Question

The time taken by 50 students to complete a 100-meter race are given below. Find its standard deviation

Time taken (sec)	8.5 − 9.5	9.5 − 10.5	10.5 − 11.5	11.5 − 12.5	12.5 − 13.5
Number of students	6	8	17	10	9

Answer 1

Assumed mean = 11

Time taken	Number of students fi	mid value xi	di = xi − A = xi − 11	fidi	fidi2
8.5 − 9.5	6	9	− 2	− 12	24
9.5 − 10.5	8	10	− 1	− 8	8
10.5 − 11.5	17	11	0	0	0
11.5 − 12.5	10	12	1	10	10
12.5 − 13.5	9	13	2	18	36
	`sumf_"i"` = 50			`sumf_"i""d"_"i"` = 8	`sumf_"i""d"_"i"^2` = 78

Here N = 50, `sumf_"i""d"_"i"` = 8, `sumf_"i""d"_"i"^2` = 78

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

= `sqrt(78/50 - (8/50)^2`

= `sqrt(1.56 - (0.16)^2`

= `sqrt(1.56 - 0.025)`

= `sqrt(1.56 - 0.03)`

= `sqrt(1.53)`

= 1.236

⇒ 1.24

∴ Standard deviation (σ) = 1.24