The table shows the distribution of masks obtained... | WAEC SSCE - Further Mathematics - 2019

Question 1 Report

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

Marks	50 - 54	55 - 59	60 - 64	65 - 69	70 - 74	75 - 79	80 - 84	85 - 89
Frequency	5	15	20	28	12	9	7	4

Using an assumed mean of 67, calculate, correct to one decimal place. the

a) Mean

b) Standard deviation of the distribution

Answer

Method (assumed mean / coding). Take assumed mean \(A = 67\) and class width \(c = 5\). For each class let \(x\) be the midpoint and \(u = \dfrac{x - A}{c}\).

Marks	Midpoint \(x\)	\(u\)	\(f\)	\(fu\)	\(fu^2\)
50 - 54	52	-3	5	-15	45
55 - 59	57	-2	15	-30	60
60 - 64	62	-1	20	-20	20
65 - 69	67	0	28	0	0
70 - 74	72	1	12	12	12
75 - 79	77	2	9	18	36
80 - 84	82	3	7	21	63
85 - 89	87	4	4	16	64
Total			100	2	300

(a) Mean.

\[ \bar{x} = A + \left(\frac{\sum fu}{\sum f}\right)c = 67 + \frac{2}{100}\times 5 = 67 + 0.1 = \mathbf{67.1} \]

(b) Standard deviation.

\[ \text{SD} = c\sqrt{\frac{\sum fu^2}{\sum f} - \left(\frac{\sum fu}{\sum f}\right)^2} = 5\sqrt{\frac{300}{100} - \left(\frac{2}{100}\right)^2} \] \[ = 5\sqrt{3 - 0.0004} = 5\sqrt{2.9996} = 5 \times 1.7319 \approx \mathbf{8.7} \]

Read lesson note on Statistics (WAEC)

Answer Details

Method (assumed mean / coding). Take assumed mean \(A = 67\) and class width \(c = 5\). For each class let \(x\) be the midpoint and \(u = \dfrac{x - A}{c}\).

Marks	Midpoint \(x\)	\(u\)	\(f\)	\(fu\)	\(fu^2\)
50 - 54	52	-3	5	-15	45
55 - 59	57	-2	15	-30	60
60 - 64	62	-1	20	-20	20
65 - 69	67	0	28	0	0
70 - 74	72	1	12	12	12
75 - 79	77	2	9	18	36
80 - 84	82	3	7	21	63
85 - 89	87	4	4	16	64
Total			100	2	300

(a) Mean.

\[ \bar{x} = A + \left(\frac{\sum fu}{\sum f}\right)c = 67 + \frac{2}{100}\times 5 = 67 + 0.1 = \mathbf{67.1} \]

(b) Standard deviation.

Read lesson note on Statistics (WAEC)

Statistics

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The table shows the distribution of masks obtained by students in an examination. Marks 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84 85 - 89 Freq...

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The table shows the distribution of masks obtained by students in an examination. Marks 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84 85 - 89 Freq...

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