Q. 3.184 - Elementary Statistics

Question

The Beatles. In the article. "Length of The Beatles' Songs",(Chance Vol. $25$. No. $1$, pp. $30-33$). T. Koyama discusses aspects and interpretations of the lengths of songs by The Beatles. Data on the length, in seconds, of $229$ Beatles' songs are presented on the Weiss Stats site.

Step-by-Step Solution

Answer

(a) The quartiles are  $125, 142, 155$

(b) The interquartile range is, $30$

(c) Five-number summary is, $106, 125, 142, 155, 202$

(d) Potential outliers are, $23, 41, 49, 52, 259, 260, 269, 274, 285, 307, 333, 388, 425, 467, 493$

(e) The required boxplot is given below.

Part (a) Step 1: Given information

We are given that,

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

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

Part (a) Step 2: Simplify

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 $153 s$.

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 $132.5 s$.

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 $181.5 s$

Part (b) Step 1: Given information

We need to find out the interquartile range 

Part (b) Step 2: Simplify

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 $49 s$.

Part (c) Step 1: Given information

We need to find out the five-number summary 

Part (c) Step 2: Explanation

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

Part (d) Step 1: Given information

We need to find out the potential outliers. 

Part (d) Step 2: Simplify

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

Therefore,

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

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 $255$

Part (e) Step 1: Given information

We need to find the boxplot which is given below 

Part(e) Step 2: Simplify

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.