Question

Find out the arithmetic mean and standard deviation from the following data:

Variable	5−10	10−15	15−20	20−25	25−30	30−35
Frequency	2	9	29	54	11	5

Answer 1

Variable (X)	Frequency (f)	Mid-Values (m)	fm	m2	fm2
5−10	2	7.5	15	56.25	112.5
10−15	9	12.5	112.5	156.25	1406.25
15−20	29	17.5	507.5	306.25	8881.25
20−25	54	22.5	1215.0	506.25	27337.5
25−30	11	27.5	302.5	756.25	8318.75
30−35	5	32.5	162.5	1056.25	5281.25
	Σf=110		Σfm=2315		Σfm2=51337.5

$$\begin{gathered} \mathrm{Mean},\overline{X}=\frac{\Sigma fm}{\Sigma f}=\frac{2315}{110}=21.05 \\ \mathrm{Standard}\mathrm{Deviation},\left(\sigma \right)=\sqrt{\frac{\Sigma f{m}^2}{\Sigma f}-{\left(\overline{X}\right)}^2} \\ or,\left(\sigma \right)=\sqrt{\frac{51337.5}{110}-{\left(21.05\right)}^2} \\ or,\left(\sigma \right)=\sqrt{466.70-443.10} \\ or,\left(\sigma \right)=\sqrt{23.6} \\ \Rightarrow \left(\sigma \right)=4.86 \end{gathered}$$