Question

Given below are the diameters of circles (in mm) drawn in a design.

Diameter	33 - 36	37 - 40	41 - 44	45 -48	49 - 52
Number of circles	15	17	21	22	25

find the mean diameter of the circles.

• A. $43.5$
• B. $42.5$
• C. $45.5$
• D. $44.5$

Answer 1

Answer: (A)

Converting the given series into an exclusive series we prepare the table given below.

Class	Frequency$f_i$	Mid point$(x_i)$	$y_i=\frac{\left(x_i-42.5\right)}{4}$	$y_i^2$	$f_i{y}_i$	$f_i{y}_i^2$
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=A	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
	N=100				25	199

$\therefore A=42.5,h=4$,
$N=\sum f_i=100,\sum f_i{y}_i=25$
and $\sum f_i{y}_i^2=199$.
$\therefore \bar{x}=\left(A+\frac{\sum f_i{y}_i}{N}\times h\right)$
$\Rightarrow \bar{x}=\left(42.5+\frac{25}{100}\times 4\right)=43.5$
Thus the mean $=43.5$