Chapter 2.2, Problem 2.29E

(a)

To determine

To Calculate: the median and explain the meaning of this means in this setting.

Answer to Problem 2.29E

  ${\text{M=Q}}_2=7$

Explanation of Solution

Given:

3 0 12 10 8 4 2 7 10 11 6 7 0 35 12 3 10 2 7 9 15 5 14 15 6

Formula used:

  ${\text{Q}}_2=2\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}$

Calculation:

Order the numbers in ascending order

0 0 2 2 3 3 4 5 6 6 7 7 7 8 9 10 10 10 11 12 12 14 15 15 35

  $\begin{array}{c}{\text{Q}}_2=2\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ =2\times \left(\frac{\text{25+1}}{4}\right)\text{th postion}\\ \text{=13th postion}\\ {\text{M=Q}}_2=7\end{array}$

I

The median or 2^nd Quartile means 50% of the data fall less than a certain value and that percentage value is known as 50^th percentile or median here 7 is the median which implies that 50% of the data fall less than 7 and 50% of the data fall greater than 7.

(b)

To determine

To Calculate: the quartiles and IQR and interpret meaning of IQR in this setting.

Answer to Problem 2.29E

  $\begin{array}{c}{\text{Q}}_1=3.5\\ {\text{Q}}_3=11.5\\ \text{IQR}=8\end{array}$

Explanation of Solution

Given:

0 0 2 2 3 3 4 5 6 6 7 7 7 8 9 10 10 10 11 12 12 14 15 15 35

Formula used:

  $\begin{array}{c}{\text{Q}}_1=1\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ {\text{Q}}_3=3\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ {\text{IQR=Q}}_3-{\text{Q}}_1\end{array}$

Calculation:

  $\begin{array}{c}{\text{Q}}_1=1\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ =1\times \left(\frac{\text{25+1}}{4}\right)\text{th postion}\\ \text{=6}\text{.5}\\ \text{=6th position +0}\text{.5}\left(7th-6th\right)\\ =3+0.5\left(4-3\right)\\ =3.5\end{array}$

  $\begin{array}{c}{\text{Q}}_3=3\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ =3\times \left(\frac{\text{25+1}}{4}\right)\text{th postion}\\ \text{=19}\text{.5}\\ \text{=19th position+0}\text{.5}\left(20th-19th\right)\\ =11+0.5\left(12-11\right)\\ =11.5\\ {\text{Q}}_3=7\end{array}$

  $\begin{array}{c}{\text{IQR=Q}}_3-{\text{Q}}_1\\ =11.5-3.5\\ =8\end{array}$

IQR stands for inter quartile range. It is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile and the quartile. Here 8 data value represent that spread of the middle 50% of the data.

(c)

To determine

To Explain: that is there outlier

Answer to Problem 2.29E

43 and 45 are extremely high outliers.

Explanation of Solution

Given:

43 45 72 73 78 80 81 82 82.5 85 85.5 86 86 87.5 87.5

88 88 88 89 91 91 92 93 93.5 93.5 94.5 94.5 95 96 98

Formula used:

For greater than ${\text{Q}}_3+1.5\left(\text{IQR}\right)$ and

For lesser than ${\text{Q}}_1-1.5\left(\text{IQR}\right)$

Calculation:

  $\begin{array}{c}{\text{Q}}_3+1.5\left(\text{IQR}\right)=11.5+1.5\left(8\right)\\ =23.5\\ {\text{Q}}_1-1.5\left(\text{IQR}\right)=3.5-1.5\left(8\right)\\ =-8.5\end{array}$

An outlier is an extremely high or extremely low value in the data set. It can be identify an outlier if it is greater than ${\text{Q}}_3+1.5\left(\text{IQR}\right)$ and lesser then ${\text{Q}}_1-1.5\left(\text{IQR}\right)$ and there is only one outlier which is extremely high outliers 35.