{VERY URGENT}Two samples of 100 and 150 obsevations have means 15 and 16 and standrd deviation 3 and 4. Find their combined mean and standard deviation.

given : the mean of 100 observation is 15, and the mean of 150 observation is 16,
and their standard deviation is 3 and 4.
then
$n_1=100, \overline{X_1} =15 , and {\sigma }_1=3$
$n_2=150, \overline{X_2} =16 , and {\sigma }_2=4$

combined mean  $\overline{X} =\frac{n_1\overline{X_1} +n_2\overline{X_2}}{n_1+n_2}$
$=\frac{100*15+150*16}{100+150}$
= $\frac{1500+2400}{250}$
$$\begin{gathered} =\frac{3900}{250} \\ =15.6 \end{gathered}$$

$$\begin{gathered} d_1=\overline{X_1} -\overline{X} =15-15.6=-0.6 \\ {d}_2=\overline{X_2} -\overline{X} =16-15.6=0.4 \end{gathered}$$

let $\sigma$ be the standard deviation for the combined data. then,
combined variance ${\sigma }^2=\frac{n_1({{\sigma }_1}^2+{d_1}^2)+n_2({{\sigma }_2}^2+{d_2}^2)}{n_1+n_2}$
$$\begin{gathered} =\frac{100.(3^2+0.6^2)+150.(4^2+0.4^2)}{100+150} \\ =\frac{100.(9+0.36)+150.(16+0.16)}{250} \\ =\frac{100*9.36+150*16.16}{250} \\ =\frac{936+2424}{250} \\ =\frac{3360}{250}=13.44 \end{gathered}$$
thus the combined standard deviation is $\sigma =\sqrt{13.44}=3.67$

hope this helps you


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