Marks 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99 Frequency 2 2 2 8 13 11 12 10 4 The table shows the distribution of marks scor...
Question 1 Report
| Marks | 10 - 19 | 20 - 29 | 30 - 39 | 40 - 49 | 50 - 59 | 60 - 69 | 70 - 79 | 80 - 89 | 90 - 99 |
| Frequency | 2 | 2 | 2 | 8 | 13 | 11 | 12 | 10 | 4 |
The table shows the distribution of marks scored by 64 students in a test
(a) Draw a histogram for the distribution.
(b) Use the histogram to estimate the modal score.
Answer Details
The class widths are all equal to 10, so the heights of the histogram bars are the frequencies.
| Marks | Class boundaries | Frequency |
|---|---|---|
| 10–19 | 9.5–19.5 | 2 |
| 20–29 | 19.5–29.5 | 2 |
| 30–39 | 29.5–39.5 | 2 |
| 40–49 | 39.5–49.5 | 8 |
| 50–59 | 49.5–59.5 | 13 |
| 60–69 | 59.5–69.5 | 11 |
| 70–79 | 69.5–79.5 | 12 |
| 80–89 | 79.5–89.5 | 10 |
| 90–99 | 89.5–99.5 | 4 |
(a) Histogram:
(b) The modal class is 50–59, since it has the greatest frequency, 13.
Using the intersecting diagonals from the top corners of the modal rectangle to the top corners of the adjacent rectangles gives
\[\text{Mode}=L+\frac{f_m-f_1}{(f_m-f_1)+(f_m-f_2)}\times c\]
where \(L=49.5\), \(f_m=13\), \(f_1=8\), \(f_2=11\), and \(c=10\).
\[\text{Mode}=49.5+\frac{13-8}{(13-8)+(13-11)}\times10\]
\[=49.5+\frac{5}{7}\times10=56.64\]
Hence, the estimated modal score is \(\boxed{57\text{ marks}}\).
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