Question

For a group of 50 male workers , the mean and standard deviation of their weekly wages are ₹ 63 and ₹ 9 respectively. For a group of 40 female workers , these are ₹ 54 and ₹ 6 respectively. Find mean and standard deviation for a combined group of 90 workers.

Answer 1

Given,
$$\begin{gathered} N_1=50,{\overline{\mathit{X}}}_1=63,{\sigma }_1=9 \\ {N}_2=40,{\overline{\mathit{X}}}_2=54,{\sigma }_2=6 \\ \mathbf{Combined}\mathbf{}\mathbf{Mean} \\ {\overline{\mathit{X}}}_{1,2}=\frac{{\overline{\mathit{X}}}_1{N}_1+{\overline{\mathit{X}}}_2{N}_2}{N_1+N_2} \\ =\frac{63\left(50\right)+54\left(40\right)}{50+40} \\ =\frac{3150+2160}{90} \\ =\frac{5310}{90} \\ =59 \end{gathered}$$

Combined Standard Deviation
$$\begin{gathered} d_1^2={\left({\overline{\mathrm{X}}}_1-{\overline{\mathrm{X}}}_{1,2}\right)}^2={\left(63-59\right)}^2={\left(4\right)}^2=16 \\ {d}_2^2={\left({\overline{\mathrm{X}}}_2-{\overline{\mathrm{X}}}_{1,2}\right)}^2={\left(54-59\right)}^2={\left(-5\right)}^2=25 \\ {\sigma }_{1,2}=\sqrt{\frac{N_1{\sigma }_1^2+N_2{\sigma }_2^2+N_1{d}_1^2+N_2{d}_2^2}{N_1+N_2}} \\ =\sqrt{\frac{50\left(81\right)+40\left(36\right)+50\left(16\right)+40\left(25\right)}{50+40}} \\ =\sqrt{\frac{4050+1440+800+1000}{90}} \\ =\sqrt{\frac{7290}{90}} \\ =\sqrt{81} \\ =9 \end{gathered}$$
Thus, mean and standard deviation for a combined group of 90 workers are 59 and 9 respectively.