The following table shows the distribution of... | WAEC SSCE - Further Mathematics - 2008

Question 1 Report

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

Marks	0-9	10-19	20-29	30-39	40-49	50-59	60-69	70-79	80-89	90-99
Frequency	50	50	40	60	100	100	50	25	15	10

(a) Construct a cumulative frequency table for the distribution

(b) Draw an ogive for the distribution

(c) Use your graph in (b) to determine : (i) semi- interquartile range ; (ii) number of students who failed, if the pass mark for the examination is 37 ; (iii) probability that a student selected at random scored between 20% and 60%.

Answer

(a) Cumulative frequency table

Marks	Frequency, \(f\)	Class boundaries	Cumulative frequency
0–9	50	−0.5–9.5	50
10–19	50	9.5–19.5	100
20–29	40	19.5–29.5	140
30–39	60	29.5–39.5	200
40–49	100	39.5–49.5	300
50–59	100	49.5–59.5	400
60–69	50	59.5–69.5	450
70–79	25	69.5–79.5	475
80–89	15	79.5–89.5	490
90–99	10	89.5–99.5	500

Total number of students, \(N=500\).

(b) Ogive

The cumulative frequencies are plotted against the upper class boundaries, beginning with \((-0.5,0)\).

graph — Ogive obtained by plotting cumulative frequency against the upper class boundaries and joining the points with a smooth increasing curve.

(c)

(i) Semi-interquartile range

\(Q_1\) is the \(\frac{N}{4}=125\)th value and \(Q_3\) is the \(\frac{3N}{4}=375\)th value. From the ogive,

\[Q_1\approx 26.0,\qquad Q_3\approx 56.3.\]

\[\text{Semi-interquartile range}=\frac{Q_3-Q_1}{2}=\frac{56.3-26.0}{2}=\boxed{15.15\text{ marks}}.\]

(ii) Number who failed

At a mark of \(37\), the cumulative frequency read from the ogive is approximately \(190\). Hence, \(\boxed{190\text{ students}}\) failed.

(iii) Probability of scoring between \(20\%\) and \(60\%\)

From the cumulative-frequency curve, the number scoring below \(20\%\) is \(100\), while the number scoring below \(60\%\) is \(400\).

\[\text{Number scoring between }20\%\text{ and }60\%=400-100=300.\]

\[P(20\%<X<60\%)=\frac{300}{500}=\boxed{\frac35}.\]

Read lesson note on Statistics (WAEC)

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