Formula

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

2. Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

3. Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`

4. Coefficient of Variation (Population) `=sigma / bar x * 100 %`

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Examples
1. Calculate Population Variance `(sigma^2)`, Population Standard deviation `(sigma)`, Population Coefficient of Variation 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 `1=1`	3 `3=3xx1` `(4)=(2)xx(3)`	3 `3=3xx1` `(5)=(4)xx(3)`
2	4	2 `2=2`	8 `8=4xx2` `(4)=(2)xx(3)`	16 `16=8xx2` `(5)=(4)xx(3)`
5	10	5 `5=5`	50 `50=10xx5` `(4)=(2)xx(3)`	250 `250=50xx5` `(5)=(4)xx(3)`
6 - 10	23	8 `8=(6+10)/2`	184 `184=23xx8` `(4)=(2)xx(3)`	1472 `1472=184xx8` `(5)=(4)xx(3)`
10 - 20	20	15 `15=(10+20)/2`	300 `300=20xx15` `(4)=(2)xx(3)`	4500 `4500=300xx15` `(5)=(4)xx(3)`
20 - 30	20	25 `25=(20+30)/2`	500 `500=20xx25` `(4)=(2)xx(3)`	12500 `12500=500xx25` `(5)=(4)xx(3)`
30 - 50	15	40 `40=(30+50)/2`	600 `600=15xx40` `(4)=(2)xx(3)`	24000 `24000=600xx40` `(5)=(4)xx(3)`
50 - 70	3	60 `60=(50+70)/2`	180 `180=3xx60` `(4)=(2)xx(3)`	10800 `10800=180xx60` `(5)=(4)xx(3)`
70 - 100	2	85 `85=(70+100)/2`	170 `170=2xx85` `(4)=(2)xx(3)`	14450 `14450=170xx85` `(5)=(4)xx(3)`
---	---	---	---	---
	`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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Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

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

`=(67991 - 39800.25)/100`

`=28190.75/100`

`=281.9075`

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Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`

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

`=sqrt((67991 - 39800.25)/100)`

`=sqrt(28190.75/100)`

`=sqrt(281.9075)`

`=16.7901`

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Coefficient of Variation (Population) `=sigma / bar x * 100 %`

`=16.7901/19.95 * 100 %`

`=84.16 %`

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2. Calculate Population Variance `(sigma^2)`, Population Standard deviation `(sigma)`, Population Coefficient of Variation 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`

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Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

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

`=(7319 - 5904.9)/40`

`=1414.1/40`

`=35.3525`

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Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`

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

`=sqrt((7319 - 5904.9)/40)`

`=sqrt(1414.1/40)`

`=sqrt(35.3525)`

`=5.9458`

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Coefficient of Variation (Population) `=sigma / bar x * 100 %`

`=5.9458/12.15 * 100 %`

`=48.94 %`

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