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 grouped data

Class	Frequency
2 - 4	3
4 - 6	4
6 - 8	2
8 - 10	1

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)`
2-4	3	3 `3=(2+4)/2`	9 `9=3xx3` `(4)=(2)xx(3)`	27 `27=9xx3` `(5)=(4)xx(3)`
4-6	4	5 `5=(4+6)/2`	20 `20=4xx5` `(4)=(2)xx(3)`	100 `100=20xx5` `(5)=(4)xx(3)`
6-8	2	7 `7=(6+8)/2`	14 `14=2xx7` `(4)=(2)xx(3)`	98 `98=14xx7` `(5)=(4)xx(3)`
8-10	1	9 `9=(8+10)/2`	9 `9=1xx9` `(4)=(2)xx(3)`	81 `81=9xx9` `(5)=(4)xx(3)`
---	---	---	---	---
--	`n = 10`	--	`sum f*x=52`	`sum f*x^2=306`

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

`=52/10`

`=5.2`

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

`=(306 - (52)^2/10)/10`

`=(306 - 270.4)/10`

`=35.6/10`

`=3.56`

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

`=sqrt((306 - (52)^2/10)/10)`

`=sqrt((306 - 270.4)/10)`

`=sqrt(35.6/10)`

`=sqrt(3.56)`

`=1.8868`

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

`=1.8868/5.2 * 100 %`

`=36.28 %`

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2. Calculate Population Variance `(sigma^2)`, Population Standard deviation `(sigma)`, Population Coefficient of Variation from the following grouped data

X	Frequency
0	1
1	5
2	10
3	6
4	3

Solution:

`x` `(1)`	Frequency `(f)` `(2)`	`f*x` `(3)=(2)xx(1)`	`f*x^2=(f*x)xx(x)` `(4)=(3)xx(1)`
0	1	0 `0=1xx0` `(3)=(2)xx(1)`	0 `0=0xx0` `(4)=(3)xx(1)`
1	5	5 `5=5xx1` `(3)=(2)xx(1)`	5 `5=5xx1` `(4)=(3)xx(1)`
2	10	20 `20=10xx2` `(3)=(2)xx(1)`	40 `40=20xx2` `(4)=(3)xx(1)`
3	6	18 `18=6xx3` `(3)=(2)xx(1)`	54 `54=18xx3` `(4)=(3)xx(1)`
4	3	12 `12=3xx4` `(3)=(2)xx(1)`	48 `48=12xx4` `(4)=(3)xx(1)`
---	---	---	---
	`n=25`	`sum f*x=55`	`sum f*x^2=147`

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

`=55/25`

`=2.2`

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

`=(147 - (55)^2/25)/25`

`=(147 - 121)/25`

`=26/25`

`=1.04`

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

`=sqrt((147 - (55)^2/25)/25)`

`=sqrt((147 - 121)/25)`

`=sqrt(26/25)`

`=sqrt(1.04)`

`=1.0198`

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

`=1.0198/2.2 * 100 %`

`=46.35 %`

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