Explain the meaning of this statement.
b.
Explain why the median is used instead of the mean.
True or False:
2. If there are only three data values, the median must equal the mean.
3.
The upper quartile is always larger than or equal to the median.
4.
The upper quartile is always larger than or equal to the mean.
5.
For summarizing a distribution of incomes by a single number, which is generally better to use, the
median or the mean? Why?
6.
A data set has a lower extreme = 18, lower quartile = 30, median = 37, upper quartile = 40, mean = 42,
and upper extreme = 70. Using the 1.5xIQR rule, tell whether each of the following observations is an
outlier.
a. 18
b. 24
c. 53
d. 60
e. 70
7.
If a distribution is mound-shaped except for one outlier at the upper extreme, would you expect the
mean to be larger, about the same, or smaller than the median? Explain.
8.
A data set contains five observations. Four of them are 6, 12, 12, and 14. Find the fifth observation so
that the median of all five equals the mean of all five.
9.
The following box plots show the final exam scores in algebra for students using two different
textbooks.
a.
b.
c.
d.
e.
What was the lowest score for a student using Textbook A?
What proportion of students using Textbook A got less than 50%?
Complete this statement: Half of the students using Textbook B got ____ percent or more on the
final exam.
Which textbook gave student scores that varied less?
Which textbook do you think is better? Explain your answer.
10. Box plots of the miles-per-gallon achieved by 3 different cars manufactured by A, B, and C are shown
below.
a.
b.
c.
d.
e.
f.
11.
If we compare manufacturers by looking at just the cars with the very highest miles-per-gallon,
which manufacturer does the best?
If we compare manufacturers by looking at just the highest 25% of all their cars, which
manufacturer does the best?
If we compare manufacturers by looking at just the median miles-per-gallon, which manufacturer
does the best?
Which manufacturer makes cars whose miles-per-gallon varies least?
Suppose you work for manufacturer C and you want to improve your miles-per-gallon compared to A
and B. Should you put extra effort into improving your cars with the most miles-per-gallon,
improving your cars with the fewest miles-per-gallon, or should you spread your extra effort over
all the cars? Explain your answer.
True or False: For manufacturer C, the median is not the center of the box because there are more
models above the median than below it.
Construct a box plot for the following data:
# of children in a family
0
Frequency
3
1
2
2
5
3
8
4
2
Key:
1a) 50% of the gardens are smaller than 663 square feet and 50% are larger.
1b) The data is probably skewed right. In skewed data, the median is a better measure of center than the
mean since it is resistant to outliers, while mean is sensitive to outliers.
2) False
3) True
4) False
5)
The data is skewed right. The median is not affected by extreme values.
6)
(d) and (e) are outliers.
7)
The mean will be larger than the median because the outlier in the upper extreme will pull the mean
towards it.
8)
12 = (6 + 12 + 12 + 14 + x)/5
x = 16
9a)
9b)
9c)
9d)
9e)
30
75%
80
A
B; There is more variation with B, but scores tended to be much higher with B than with A. 75% with B
scored higher than the maximum score with A (excluding the outlier).
10a) A
10b) C
10c) B
10d) B (or without outliers, A also)
10e) Choose improving your cars with the fewest miles-per-gallon. The top 25% of C’s cars are already better
than the top of A or B. But C’s bottom 50% are worse than the bottom 50% of A.
10f) False
11) see calculator