The outcomes of each of 30 items was observed; 10 items gave an outcomes of  $\frac{1}{4}-d$  each, 10 items gave outcomes of  $\frac{1}{4}$ each and the remaining 10 items gave outcomes of  $\frac{1}{4}+d$  each. If the variance of this outcome data is $\frac{4}{3}$  then  $d^2$ equals to

Outcomes are  $(\frac{1}{4}-d), (\frac{1}{4}-d),\mathrm{...},$ 10 times   $\frac{1}{4},\frac{1}{4},\mathrm{...},$10 times,   $\frac{1}{4}+d, \frac{1}{4}+d,\mathrm{...},$ 10 times
Mean = $\frac{1}{30}(\frac{1}{4}\times 30)=\frac{1}{4}$
Variance of the outcomes is, 
$$\begin{gathered} {\sigma }^2=\frac{1}{30}\sum x_i^2-{\left(\overline{x}\right)}^2 \\ =\frac{1}{30}[{(\frac{1}{4}-d)}^2\times 10+{\left(\frac{1}{4}\right)}^2\times 10+{(\frac{1}{4}+d)}^2\times 10]-\frac{1}{16} \\ \Rightarrow \frac{4}{3}=\frac{1}{30}[30\times \frac{1}{16}+20d^2]-\frac{1}{16} \Rightarrow \frac{4}{3}=\frac{1}{16}+\frac{2}{3}{d}^2-\frac{1}{16} \\ \Rightarrow d^2=2\Rightarrow \left|d\right|=\sqrt{2} \end{gathered}$$