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Chapter 18: A. Basic Statistics and Applications 235
Continued
12 12 14 15 16 16 16 16 17 17 18 18 18 1 9 19 20 20 20 20
20 20 20 21 21 21 22 22 23 23 23 24 24 25 26 26 28 28 28
a. Find the mean, mode, and median for these data.
b. Prepare the box plot for the data
c. Using the results of (a) and (b) determine if the data are symmetric or skewed.
Examine whether the conclusions made using the two methods about the shape
of the distribution are the same or not.
d. Using the box plot, check whether the data contains any outliers
e. If in part (c) the conclusion is that the data are at least approximately symmetric
then find the standard deviation and determine if the empirical rule holds or not.
Solution:
a. The sample size in this problem is n = 40. Thus, we have
Mean X – = x i /n = 800/40 = 20
Mode = 20
Median = 20.
b. To prepare the box plot we first find the quartiles Q 1 , Q 2 , and Q 3 .
Rank of Q 1 = (25/100)(40 + 1) = 10.25
Rank of Q 2 = (50/100)(40 + 1) = 20.5
Rank of Q 3 = (75/100)(40 + 1) = 30.75
Since the data presented in this problem are already in ascending order, we can
easily see that the quartiles Q 1 , Q 2 , and Q 3 are
Interquartile range
Q 1 = 17 Q 2 = 20 Q 3 = 23
IQR = Q 3 – Q 1 = 23 – 17 = 6
1.5(IQR) = 1.5(6) = 9
The box plot for the data is as shown in Figure 18.23
c. Both parts (a) and (b) lead us to the same conclusion—that the data are symmetric.
d. From the box plot in Figure 18.23 we see that the data do not contain any outliers.
e. In part (c) we conclude that the data are symmetric, we proceed to calculate the
standard deviation, and then determine if the empirical rule holds or not.
Part IV.A.4
( )
40 1 12 28 12 28
= + +
…
40
2
1
+ … +
S 2 2 2
18 1
= . 538
Continued