WAEC 2020 · Paper 2 · Q10

Marks (%)	0–9	10–19	20–29	30–39	40–49	50–59	60–69	70–79	80–89	90–99
Frequency	7	11	17	20	29	34	30	25	21	6

The table shows the distribution of marks obtained by students in an examination.

(a)

Construct a cumulative frequency table for the distribution.

Model answer

Marks (%)	Frequency	Upper class boundary	Cumulative frequency
0–9	7	9.5	7
10–19	11	19.5	18
20–29	17	29.5	35
30–39	20	39.5	55
40–49	29	49.5	84
50–59	34	59.5	118
60–69	30	69.5	148
70–79	25	79.5	173
80–89	21	89.5	194
90–99	6	99.5	200

Each cumulative frequency is the running total of the frequencies; the last one equals the total, 200.

(b)
Draw the cumulative frequency curve for the distribution.
Model answer
Plot each cumulative frequency against the upper class boundary of its class, starting from $(-0.5, 0)$ where the cumulative frequency is 0 and ending at $(99.5, 200)$ . 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.

For (c): the median is the 100th mark. Read across from 100: about 54.2. The top 5% are the last 10 of 200, so read across from 190: the lowest distinction mark is about 87.1.
(c)
Using the curve, find, correct to one decimal place, the: (i) median mark; (ii) lowest mark for distinction if $5\%$ of the students passed with distinction.
Separate values with commas, e.g. 3, −2

The ogive, with the median (purple) and distinction (red) readings.

Worked solution (try it first)

Marks	0–9	10–19	20–29	30–39	40–49	50–59	60–69	70–79	80–89	90–99
Upper boundary	9.5	19.5	29.5	39.5	49.5	59.5	69.5	79.5	89.5	99.5
Cumulative frequency	7	18	35	55	84	118	148	173	194	200

Plot each cumulative frequency at its upper boundary, starting from $(-0.5, 0)$ , and join the points with a smooth S-shaped curve.

The median is at $\frac{200}{2} = 100$ .
Go across from 100 to the curve and down: about 54.5.
(Check: 100 lies between 84 at 49.5 and 118 at 59.5, and $49.5 + \frac{100 - 84}{34} \times 10 \approx 54.2$ .)

The top $5\%$ got distinctions, so $95\%$ are below the lowest distinction mark: $0.95 \times 200 = 190$ .
Go across from 190: about 88.
(Check: $79.5 + \frac{190 - 173}{21} \times 10 \approx 87.6$ .)