Age(in years)	No of workers	Mid-point (x)	\(d = x - A\)	\(d^{2}\)	\(fd^{2}\)	\(fd\)
17 - 21	12	19	-20	400	4800	-240
22 - 26	24	24	-15	225	5400	-360
27 - 31	30	29	-10	100	3000	-300
32 - 36	37	34	-5	25	925	-185
37 - 41	45	39	0	0	0	0
42 - 46	25	44	5	25	625	125
47 - 51	10	49	10	100	1000	100
52 - 56	7	54	15	225	1575	105
Total	190				17325	-755

(a)(i) Mean, \(\bar{x}\) = \(A + \frac{\sum fd}{\sum f}\)
= \(39 + \frac{-755}{190}\)
= \(39 - 3.974 = 35.026\)
(ii) Standard deviation \(SD = \sqrt{\frac{\sum fd^{2}}{N} - (\frac{\sum fd}{N})^{2}}\)
= \(\sqrt{\frac{17325}{190} - (\frac{-755}{190})^{2}}\)
= \(\sqrt{91.184 - (-3.974)^{2}}\)
\(\sqrt{91. 184 - 15.794} = \sqrt{75.39}\)
= \(8.68\)
(b) Total number of workers = 190
No of workers who are at most 36 years old = 12 + 24 + 30 + 37 = 103
p(award to a person at most 36 years old) = \(\frac{103}{190}\)
= 0.542