Question

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

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

The standard deviation and mean diameter of the circles respectively are

• A. $5.55,44.5$
• B. $5.45,43.5$
• C. $5.55,43.5$
• D. $5.45,44.5$

Answer 1

Answer: (C)

Class	$f_i$	Mid value$(x_i)$	Deviation from mean $d_i=\frac{x_i-42.5}{4}A=42.5,h=4$	$d_i^2$	$f_i{d}_i$	$f_i{d}_i^2$
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	100				25	199

Mean $(\bar{x})=A+\frac{\Sigma f{d}_i}{\Sigma f_i}\times h$
$=42.5+\frac{25}{100}\times 4$
$=42.5+1=43.5$
Standard deviation $(\sigma )=\sqrt{\frac{\Sigma f{d}_i^2}{\Sigma f_i}-{\left(\frac{\Sigma f{d}_i}{\Sigma f_i}\right)}^2}\times h$
$=\sqrt{\frac{199}{100}-{\left(\frac{25}{100}\right)}^2}\times 4$
$=\sqrt{\frac{199\times 100-625}{100\times 100}}\times 4$
$=\frac{4}{100}\sqrt{19900-625}$
$=\frac{1}{25}\times \sqrt{19275}=\frac{138.83}{25}=5.55$