WAEC 2016 · Paper 2 · Q6

Statistics & correlation

The mean of the numbers $1, 4, k, (k + 4)$ and $11$ is $(k + 1)$ . Calculate the:

(a)
value of $k$ ;
(b)
standard deviation.

Worked solution (try it first)

(a)

The five numbers add to $2k + 20$ , so their mean is $\dfrac{2k + 20}{5} = k + 1$ .
Multiply by 5: $2k + 20 = 5k + 5$ , so $3k = 15$ and $k = 5$ .

(b)

The numbers are $1, 4, 5, 9, 11$ , with mean 6.
The deviations are $-5, -2, -1, 3, 5$ .
$\sum d^2 = 25 + 4 + 1 + 9 + 25 = 64$ , so $\sigma = \sqrt{\dfrac{64}{5}}$
$= \dfrac{8}{\sqrt5}$
$= \dfrac{8\sqrt5}{5}$
$\approx 3.578$ .

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