Question

the lengths (in kilometers) of a random sample of 18 rivers on the south island of new zealand in the pacific ocean are listed in table below.
lengths of rivers (in km)
169
93
322
209
160
208
145
121
97
209
64
48
136
140
90
66
56
161
for the data shown above, find the following.
do not round any of your answers.
a) find the 5 - number summary:
(48, 90, 138, 140, 322)
b) compute the iqr.
118
c) what is the lower fence for this data set?
-87
what is the upper fence for this data set?
385

Explanation:

Step1: Arrange data in ascending order

48, 56, 64, 66, 90, 93, 97, 121, 136, 140, 145, 160, 161, 169, 208, 209, 209, 322

Step2: Find minimum value

The minimum value is 48.

Step3: Find first - quartile ($Q_1$)

Since $n = 18$, the position of $Q_1$ is $\frac{n + 1}{4}=\frac{18+1}{4}=4.75$. So $Q_1=0.25\times66 + 0.75\times90=84$.

Step4: Find median ($Q_2$)

The position of the median is $\frac{n+1}{2}=\frac{18 + 1}{2}=9.5$. So $Q_2=\frac{136 + 140}{2}=138$.

Step5: Find third - quartile ($Q_3$)

The position of $Q_3$ is $\frac{3(n + 1)}{4}=\frac{3\times(18 + 1)}{4}=14.25$. So $Q_3=0.75\times161+0.25\times169 = 163$.

Step6: Find maximum value

The maximum value is 322.

Step7: Compute the IQR

$IQR=Q_3 - Q_1=163 - 84 = 79$.

Step8: Compute lower fence

Lower fence $=Q_1-1.5\times IQR=84-1.5\times79=84 - 118.5=-34.5$.

Step9: Compute upper fence

Upper fence $=Q_3 + 1.5\times IQR=163+1.5\times79=163+118.5 = 281.5$.

Answer:

a) [48, 84, 138, 163, 322]
b) 79
c) -34.5
d) 281.5