Question

Find the combined standard deviation of two groups $X$ and $Y$ from the data gives below:

• A. 5
• B. 5.8
• C. 5.2
• D. 6

Answer 1

Answer: (C)

Combined standard deviation $X$ and $Y$ is: ${\sigma }_{xy}=\sqrt{\frac{N_x{\sigma }_x^2+N_y{\sigma }_y^2+N_x{d}_x^2+N_y{d}_y^2}{N_x+N_y}}$ $N_x=10,N_y=10,\overline{X_x}=15,\overline{X_y}=20,{\sigma }_x=4,$ and ${\sigma }_y=5$ $\overline{X_{xy}}=\frac{\overline{X_x}{N}_x+\overline{X_y}{N}_y}{N_x+N_y}$ $\text{ Or }\overline{X_{xy}}=\frac{(15\times 10)+(20\times 15)}{10+15}=\frac{150+300}{25}$ $=\frac{450}{25}=18$ $d_x=(\overline{X_{xy}}-\overline{X_x})=18-15=3$ $d_y=(\overline{X_{xy}}-\overline{X_y})=18-20=-2$ On substituting all the values in the formula for combined standard deviation, we get. ${\sigma }_{xy}=\sqrt{\frac{10(4{)}^2+15(5{)}^2+10(18-15{)}^2+15(18-20{)}^2}{10+15}}$ $=\sqrt{\frac{(10\times 16)+(15\times 25)+(10\times 9)+(15\times 4)}{25}}$ $=\sqrt{\frac{160+375+90+60}{25}}$ $=\sqrt{\frac{68.5}{25}}=5.2\text{ (approximately) }$