Question

The diameter of circles (in mm) drawn in the design are given below.

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

Calculate the standard deviation.

Answer 1

Assumed mean = 42.5

Diameters	Number of Circles (fi)	Mid value xi	d = xi − A = xi − 42.5	fidi	fidi2
33 − 36	15	34.5	− 8	− 120	960
37 − 40	17	38.5	− 4	− 68	272
41 − 44	21	42.5	0	0	0
45 − 48	22	46.5	4	88	352
49 − 52	25	50.5	8	200	1600
	`sumf_"i"` = 100			`sumf_"i""d"_"i"` = 100	`sumf_"i""d"_"i"^2` = 3184

Here `sumf_"i"` = 100, `sumf_"i""d"_"i"` = 100, `sumf_"i""d"_"i"^2` = 3184

Standard deviation (σ) = `sqrt((sumf_"i""d"_"i"^2)/(sumf_"i") - ((sumf_"i""d"_"i")/(sumf_"i"))^2`

= `sqrt(3184/100 - (100/100)^2`

= `sqrt(31.84 - 1)`

σ = `sqrt(30.84)`

Standard deviation (σ) = 5.55