Explanation

(d)
Let $n_{1}$ and $n_{2}$ be the number of observations in two groups having means ${\overline{X}}_{1}\ and\ {\overline{X}}_{2}$ respectively
Then, $\overline{X} = \frac{n_{1}{\overline{X}}_{1} + n_{2}{\overline{X}}_{2}}{n_{1} + n_{2}}$
Now, $\overline{X} - {\overline{X}}_{1} = \frac{n_{1}{\overline{X}}_{1} + n_{2}{\overline{X}}_{2}}{n_{1} + n_{2}} - \overline{{X_{\ }}_{1}}$
$= n_{2}\frac{({\overline{X}}_{2} - {\overline{X}}_{1})}{n_{1} + n_{2}} > 0\ \ \ \ \ (\because\ {\overline{X}}_{2} > {\overline{X}}_{1})$
$\Rightarrow \ \overline{X} > {\overline{X}}_{1}$ ...(i)
And $\overline{X} - {\overline{X}}_{2} = \frac{n_{1}\left( {\overline{X}}_{1} - {\overline{X}}_{2} \right)}{n_{1} + n_{2}} < 0\ \ \ \ \because{\overline{X}}_{2} > {\overline{X}}_{2}$
$\Rightarrow \ \ \overline{X} < {\overline{X}}_{2}$ ...(ii)
From relations (i) and (ii), we get
${\overline{X}}_{1} < \overline{X} < {\overline{X}}_{2}$