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

Question 7:
A set of data consists of twelve positive numbers.
$\text{It is given that }\Sigma {\left(x-\overline{x}\right)}^2=600\text{ and }\Sigma x^2=1032.$
Find
(a) the variance
(b) the mean

Solution:
(a)
$\begin{array}{l}\text{Variance}=\frac{\Sigma {\left(x-\overline{x}\right)}^2}{N}\\ =\frac{600}{12}\\ =50\end{array}$

(b)
$\begin{array}{l}\text{Variance}=\frac{\Sigma x^2}{N}-{\left(\overline{x}\right)}^2\\ 50=\frac{1032}{12}-{\left(\overline{x}\right)}^2\\ {\left(\overline{x}\right)}^2=86-50\\ =36\\ \overline{x}=36\end{array}$

Question 8:
Solution:
(a)(i)
$\begin{array}{l}\text{Given mean}=6\\ \frac{\Sigma x}{6}=6\\ \Sigma x=36\end{array}$

(a)(ii)
$\begin{array}{l}\text{Given }\sigma =2.4\\ {\sigma }^2={2.4}^2\\ \frac{\Sigma x^2}{n}-{\overline{X}}^2=5.76\\ \frac{\Sigma x^2}{6}-6^2=5.76\\ \frac{\Sigma x^2}{6}=41.76\\ \Sigma x^2=250.56\end{array}$

(b)(i)