Question

A sample of 35 values has mean 80 and Standard Deviation 4 . A second sample of 65 values from same population has mean 70 and standard deviation 3. Find the mean and standard deviation of combined sample of 100 values.

Answer 1

Given,
$$\begin{gathered} N_1=35,{\overline{\mathit{X}}}_1=80,{\sigma }_1=4 \\ {N}_2=65,{\overline{\mathit{X}}}_2=70,\mathit{}{\sigma }_2=3 \\ \mathbf{Combined}\mathbf{}\mathbf{Mean} \\ {\overline{\mathit{X}}}_{1,2}=\frac{{\overline{\mathit{X}}}_1{N}_1+{\overline{\mathit{X}}}_2{N}_2}{N_1+N_2}=\frac{80\left(35\right)+70\left(65\right)}{35+65} \\ =\frac{2800+4550}{100}=\frac{7350}{100} \\ =73.5 \end{gathered}$$
Combined Standard Deviation
$$\begin{gathered} d_1^2={\left({\overline{\mathit{X}}}_1-{\overline{\mathit{X}}}_{1,2}\right)}^2={\left(80-73.5\right)}^2={\left(6.5\right)}^2=42.25 \\ {d}_2^2={\left({\overline{\mathit{X}}}_2-{\overline{\mathit{X}}}_{1,2}\right)}^2={\left(70-73.5\right)}^2={\left(-3.5\right)}^2=12.25 \\ {\sigma }_{1,2}=\sqrt{\frac{{\sigma }_1^2{N}_1+{\sigma }_2^2{N}_2+N_1{d}_1^2+N_2{d}_2^2}{N_1+N_2}} \\ =\sqrt{\frac{4^2\left(35\right)+3^2\left(65\right)+35\left(42.25\right)+65\left(12.25\right)}{35+65}} \\ =\sqrt{\frac{560+585+1478.75+796.25}{100}} \\ =\sqrt{\frac{3420}{100}}=\sqrt{34.20} \\ =5.84 \end{gathered}$$
Thus, mean and standard deviation of combined sample of 100 values are 73.5 and 5.84 respectively.