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, 84, 138, 163, 322
b) compute the iqr.
79
c) what is the lower fence for this data set?
-34.5
what is the upper fence for this data set?
281.5

Explanation:

Step1: Sort the data

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

Step2: Find the minimum and maximum

The minimum value is $48$ and the maximum value is $322$.

Step3: Find the median (Q2)

Since $n = 18$ (even), the median is the average of the $\frac{n}{2}=9$ - th and $(\frac{n}{2}+ 1)=10$ - th ordered values. So, $Q2=\frac{136 + 140}{2}=138$.

Step4: Find Q1

The lower - half of the data is $48,56,64,66,90,93,97,121,136$. Since $n_1 = 9$ (odd), $Q1$ is the $(\frac{9 + 1}{2})=5$ - th value, so $Q1 = 90$.

Step5: Find Q3

The upper - half of the data is $140,145,160,161,169,208,209,209,322$. Since $n_2=9$ (odd), $Q3$ is the $(\frac{9 + 1}{2}) = 5$ - th value, so $Q3=169$.

Step6: Compute the IQR

$IQR=Q3 - Q1=169 - 90 = 79$.

Step7: Compute the lower fence

Lower fence $=Q1-1.5\times IQR=90-1.5\times79=90 - 118.5=-28.5$.

Step8: Compute the upper fence

Upper fence $=Q3 + 1.5\times IQR=169+1.5\times79=169 + 118.5=287.5$.

Answer:

a) Minimum: $48$, Q1: $90$, Median: $138$, Q3: $169$, Maximum: $322$
b) $79$
c) $-28.5$
d) $287.5$