Find the Interquartile Range (H-Spread) 56 , 80 , 42 , 22 , 31 , 60 , 14 , 7

$56$ , $80$ , $42$ , $22$ , $31$ , $60$ , $14$ , $7$

Step 1

There are $8$ observations, so the median is the mean of the two middle numbers of the arranged set of data. Splitting the observations either side of the median gives two groups of observations. The median of the lower half of data is the lower or first quartile. The median of the upper half of data is the upper or third quartile.

The median of the lower half of data is the lower or first quartile

The median of the upper half of data is the upper or third quartile

Step 2

Arrange the terms in ascending order.

$7, 14, 22, 31, 42, 56, 60, 80$

Step 3

Find the median of $7, 14, 22, 31, 42, 56, 60, 80$ .

The median is the middle term in the arranged data set. In the case of an even number of terms, the median is the average of the two middle terms.

$\frac{31 + 42}{2}$

Remove parentheses.

$\frac{31 + 42}{2}$

Add $31$ and $42$ .

$\frac{73}{2}$

Convert the median $\frac{73}{2}$ to decimal.

$36.5$

Step 4

The lower half of data is the set below the median.

$7, 14, 22, 31$

Step 5

The median for the lower half of data $7, 14, 22, 31$ is the lower or first quartile. In this case, the first quartile is $18$ .

The median is the middle term in the arranged data set. In the case of an even number of terms, the median is the average of the two middle terms.

$\frac{14 + 22}{2}$

Remove parentheses.

$\frac{14 + 22}{2}$

Cancel the common factor of $14 + 22$ and $2$ .

Factor $2$ out of $14$ .

$\frac{2 \cdot 7 + 22}{2}$

Factor $2$ out of $22$ .

$\frac{2 \cdot 7 + 2 \cdot 11}{2}$

Factor $2$ out of $2 \cdot 7 + 2 \cdot 11$ .

$\frac{2 \cdot (7 + 11)}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 \cdot (7 + 11)}{2 (1)}$

Cancel the common factor.

$\frac{2 \cdot (7 + 11)}{2 \cdot 1}$

Rewrite the expression.

$\frac{7 + 11}{1}$

Divide $7 + 11$ by $1$ .

$7 + 11$

Add $7$ and $11$ .

$18$

Convert the median $18$ to decimal.

$18$

Step 6

The upper half of data is the set above the median.

$42, 56, 60, 80$

Step 7

The median for the upper half of data $42, 56, 60, 80$ is the upper or third quartile. In this case, the third quartile is $58$ .

The median is the middle term in the arranged data set. In the case of an even number of terms, the median is the average of the two middle terms.

$\frac{56 + 60}{2}$

Remove parentheses.

$\frac{56 + 60}{2}$

Cancel the common factor of $56 + 60$ and $2$ .

Factor $2$ out of $56$ .

$\frac{2 \cdot 28 + 60}{2}$

Factor $2$ out of $60$ .

$\frac{2 \cdot 28 + 2 \cdot 30}{2}$

Factor $2$ out of $2 \cdot 28 + 2 \cdot 30$ .

$\frac{2 \cdot (28 + 30)}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$\frac{2 \cdot (28 + 30)}{2 (1)}$

Cancel the common factor.

$\frac{2 \cdot (28 + 30)}{2 \cdot 1}$

Rewrite the expression.

$\frac{28 + 30}{1}$

Divide $28 + 30$ by $1$ .

$28 + 30$

Add $28$ and $30$ .

$58$

Convert the median $58$ to decimal.

$58$

Step 8

The interquartile range is the difference between the first quartile $18$ and the third quartile $58$ . In this case, the difference between the first quartile $18$ and the third quartile $58$ is $58 - (18)$ .

$58 - (18)$

Step 9

Simplify $58 - (18)$ .

Multiply $- 1$ by $18$ .

$58 - 18$

Subtract $18$ from $58$ .

$40$

Find the Interquartile Range (H-Spread) 56 , 80 , 42 , 22 , 31 , 60 , 14 , 7

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