Question

Number of Pairs of Shoes Number of 6th Graders
2 4
3 3
4 1
5 1
6 3
7 7
8 2
10 1
Michael wants to know how many pairs of shoes 6th graders own. He surveys the 6th graders at his summer camp. His results are given in the table. After Michael creates the table, he realizes that he forgot to include himself. Michael owns 4 pairs of shoes.

Determine the interquartile range (IQR) of the original dataset and the interquartile range of the new dataset with the data point for Michael’s 4 pairs of shoes added.

Answer 1

First dataset (22 students)

Ordered positions (value: positions)
- 2: positions 1–4
- 3: positions 5–7
- 4: position 8
- 5: position 9
- 6: positions 10–12
- 7: positions 13–19
- 8: positions 20–21
- 10: position 22

For N = 22, lower half = positions 1–11, upper half = positions 12–22.
- Q1 = median of positions 1–11 = 6th value = 3
- Q3 = median of positions 12–22 = 6th value of the upper half = 7
IQR = Q3 − Q1 = 7 − 3 = 4

After adding Michael (one more 4), N = 23

New ordered positions
- 2: 1–4
- 3: 5–7
- 4: 8–9
- 5: 10
- 6: 11–13
- 7: 14–20
- 8: 21–22
- 10: 23

For N = 23, exclude the median (position 12) and take lower half 1–11 and upper half 13–23.
- Q1 = median of 1–11 = 6th value = 3
- Q3 = median of 13–23 = 6th value = 7
IQR = 7 − 3 = 4

Answer: original IQR = 4; new IQR = 4.