WAEC 2016 · Paper 2 · Q12

Marks	3	4	5	6	7	8	9
Frequency	1	4	3	5	2	$x$	2

The table shows the distribution of marks of students in a Mathematics test. If the mean of the distribution is 6, calculate the:

(a)
value of $x$ ;
(b)
standard deviation, correct to 2 decimal places.

Worked solution (try it first)

(a)

$\sum f = 1 + 4 + 3 + 5 + 2 + x + 2$
$= 17 + x$ and $\sum fx = 3 + 16 + 15 + 30 + 14 + 8x + 18$
$= 96 + 8x$ .
The mean is 6, so $\frac{96 + 8x}{17 + x} = 6$ .
Then $96 + 8x = 102 + 6x$ , so $2x = 6$ and $x = 3$ .
Check: $\sum f = 20$ , $\sum fx = 120$ , and $\frac{120}{20} = 6$ .

(b)

Set out the table with the mean 6:
$x$ 3 4 5 6 7 8 9 Total

$f$ 1 4 3 5 2 3 2 20

$(x - 6)^2$ 9 4 1 0 1 4 9

$f(x - 6)^2$ 9 16 3 0 2 12 18 60
Standard deviation $= \sqrt{\frac{\sum f(x - \bar x)^2}{\sum f}}$
$= \sqrt{\frac{60}{20}}$
$= \sqrt 3$
$\approx 1.73$ .

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