d Which measure of center is greater? Which box-and-whisker plot shows more variation?
A
aBox-and-Whisker Plot:
B
bBrand A Distribution: Symmetric Brand B Distribution: Skewed left
C
c Brand A, see solution.
D
d See solution.
Practice makes perfect
a We will begin by looking at the table that shows the battery life (in hours) for Brand A.
Brand A
20.75
18.5
13.5
16.25
8.5
13.5
14.5
15.5
11.5
16.75
Now, let's look at the table for Brand B.
Brand B
10.5
12.5
9.5
10.25
9.0
9.75
8.5
8.5
9.0
7.0
We are asked to make a double box-and-whisker plot that represents the given data sets. To do so, we will first find the five number summary for Brand A. Let's write the Brand A data in numerical order.
8.5,11.5, 13.5,13.5,14.5,15.5, 16.25, 16.75,18.5, 20.75
We can see that the least and greatest value are 8.5 and 20.75, respectively. The first quartile is 13.5 and the third quartile is 16.75. The median is the mean of the middle values.
14.5+15.5/2= 15
In the same way, let's first order the data for Brand B.
7.0,8.5, 8.5,9.0,9.0,9.5, 9.75, 10.25,10.5, 12.5
The least and greatest value are 7.0 and 12.5, respectively. The first quartile is 8.5 and the third quartile is 10.25. The median is the mean of the middle values.
9.0+9.5/2= 9.25
Now, let's draw a number line that includes the least and greatest value of each data set. Then we will graph points above the number line for the five-number summary.
Finally, let's draw the box for each plot by using Q_1 and Q_3. Then we will draw a line through the median and the whiskers from the box to the least and greatest values of each data set.
b Note that for Brand A, the length of the whisker and box to the right of the median are about the same as the box and whisker to the left. Therefore, its distribution is symmetric. For Brand B, the whisker and box to the right of the median are larger than those to the right of the median. Therefore, its distribution is skewed left.
c To find which brand's battery lives are more spread out, let's find the range and interquartile range of each data. The range is the difference of the greatest value and the least value.
Brand A's Range: &20.75-8.5=12.25
Brand B's Range: &12.5-7.0=5.5
The interquartile range (IQR) is given by the difference of the third quartile and the first quartile. Let's calculate it for each data set.
Brand A's IQR: &16.75-13.5=3.25
Brand B's IQR: &10.25-8.5=1.75
Note that the range and interquartile range of Brand A are greater than the range and interquartile range of Brand B. Therefore, Brand A's battery lives are more spread out.
d In Part B, we have found that Brand B distribution is skewed left. Therefore, the median best represents the center of Brand B, which is 9.25. Additionally, Brand A's distribution is symmetric. Therefore, the mean best represents the center of Brand A. Let's find the mean for Brand A.
Since Brand A's mean is greater than Brand B's median, the battery life for Brand A is typically longer than the battery life for Brand B. Additionally, from the double box-and-whisker plot, we can see that Brand A is more variable. Therefore, Brand A tends to differ more from one battery to the next.
