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

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

Calculate the standard deviation and mean diameter of the circles.

[ Hint : First make the data continuous by making the classes as $32.5-36.5,36.5-40.5,$ $40.5-44.5,44.5-48.5,48.5-52.5 $ and then proceed.]

here, $N=100,$ $h=4$

Let the assumed mean, $A,$ be $42.5$

Mean,   $\bar x = A + \frac{{\sum\limits_{i = 1}^5 {{f_i}{y_i}} }}{N} \times h$

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

Variance,  $\left( {{\sigma ^2}} \right) = \frac{{{h^2}}}{{{N^2}}}\left[ {N\sum\limits_{i = 1}^5 {{f_i}{y_i}^2 - {{\left( {\sum\limits_{i = 1}^5 {{f_i}{y_i}} } \right)}^2}} } \right]$

$=\frac{16}{10000}\left[100 \times 199-(25)^{2}\right]$

$=\frac{16}{10000}[19900-625]$

$=\frac{16}{10000} \times 19275$

$=30.84$

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