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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.


Answer :


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

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