The table shows the distribution of marks scored by some students in a test. Marks 1-10 11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 No. of studen...
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The table shows the distribution of marks scored by some students in a test.
| Marks | 1-10 | 11-20 | 21-30 | 31-40 | 41-50 | 51-60 | 61-70 | 71-80 | 81-90 | 91-100 |
| No. of students | 3 | 17 | 41 | 85 | 97 | 115 | 101 | 64 | 21 | 6 |
(a)(i) Construct a cumulative frequency table for the distribution ; (ii) Draw a cumulative frequency curve for the distribution.
(b) Use the curve to estimate the :
(i) number of students who scored marks between 32 and 74 ; (ii) pass mark, if 18% of the students failed ; (iii) lowest mark for distinction, if 8% of the students passed with distinction.
Answer Details
(a)(i) Cumulative frequency table
The cumulative frequency is obtained by adding each class frequency to the running total. The curve is plotted against the upper class boundary of each class, so these are tabulated as well. The total number of students is \(N = 550\).
| Marks | Class boundaries | Frequency \(f\) | Cumulative frequency \(\sum f\) |
|---|---|---|---|
| 1 - 10 | 0.5 - 10.5 | 3 | 3 |
| 11 - 20 | 10.5 - 20.5 | 17 | 20 |
| 21 - 30 | 20.5 - 30.5 | 41 | 61 |
| 31 - 40 | 30.5 - 40.5 | 85 | 146 |
| 41 - 50 | 40.5 - 50.5 | 97 | 243 |
| 51 - 60 | 50.5 - 60.5 | 115 | 358 |
| 61 - 70 | 60.5 - 70.5 | 101 | 459 |
| 71 - 80 | 70.5 - 80.5 | 64 | 523 |
| 81 - 90 | 80.5 - 90.5 | 21 | 544 |
| 91 - 100 | 90.5 - 100.5 | 6 | 550 |
(a)(ii) Cumulative frequency curve (ogive)
Plot the cumulative frequency (vertical axis) against the upper class boundary (horizontal axis), starting the curve from \((0.5,\,0)\), and join the points with a smooth S-shaped curve.
(b) Estimates read from the curve
(i) Number of students scoring between 32 and 74. Read the cumulative frequency at each mark from the ogive (dashed guide lines):
\[\text{CF}(32)\approx 61+\frac{32-30.5}{10}\times 85 \approx 74\]\[\text{CF}(74)\approx 459+\frac{74-70.5}{10}\times 64 \approx 481\]Number scoring between 32 and 74:
\[481-74 = 407\ \text{students (approximately)}.\](ii) Pass mark, if 18% failed. The number who failed is
\[18\%\ \text{of}\ 550 = \frac{18}{100}\times 550 = 99.\]The pass mark is the mark below which 99 students lie, i.e. the mark corresponding to \(\text{CF}=99\). Reading horizontally from \(99\) on the vertical axis to the curve and down to the mark axis:
\[\text{pass mark} \approx 30.5+\frac{99-61}{85}\times 10 \approx 35.\](iii) Lowest mark for distinction, if 8% passed with distinction. The distinction group is the top 8%, so the number below the distinction mark is
\[550-\Big(8\%\ \text{of}\ 550\Big)=550-44=506.\]Read the mark corresponding to \(\text{CF}=506\):
\[\text{lowest distinction mark} \approx 70.5+\frac{506-459}{64}\times 10 \approx 78.\]Download The App On Google Playstore
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