Question

2 the data set represents the shoe sizes of 19 students in a fifth grade physical education class. 4 5 5 5 6 6 6 6 7 7 7 7 7.5 7.5 8 8 8.5 8.5 9 create a box plot to represent the distribution of the data.

Explanation:

Step1: Order the data (already ordered)

Data: 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7.5, 7.5, 8, 8, 8.5, 8.5, 9 (n = 19)

Step2: Find the median (Q2)

Position of median: $\frac{n + 1}{2}=\frac{19 + 1}{2}=10$th term. The 10th term is 7, so $Q2 = 7$.

Step3: Find Q1 (median of lower half)

Lower half: 4, 5, 5, 5, 6, 6, 6, 6, 7 (first 9 terms, n = 9). Position of Q1: $\frac{9 + 1}{2}=5$th term. 5th term is 6, so $Q1 = 6$.

Step4: Find Q3 (median of upper half)

Upper half: 7.5, 7.5, 8, 8, 8.5, 8.5, 9 (last 9 terms? Wait, n = 19, after median (10th term), upper half has 19 - 10 = 9 terms? Wait, no: for n odd, lower half is first $\frac{n - 1}{2}=9$ terms, upper half is last 9 terms. Wait, data after median (10th term: 7) is terms 11 to 19: 7.5, 7.5, 8, 8, 8.5, 8.5, 9? Wait, no, original data: positions 1 - 19. Median at 10: term 10 is 7. Lower half: terms 1 - 9: 4,5,5,5,6,6,6,6,7. Upper half: terms 11 - 19: 7.5,7.5,8,8,8.5,8.5,9? Wait, no, term 11 is 7.5, term 12 is 7.5, term 13 is 8, term 14 is 8, term 15 is 8.5, term 16 is 8.5, term 17 is 9? Wait, original data: let's count again: 4 (1),5(2),5(3),5(4),6(5),6(6),6(7),6(8),7(9),7(10),7(11),7(12),7.5(13),7.5(14),8(15),8(16),8.5(17),8.5(18),9(19). Oh! I made a mistake earlier. So median (Q2) is term 10: 7. Lower half: terms 1 - 9: 4,5,5,5,6,6,6,6,7. Upper half: terms 11 - 19: 7,7,7.5,7.5,8,8,8.5,8.5,9. Now, for upper half (n = 9), median (Q3) is term $\frac{9 + 1}{2}=5$th term of upper half. Upper half terms: 7(11),7(12),7.5(13),7.5(14),8(15),8(16),8.5(17),8.5(18),9(19). 5th term of upper half is 8, so $Q3 = 8$.

Step5: Find minimum and maximum

Minimum: 4, Maximum: 9.

Step6: Draw the box plot

• Draw a number line from 4 to 9.
• Plot the minimum (4), Q1 (6), median (7), Q3 (8), maximum (9).
• Draw a box from Q1 to Q3, with a line inside the box at the median.
• Draw whiskers from the box to the minimum and maximum.

Answer:

The box plot has:

• Minimum: 4
• Q1: 6
• Median (Q2): 7
• Q3: 8
• Maximum: 9

To draw it:

• On a number line, mark 4 (min), 6 (Q1), 7 (median), 8 (Q3), 9 (max).
• Draw a box between 6 and 8, with a vertical line at 7.
• Draw horizontal lines (whiskers) from 6 to 4 and from 8 to 9.