Question

Calculate the mean, median and standard deviation of the following distribution:

Class-interval:	31-35	36-40	41-45	46-50	51-55	56-60	61-65	66-70
Frequency:	2	3	8	12	16	5	2	3

Answer 1

Class Interval	$f_i$	Midpoint $x_i$	$u_i=\frac{x_i-53}{4}$	ui 2	$f_i{u}_i$	$f_i{u_i}^2$
31−35	2	33	$-$5	25	$-$10	50
36−40	3	38	$-$3.75	14.06	$-$11.25	42.18
41−45	8	43	$-$2.5	6.25	$-$20	50
46−50	12	48	$-$1.25	1.56	$-$15	18.72
51−55	16	53	0	0	0	0
56−60	5	58	1.25	1.56	6.25	7.8
61−65	2	63	2.5	6.25	5	12.5
66−70	3	68	3.75	14.06	11.25	42.18
	N = 51				$\sum _{i=1}^n{f}_i{u}_i=-33.75$	$\sum _{i=1}^n{f}_i{u_i}^2=223.38$

$$\begin{gathered} \stackrel{}{\overline{X}}=a+h\left(\frac{{\displaystyle \sum _{i=1}^n}{f}_i{u}_i}{N}\right) \\ =53+4\left(\frac{-33.75}{51}\right) \\ =50.36 \end{gathered}$$

$$\begin{gathered} {\sigma }^2=h^2\left(\frac{{\displaystyle \sum _{i=1}^n}{f}_i{u_i}^2}{N}-{\left(\frac{{\displaystyle \sum _{i=1}^n}{f}_i{u}_i}{N}\right)}^2\right) \\ =16\left(\frac{223.38}{51}-\frac{1139.06}{2601}\right) \\ =63.07 \\ \sigma =\sqrt{63.07} \\ =7.94 \end{gathered}$$

$f_i$	$CF$ (Cumulative frequency)
2	2
3	5
8	13
12	25
16	41
5	46
2	48
3	51

$$\begin{gathered} \sum f_i=51=N \\ \frac{N}{2}=25.5 \end{gathered}$$

Median class interval is 51−55.
$$\begin{gathered} L=51 \\ F=25 \\ f=16 \\ h=4 \end{gathered}$$

$\mathrm{Median}=L+\frac{{\displaystyle \frac{N}{2}}-F}{f}\times h$

$$\begin{gathered} =51+\frac{25.5-25}{16}\times 4 \\ =51+\frac{0.5}{4} \\ =51.125 \end{gathered}$$