Question

The following table gives the distribution of wages in the two branches of a factory:

Monthly wages ( in ₹)	:	100-150	150-200	200-250	250-300	300-350	Total
Number of workers	:
Branch A	:	167	207	253	205	168	1000
Branch B	:	63	93	157	105	82	500

Find the mean and standard deviation for the two branches for the wages separately.
(a) Which branch pays higher average wages?
(b) Which branch has greater variability in wages in relation to the average wages?
(c) What is the average monthly wage for the factory as a whole?
(d) What is the variance of wages of all the workers in the two branches$—$A and B taken together?

Answer 1

For Branch A

Monthly Wages	Mid Value (m)	f	fm	Deviation from mean $$\begin{gathered} {\mathit{x}}_{\mathbf{A}}\mathbf{=}\mathit{m}\mathbf{-}{\overline{\mathit{X}}}_{\mathbf{A}} \\ {\overline{\mathit{X}}}_{\mathbf{A}}\mathbf{=}\mathbf{225} \end{gathered}$$	${\mathit{x}}_{\mathbf{A}}^{\mathbf{2}}$	$\mathit{f}{\mathit{x}}_{\mathbf{A}}^{\mathbf{2}}$
100 − 150 150 − 200 200 − 250 250 − 300 300 − 350	125 175 225 275 325	167 207 253 205 168	20875 36225 56925 56375 54600	−100 −50 0 50 100	10000 2500 0 2500 10000	1670000 517500 0 512500 1680000
		Σf = 1000	Σfm = 225000			$\mathrm{\Sigma }f{x}_{\mathrm{A}}^2=4380000$

$$\begin{gathered} {\overline{X}}_{\mathrm{A}}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{225000}{1000}=225 \\ S{D}_{\mathrm{A}}=\sqrt{\frac{\mathrm{\Sigma }f{x}_{\mathrm{A}}^2}{\mathrm{\Sigma }f}}=\sqrt{\frac{4380000}{1000}}=\sqrt{4380}=66.19 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{variation}=\frac{S{D}_{\mathrm{A}}}{{\overline{X}}_{\mathrm{A}}}\times 100 \\ =\frac{66.19}{225}\times 100 \\ =29.42\% \end{gathered}$$


For Branch B

Monthly wage	Mid value (m)	f	fm	Deviation from mean $$\begin{gathered} {\mathit{x}}_{\mathbf{B}}\mathbf{=}\mathit{m}\mathbf{-}{\overline{\mathbf{X}}}_{\mathbf{B}} \\ {\overline{\mathbf{X}}}_{\mathbf{B}}\mathbf{=}\mathbf{230} \end{gathered}$$	${\mathit{x}}_{\mathbf{B}}^{\mathbf{2}}$	$\mathit{f}{\mathit{x}}_{\mathbf{B}}^{\mathbf{2}}$
100 − 150 150 − 200 200 − 250 250 − 300 300 − 350	125 175 225 275 325	63 93 157 105 82	7875 16275 35325 28875 26650	−105 −55 −5 45 95	11025 3025 25 2025 9025	694575 281325 3925 212625 740050
		Σf = 500	Σfm = 115000			$\mathrm{\Sigma }f{x}_{\mathrm{B}}^2=1932500$

$$\begin{gathered} {\overline{\mathit{X}}}_{\mathrm{B}}=\frac{\mathrm{\Sigma }fm}{\mathrm{\Sigma }f}=\frac{115000}{500}=230 \\ S{D}_{\mathrm{B}}=\sqrt{\frac{\mathrm{\Sigma }f{x}_{\mathrm{B}}^2}{\mathrm{\Sigma }f}}=\sqrt{\frac{1932500}{500}}=\sqrt{3865}=62.16 \\ \mathrm{Coefficient}\mathrm{of}\mathrm{variation}=\frac{S{D}_{\mathrm{B}}}{{\overline{\mathit{X}}}_{\mathrm{B}}}\times 100=\frac{62.16}{230}\times 100=27.02\% \end{gathered}$$

(a) Branch B pays higher average monthly wages i.e. Rs 230.
(b) Branch A has greater variability since its coefficient of variation is higher than that of B.
(c)
$$\begin{gathered} \mathrm{Combined}\mathrm{mean}\left({\overline{X}}_{\mathrm{AB}}\right)=\frac{{\overline{X}}_{\mathrm{A}}{N}_{\mathrm{A}}+{\overline{X}}_B\mathit{}{N}_{\mathrm{B}}}{N_{\mathrm{A}}+N_{\mathrm{B}}} \\ =\frac{225\times 1000+230\times 500}{1000+500} \\ =\frac{225000+115000}{1500} \\ =\frac{340000}{1500} \\ =226.67 \end{gathered}$$
Thus, the average monthly wage for the factory as a whole is Rs 226.67.

(d)
$$\begin{gathered} {\sigma }_{\mathrm{A},\mathrm{B}}=\sqrt{\frac{N_{\mathrm{A}}{\sigma }_{\mathrm{A}}^2+N_{\mathrm{B}}{\sigma }_{\mathrm{B}}^2+N_{\mathrm{A}}{d}_{\mathrm{A}}^2+N_{\mathrm{B}}{d}_{\mathrm{B}}^2}{N_{\mathrm{A}}+N_{\mathrm{B}}}} \\ {d}_{\mathrm{A}}^2={\left({\overline{X}}_{\mathrm{A}}-{\overline{X}}_{\mathrm{A},\mathrm{B}}\right)}^2={\left(225-226.67\right)}^2 \\ =2.79 \\ {d}_{\mathrm{B}}^2={\left({\overline{X}}_{\mathrm{B}}-{\overline{X}}_{\mathrm{A},\mathrm{B}}\right)}^2={\left(230-226.67\right)}^2 \\ =11.09 \\ {\sigma }_{\mathrm{A},\mathrm{B}}=\sqrt{\frac{1000\times 4381.12+500\times 3863.86+1000\times 2.79+1000\times 11.09}{1000+500}} \\ =\sqrt{\frac{4381120+1931930+2790+11090}{1500}} \\ =\sqrt{\frac{6326930}{1500}}=\sqrt{4217.95} \\ \mathrm{Variance}={\left(SD\right)}^2 \\ ={\left(\sqrt{4217.95}\right)}^2 \\ =4217.95 \end{gathered}$$
Thus, the variance of wages of all the workers in the two branches- A and B taken together is Rs 4217.95.