1. Calculate Sample Standard deviation `(S)` from the following mixed data

Class	Frequency
1	3
2	4
5	10
6 - 10	23
10 - 20	20
20 - 30	20
30 - 50	15
50 - 70	3
70 - 100	2

Solution:

Class `(1)`	Frequency `(f)` `(2)`	Mid value `(x)` `(3)`	`f*x` `(4)=(2)xx(3)`	`f*x^2=(f*x)xx(x)` `(5)=(4)xx(3)`
1	3	1	3	3
2	4	2	8	16
5	10	5	50	250
6 - 10	23	8	184	1472
10 - 20	20	15	300	4500
20 - 30	20	25	500	12500
30 - 50	15	40	600	24000
50 - 70	3	60	180	10800
70 - 100	2	85	170	14450
---	---	---	---	---
	`n = 100`	-----	`sum f*x=1995`	`sum f*x^2=67991`

Mean `bar x = (sum fx)/n`

`=1995/100`

`=19.95`

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Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`

`=sqrt((67991 - (1995)^2/100)/99)`

`=sqrt((67991 - 39800.25)/99)`

`=sqrt(28190.75/99)`

`=sqrt(284.7551)`

`=16.8747`

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2. Calculate Sample Standard deviation `(S)` from the following mixed data

Class	Frequency
2	1
3	2
4	2
5 - 9	8
10 - 14	15
15 - 19	8
20 - 29	4

Solution:

Class `(1)`	Frequency `(f)` `(2)`	Mid value `(x)` `(3)`	`f*x` `(4)=(2)xx(3)`	`f*x^2=(f*x)xx(x)` `(5)=(4)xx(3)`
2	1	2	2	4
3	2	3	6	18
4	2	4	8	32
5 - 9	8	7	56	392
10 - 14	15	12	180	2160
15 - 19	8	17	136	2312
20 - 29	4	24.5	98	2401
---	---	---	---	---
	`n = 40`	-----	`sum f*x=486`	`sum f*x^2=7319`

Mean `bar x = (sum fx)/n`

`=486/40`

`=12.15`

---

Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`

`=sqrt((7319 - (486)^2/40)/39)`

`=sqrt((7319 - 5904.9)/39)`

`=sqrt(1414.1/39)`

`=sqrt(36.259)`

`=6.0215`

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