Question

the lengths (in kilometers) of a random sample of 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 √, 64.5 ×, 138 √, 163 ×, 322 √ b) compute the iqr. 98.5 × c) what is the lower fence for this data set? -83.25 × what is the upper fence for this data set? 310.75 ×

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

The minimum value is $48$.

Step3: Find the first - quartile ($Q_1$)

There are $n = 19$ data points. The position of $Q_1$ is $i=\frac{n + 1}{4}=\frac{19+1}{4}=5$. So $Q_1$ is the 5th value in the sorted list, $Q_1 = 90$.

Step4: Find the median ($Q_2$)

The position of the median is $i=\frac{n + 1}{2}=\frac{19+1}{2}=10$. So the median $Q_2=140$.

Step5: Find the third - quartile ($Q_3$)

The position of $Q_3$ is $i=\frac{3(n + 1)}{4}=\frac{3\times(19 + 1)}{4}=15$. So $Q_3$ is the 15th value in the sorted list, $Q_3=208$.

Step6: Find the maximum

The maximum value is $322$.

Step7: Compute the IQR

$IQR=Q_3 - Q_1=208 - 90 = 118$.

Step8: Compute the lower fence

Lower fence $=Q_1-1.5\times IQR=90-1.5\times118=90 - 177=- 87$.

Step9: Compute the upper fence

Upper fence $=Q_3+1.5\times IQR=208+1.5\times118=208 + 177=385$.

Answer:

a) $[48,90,140,208,322]$
b) $118$
c) $-87$
d) $385$