91Ó°ÊÓ

We are given that,

Sorted data is given in the Weiss stats which are as follows,

$$\begin{gathered} 23,41,49,52,66,72,80,91,96,100,105,106,108,109,110,112,114, \\ 116,117,118,119,119,119,121,121,121,121,122,123,123,123,123, \\ 124,124,124,124,124,124,125,125,125,125,125,125,125,127,127, \\ 128,128,129,129,129,130,131,131,132,132,133,133,135,135,135, \\ 135,136,136,136,136,137,137,137,138,138,138,138,138,138, \\ 138,139,140,140,141,142,142,143,143,144,144,144,144,144, \\ 145,145,146,146,147,147,147,147,148,148,149,149,149,149, \\ 150,150,150,151,152,152,152,152,153,153,153,153,153,153, \\ 153,153,155,156,156,157,157,157,158,158,158,158,158,158, \\ 158,158,158,159,160,161,162,162,163,163,163,163,163,164, \\ 164,165,166,166,166,167,167,167,169,170,171,171,172,173, \\ 174,174,175,175,176,177,177,179,179,180,181,181,182,182, \\ 183,183,183,183,183,183,185,187,188,190,190,191,193,194, \\ 194,195,196,201,202,205,207,207,207,210,212,213,213,217, \\ 217,217,227,227,227,230,230,232,236,236,240,241,242,250, \\ 255,259,260,260,269,274,285,307,333,388,425,467,493 \end{gathered}$$

As we know that median is the middle value of a sorted data set. Since the number of data values is odd, the median is:-

$Q_2=153$

So, roughly $50\%$the lengths of Beetles' songs below $153s$.

Now, the first quartile is the median of values below $Q_2$i.e.

$Q_1=\frac{132+133}{2}=132.5$

So, roughly $25\%$the lengths of Beetles' songs below $132.5s$.

Now, the first quartile is the median of values below $Q_2$i.e.

$Q_3=\frac{181+182}{2}=181.5$

So, roughly $75\%$ the lengths of Beetles' songs below$181.5s$

We need to find out the interquartile range

The interquartile range is the difference between $Q_1$and $Q_3$$Q_3$

$IQR=Q_3-Q_1=181.5-132.5=49$

The middle $50\%$the lengths of Beetles' songs vary by$49s$.

We need to find out the five-number summary

The five-number summary is minimum$=23$, first quartile$=132.5$, second quartile$=153$, third quartile$=181.5$ and maximum$=493$.

We need to find out the potential outliers.

An outlier is more than $1.5IQR$or greater than $Q_3$or less than$Q_1$

Therefore,

$$\begin{gathered} Q_3+1.5IQR=181.5+(1.5×49)=255 \\ {Q}_3-1.5IQR=132.5-(1.5×49)=59 \end{gathered}$$

Hence, there are many outliers i.e. $23,41,49,52,259,260,260,269,274,285,307,333,388,425,467,493$because it does not lie between $59$and$255$

We need to find the boxplot which is given below

The whiskers of the boxplot are at a low and high value. The box starts at $Q_1$and ends at $Q_3$and has a straight line at the median.