Question

The diameters of circles (in $mm$) drawn in a design are given below :

Diameters	No. of circles
$33-36$	$15$
$37-40$	$17$
$41-44$	$21$
$45-48$	$22$
$49-52$	$25$

Answer 1

Since, the given frequency distribution is not continuous, so to make it continuous, we will subtract $0.5$ from lower limit and add $0.5$ to the upper limit of each class.

Class	$f_i$	$x_i$	$u_i=\frac{x_i-A}{h}$	$u_i^2$	$f_i{u}_i$	$f_i{u}_i^2$
32.5-36.5	15	34.5	-2	4	-30	60
36.5-40.5	17	38.5	-1	1	-17	17
40.5-44.5	21	42.5	0	0	0	0
44.5-48.5	22	46.5	1	1	22	22
48.5-52.5	25	50.5	2	4	50	100
Total	10 0				25	199

Here, $N=100,h=4$
Let the assumed mean $A=42.5$
Mean $\overline{x}=A+\frac{{\sum }_{i=1}^5{f}_i{u}_i}{N}\times h$

$=42.5+\frac{25}{100}\times 4=43.5$

Variance $\left({\sigma }^2\right)=\frac{h^2}{N^2}[N\sum _{i=1}^5{f}_i{u}_i^2-{\left(\sum _{i=1}^5{f}_i{u}_i\right)}^2]$
=$\frac{16}{10000}[100\times 199-{\left(25\right)}^2]$
$=\frac{16}{10000}[19900-625]$
$=\frac{16}{10000}\times 19275$
$=30.84$

$\therefore$ Standard deviation $\left(\sigma \right)=5.55$