Question

Calculate standard deviation of the following data:

Age in years (below)	10	20	30	40	50	60	70	80
No. of persons	15	30	53	75	100	110	115	125

Answer 1

Age (X)	Cumulative Frequency (c.f.)	Frequency (f)	Mid-Values (m)	fm	m2	fm2
0−10	15	15-0=15	5	75	25	375
10−20	30	30-15=15	15	225	225	3375
20−30	53	53-15=23	25	575	625	14375
30−40	75	75-23=22	35	770	1225	26950
40−50	100	100-75=25	45	1125	2025	50625
50−60	110	110-100=10	55	550	3025	30250
60−70	115	115-110=5	65	325	4225	21125
70−80	125	125-115=10	75	750	5625	56250
		Σf=125		Σfm=4395		Σfm2=203325

$$\begin{gathered} Mean,\overline{X}=\frac{\Sigma fm}{\Sigma f}=\frac{4395}{125}=35.16 \\ \mathrm{Standard}\mathrm{deviation}\left(\sigma \right)=\sqrt{\frac{\Sigma f{m}^2}{\Sigma f}-{\left(\overline{X}\right)}^2} \\ or,\left(\sigma \right)=\sqrt{\frac{203325}{125}-{\left(35.16\right)}^2} \\ or,\left(\sigma \right)=\sqrt{1626.6-1236.23} \\ or,\left(\sigma \right)=\sqrt{390.37} \\ \Rightarrow \left(\sigma \right)=19.76 \end{gathered}$$

Hence, standard deviation of the above series is 19.76 years.