8.4.3 Measures of Dispersion for Ungrouped Data, SPM Paper (Long Questions)

Question 5:
A set of data consists of 9, 2, 7, x2 – 1 and 4. Given the mean is 6, find
(a) the positive value of x,
(b) the median using the value of x in part (a).

Solution:
(a)

(b)
Arrange the numbers in ascending order
2, 4, 7, 8, 9
Median = 7

Question 6:
A set of seven numbers has a standard deviation of 3 and another set of three numbers has a standard deviation of 4. Both sets of numbers have an equal mean.
If the two sets of numbers are combined, find the variance.

Solution:
$\begin{array}{l}{\overline{X}}_{1}=\frac{\Sigma {X}_{1}}{{n}_{1}}\\ m=\frac{\Sigma {X}_{1}}{7}\\ \Sigma {X}_{1}=7m\\ \\ m=\frac{\Sigma {X}_{2}}{3}\\ \Sigma {X}_{2}=3m\\ \\ \sigma =\sqrt{\frac{\Sigma {X}^{2}}{N}-{\left(\overline{X}\right)}^{2}}\\ {\sigma }^{2}=\frac{\Sigma {X}^{2}}{N}-{\left(\overline{X}\right)}^{2}\\ 9=\frac{\Sigma {X}_{1}{}^{2}}{7}-{m}^{2}\\ 63=\Sigma {X}_{1}{}^{2}-7{m}^{2}\\ \Sigma {X}_{1}{}^{2}=7{m}^{2}+63\end{array}$