# What is the interquartile range of the data set? {33, 38, 45, 56, 57, 63, 72, 91}

###### Question:

## Answers

The IQR = 26.

The interquartile range (IQR) is found by subtracting the first quartile from the third quartile (Q3 - Q1 = IQR).

First Quartile (Q1) = 41.5

To find the first quartile, you find the middle value between the minimum value of the set and the set's median. The minimum value of the set is 33, and the median is 56.5 (56 + 57 = 113, 113/2 = 56.5).

Since there are an even number of values in the set, split it in half. Find the median in the lesser half of values: {33, 38, 45, 56}. The two middle values are 38 and 45, which you add together and divide by 2 to get the first quartile. 33 + 38 = 83, 83/2 = 41.5

Third Quartile (Q3) = 67.5

The second half of the data values from when you split the set in half earlier is composed of the numbers {57, 63, 72, 91}. Add 63 and 72 together to get 135 and divide by 2 to find the 3rd Quartile value. 135/2 = 67.5

Interquartile Range: 26

The IQR is found by subtracting Q3 - Q1. You get 67.5 - 41.5 which equals 26. The IQR is 26.

[tex]IQR =Q_3 -Q_1[/tex]

We have for this case n= 8 values. In order to find the Q1 we can select the first 4 values {33, 38, 45, 56 and the Q1 would be:

[tex]Q_1 = \frac{38+45}{2}= 41.5[/tex]

And for the Q3 we can select the last 4 values 57, 63, 72, 91 and for Q3 we got:

[tex]Q_3= \frac{63+72}{2}= 67.5[/tex]

And then the interquartile range would be:

[tex]IQR =Q_3 -Q_1 = 67.5-41.5 = 26[/tex]

Step-by-step explanation:

For this problem we have the following dataset ordered:

{33, 38, 45, 56, 57, 63, 72, 91}

And we want to find the interquartile range defined as:

[tex]IQR =Q_3 -Q_1[/tex]

We have for this case n= 8 values. In order to find the Q1 we can select the first 4 values {33, 38, 45, 56 and the Q1 would be:

[tex]Q_1 = \frac{38+45}{2}= 41.5[/tex]

And for the Q3 we can select the last 4 values 57, 63, 72, 91 and for Q3 we got:

[tex]Q_3= \frac{63+72}{2}= 67.5[/tex]

And then the interquartile range would be:

[tex]IQR =Q_3 -Q_1 = 67.5-41.5 = 26[/tex]