Q 90. - The Practice of Statistics for AP

Question

Cowboys Refer to Exercise $80$

(a) Find and interpret the interquartile range (IQR). 

(b) Determine whether there are any outliers. Show your work.

Step-by-Step Solution

Answer

Part (a) The flier interquartile range is $12$

Part (b) All the items are within the range thus there are no outliers.

Part (a) Step 1: Given information

The table is 

S. No.	Quiz grade
1	285
2	303
3	305
4	306
5	309
6	315
7	318

Part (a) Step 2: Concept

The interquartile range describes the spread of your distribution's middle half. Any distribution that is sorted from low to high is divided into four equal sections by quartiles.

Part (a) Step 3: Calculation

The flier interquartile range (IQR) is calculated as $$\begin{gathered} {Q }_1=\frac{N+1}{4} th item \\ {Q }_1=\frac{7+1}{4} th item \\ {Q }_1=\frac{8}{4} th item \\ {Q }_1=2 th item=303 \\ {Q }_3=\frac{3(N+1)}{4} th item \\ {Q }_3=\frac{3(7+1)}{4} th item \\ {Q }_3=\frac{24}{4} th item \\ {Q }_3=6 th item=315 \\ IQR ={ Q }_3- Q_1 \\ = 315 - 303 \\ = 12 \\ = 12 \end{gathered}$$

Therefore, the flier interquartile range is $12$

Part (b) Step 1: Calculation

We can conclude that there are no outliers in Joe's first $14$ quiz grades in a marking session. If an observation falls more than $1.5$ IQR above or below the First quartile, it is considered an outlier.$$\begin{gathered} IQR \times 1.5 \\ 12 \times 1.5 = 18 \\ {Q}_1- 18 = 303 - 18 \\ = 285 \\ {Q}_3 -18 = 315 + 18 \\ = 333 \end{gathered}$$

There are no outliers because all of the items are within the range. Thus there are no outliers.