Question

The means of two samples of sizes 50 and 100 respectively are 54.1 and 50.3 and the standard deviations are 8 and 7. Find the mean and the standard deviation of the sample of size 150 obtained by combining the two samples.

Answer 1

Calculating combined standard deviation,

$$\begin{gathered} { \\ }_{1,2}^{\sigma }=\sqrt{\frac{N_1{\sigma }_1^2+N_2{\sigma }_2^2+N_1{d}_1^2+N_2{d}_2^2}{N_1+N_2}} \\ where, \\ {N}_1=50N_2=100 \\ {\sigma }_1=8{\sigma }_2=7 \\ {\overline{X}}_1=54.1{\overline{X}}_2=50.3 \end{gathered}$$

To calculate combined standard deviation, we need to find the combined mean of the observations.

Thus, Combined mean is

$$\begin{gathered} {\overline{X}}_{12}=\frac{N_1{\overline{X}}_1+N_2{\overline{X}}_2}{N_1+N_2} \\ =\frac{50\times 54.1+100\times 50.3}{150} \\ =\frac{2705+5030}{150}=\frac{7735}{150}=51.57 \\ {d}_1^2={\left({\overline{X}}_1-{\overline{X}}_{12}\right)}^2={\left(54.1-51.57\right)}^2={\left(2.53\right)}^2=6.40 \\ {d}_2^2={\left(X_2-{\overline{X}}_{1,2}\right)}^2={\left(50.3-51.57\right)}^2={\left(-1.27\right)}^2=1.61 \\ Thus,Combineds\tan darddeviationwillbe: \\ {\sigma }_{1,2}=\sqrt{\frac{50{\left(8\right)}^2+100{\left(7\right)}^2+50\left(6.4\right)+100\left(1.61\right)}{150}} \\ or,{\sigma }_{1,2}=\sqrt{\frac{3200+4900+320+161}{150}} \\ or,{\sigma }_{1,2}=\sqrt{\frac{8581}{150}} \\ or,{\sigma }_{1,2}=\sqrt{57.20} \\ \Rightarrow {\sigma }_{1,2}=7.56 \end{gathered}$$

Hence, the combined standard deviation is 7.56