Question

find the iqr of the data set:
23, 22, 17, 15, 26, 24, 22
20, 40, 45, 20, 25, 45
32, 37, 34, 36, 32, 37
73, 93, 85, 92, 52, 57, 85
4, 3, 4, 2, 5, 2, 2, 1, 6
13, 13, 8, 10, 7, 15, 17, 8, 11

Explanation:

Step1: Sort the first data set

Sort \( 23, 22, 17, 15, 26, 24, 22 \) to get \( 15, 17, 22, 22, 23, 24, 26 \).

Step2: Find Q1 and Q3 for first set

For \( n = 7 \) (odd), median is the 4th term: \( 22 \). Lower half: \( 15, 17, 22 \), Q1 (median of lower half) is \( 17 \). Upper half: \( 23, 24, 26 \), Q3 (median of upper half) is \( 24 \). IQR = \( 24 - 17 = 7 \).

Step3: Sort the second data set

Sort \( 20, 40, 45, 20, 25, 45 \) (note: assume typo, should be \( 20, 40, 45, 20, 25, 45 \) → \( 20, 20, 25, 40, 45, 45 \)).

Step4: Find Q1 and Q3 for second set

\( n = 6 \) (even), median is average of 3rd and 4th: \( \frac{25 + 40}{2} = 32.5 \). Lower half: \( 20, 20, 25 \), Q1 = \( 20 \). Upper half: \( 40, 45, 45 \), Q3 = \( 45 \). IQR = \( 45 - 20 = 25 \).

Step5: Sort the third data set

Sort \( 32, 37, 34, 36, 32, 37 \) to get \( 32, 32, 34, 36, 37, 37 \).

Step6: Find Q1 and Q3 for third set

\( n = 6 \), median: \( \frac{34 + 36}{2} = 35 \). Lower half: \( 32, 32, 34 \), Q1 = \( 32 \). Upper half: \( 36, 37, 37 \), Q3 = \( 37 \). IQR = \( 37 - 32 = 5 \).

Step7: Sort the fourth data set

Sort \( 73, 93, 85, 92, 52, 57, 85 \) (note: assume typo, should be \( 73, 93, 85, 92, 52, 57, 85 \) → \( 52, 57, 73, 85, 85, 92, 93 \)).

Step8: Find Q1 and Q3 for fourth set

\( n = 7 \), median (4th term): \( 85 \). Lower half: \( 52, 57, 73 \), Q1 = \( 57 \). Upper half: \( 85, 92, 93 \), Q3 = \( 92 \). IQR = \( 92 - 57 = 35 \).

Step9: Sort the fifth data set

Sort \( 4, 3, 4, 2, 5, 2, 2, 1, 6 \) to get \( 1, 2, 2, 2, 3, 4, 4, 5, 6 \).

Step10: Find Q1 and Q3 for fifth set

\( n = 9 \), median (5th term): \( 3 \). Lower half: \( 1, 2, 2, 2 \), Q1 (median of lower half) is \( 2 \). Upper half: \( 4, 4, 5, 6 \), Q3 (median of upper half) is \( 4.5 \). IQR = \( 4.5 - 2 = 2.5 \).

Step11: Sort the sixth data set

Sort \( 13, 13, 8, 10, 7, 15, 17, 8, 11 \) to get \( 7, 8, 8, 10, 11, 13, 13, 15, 17 \).

Step12: Find Q1 and Q3 for sixth set

\( n = 9 \), median (5th term): \( 11 \). Lower half: \( 7, 8, 8, 10 \), Q1 = \( 8 \). Upper half: \( 13, 13, 15, 17 \), Q3 = \( 14 \) (median of \( 13,13,15,17 \) is \( \frac{13 + 15}{2}=14 \)). IQR = \( 14 - 8 = 6 \).

Answer:

1. For \( 15, 17, 22, 22, 23, 24, 26 \): \( 7 \)
2. For \( 20, 20, 25, 40, 45, 45 \): \( 25 \)
3. For \( 32, 32, 34, 36, 37, 37 \): \( 5 \)
4. For \( 52, 57, 73, 85, 85, 92, 93 \): \( 35 \)
5. For \( 1, 2, 2, 2, 3, 4, 4, 5, 6 \): \( 2.5 \)
6. For \( 7, 8, 8, 10, 11, 13, 13, 15, 17 \): \( 6 \)