WAEC 2025 · Paper 2 · Q11✱✱

The table shows the scores of 2000 candidates in an examination.

Marks (%)	11–20	21–30	31–40	41–50	51–60	61–70	71–80	81–90
Frequency	68	184	294	402	480	310	164	98

(a)
Prepare a cumulative frequency table.
Separate values with commas, e.g. 3, −2
(b)
Draw the cumulative frequency curve.
Model answer
Plot each cumulative frequency against the upper class boundary of its class, starting from $(10.5, 0)$ where the cumulative frequency is 0 and ending at $(90.5, 2000)$ . Join the points with a smooth rising S-shaped curve (an ogive), not straight lines. Label both axes. Readings from a hand-drawn curve differ a little from person to person; examiners accept a small range, usually about ±1. Use a scale that fits 2000 on the vertical axis, e.g. 2 cm to 250 candidates.

Worked solution (try it first)

Add the frequencies one class at a time, and write the upper class boundary beside each running total:
Marks 11–20 21–30 31–40 41–50 51–60 61–70 71–80 81–90

Upper boundary 20.5 30.5 40.5 50.5 60.5 70.5 80.5 90.5

Cumulative frequency 68 252 546 948 1428 1738 1902 2000
The last cumulative frequency is 2000, the number of candidates, which checks the additions.

Marks	11–20	21–30	31–40	41–50	51–60	61–70	71–80	81–90
Upper boundary	20.5	30.5	40.5	50.5	60.5	70.5	80.5	90.5
Cumulative frequency	68	252	546	948	1428	1738	1902	2000

Plot each cumulative frequency against its upper class boundary, starting from $(10.5, 0)$ , the lower boundary of the first class.
Join the points with one smooth S-shaped curve, drawn freehand.