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# Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in Rs.) 100-120 120-140 140-160 160-180 180-200 Number of workers 12 14 8 6 10 Find the mean daily wages of the workers of the factory by using an appropriate method.

The midpoint of the given interval is found by formula: $$Midpoint \ (x_i) \ = \ \frac{upper \ limit \ + \ lower \ limit}{2}$$

In this case, the value of mid-point ( $$x_i$$) is very large,so let us assume the mean value, A = 150 and class interval is h = 20.

So, $$u_i \ = \ \frac{x_i \ - \ A}{h} \$$
$$\Rightarrow\ u_i \ = \ \frac{x_i \ - \ 150}{20}$$

 Daily wages (Class interval) Number of workers (frequency ($$f_i$$) ) Mid-point ( $$x_i$$ ) $$u_i \ = \ \frac{x_i \ - \ 150}{20}$$ $$f_i u_i$$ 100-120 12 110 -2 -24 120-140 14 130 -1 -14 140-160 8 150 0 0 160-180 6 170 1 6 180-200 10 190 2 20 Total $$\sum f_i \ = \ 50$$ $$\sum f_i u_i \ = \ -12$$

So, the formula to find out the mean is : $$\ \overline{x} \ = \ A \ + h \frac{ \sum \ f_i x_i}{ \sum f_i}$$
$$= \ 150 \ + \ 20 \ × \ \frac{-12}{50}$$
$$= \ 150 \ - \ 4.8$$
$$= \ 145.20$$
$$\therefore$$ Mean daily wage of the workers = Rs. 145.20