Q 89. - The Practice of Statistics for AP

Question

Refer to Exercise 79 

(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 interquartile range is 14

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

Part (a) Step 1: Given information

Given data : 

S. no.	Quiz grades
1	74
2	75
3	76
4	78
5	80
6	82
7	84
8	86
9	87
10	90
11	91
12	93
13	96
14	98

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 inter-quartile range is calculated as $$\begin{gathered} Q_1=\frac{N+1}{4}th item \\ {Q}_1=\frac{14+1}{4}th item \\ {Q}_1=\frac{15}{4}th item \\ {Q}_1=3.75 th item \\ = \frac{1}{4} x3rd item + \frac{3}{4}x4th item \\ =77.5 \\ Q1 =\frac{1}{4} x76 +\frac{3}{4} x78 \\ =77.5 \\ Q3 = \frac{3(N+1)}{4}th item \\ Q3 = \frac{3(14+1)}{4}th item \\ Q3 = \frac{45}{4}th item \\ Q3 = 11.25 th item =\frac{3}{4} x11th item +\frac{1}{4} x12th item \\ Q3 =\frac{3}{4} x91 + \frac{1}{4}x93 \\ = 91.5 \\ IQR = Q_3- Q_1 \\ = 91.5 - 77.5 \\ = 14 \end{gathered}$$

The interquartile range is $14$

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. $$\begin{gathered} IQR \times 1.5 \\ 14 \times 1.5 = 21 \\ {Q}_1-21=77.5-21 \\ =56.5 \\ {Q}_3-21=91.5+21 \\ =112.5 \end{gathered}$$

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