Question

3 find the interquartile range.
number of miles run per week

Explanation:

Step1: Count total data points

First, we count the number of dots (data points):

• At 4: 1
• At 6.5 (between 6 and 7): 1
• At 7: 3
• At 7.5 (between 7 and 8): 2
• At 8: 2
• At 9: 2
• At 10: 4
• At 10.5 (between 10 and 11): 1
• At 11: 3
• Total: \(1 + 1 + 3 + 2 + 2 + 2 + 4 + 1 + 3 = 19\)

Step2: Find position of Q1 and Q3

For \(n = 19\) (odd), the median is at position \(\frac{n + 1}{2} = 10\).

• Lower half (first 9 data points, positions 1–9) for Q1: position \(\frac{9 + 1}{2} = 5\) of the lower half.
• Upper half (last 9 data points, positions 11–19) for Q3: position \(\frac{9 + 1}{2} = 5\) of the upper half.

Step3: Identify data values

List the data in order:
4, 6.5, 7, 7, 7, 7.5, 7.5, 8, 8, 8 (median), 9, 9, 10, 10, 10, 10, 10.5, 11, 11, 11

Wait, correction: Let's list each data point with their counts:

• 4: 1 → [4]
• 6.5: 1 → [4, 6.5]
• 7: 3 → [4, 6.5, 7, 7, 7]
• 7.5: 2 → [4, 6.5, 7, 7, 7, 7.5, 7.5]
• 8: 2 → [4, 6.5, 7, 7, 7, 7.5, 7.5, 8, 8]
• 9: 2 → [4, 6.5, 7, 7, 7, 7.5, 7.5, 8, 8, 9, 9] (wait, no, total is 19. Wait, earlier count was wrong. Let's recount:

4:1, 6.5:1, 7:3 (total 5), 7.5:2 (7), 8:2 (9), 9:2 (11), 10:4 (15), 10.5:1 (16), 11:3 (19). Ah, correct. So positions:
1:4, 2:6.5, 3:7, 4:7, 5:7, 6:7.5, 7:7.5, 8:8, 9:8, 10:9 (median), 11:9, 12:10, 13:10, 14:10, 15:10, 16:10.5, 17:11, 18:11, 19:11.

Now, lower half: positions 1–9 (values: 4, 6.5, 7, 7, 7, 7.5, 7.5, 8, 8). The 5th value of the lower half (Q1) is the 5th term: 7.

Upper half: positions 11–19 (values: 9, 10, 10, 10, 10, 10.5, 11, 11, 11). The 5th value of the upper half (Q3) is the 5th term (position 15: 10).

Step4: Calculate IQR = Q3 - Q1

\(IQR = 10 - 7 = 3\)

Answer:

3