1. Calculate Population Variance `(sigma^2)` from the following data
`3,13,11,15,5,4,2`

Solution:

`x`	`x^2`
3	9
13	169
11	121
15	225
5	25
4	16
2	4
---	---
`sum x=53`	`sum x^2=569`

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

`=(3+13+11+15+5+4+2)/7`

`=53/7`

`=7.5714`

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

`=(569 - (53)^2/7)/7`

`=(569 - 401.2857)/7`

`=167.7143/7`

`=23.9592`

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2. Calculate Population Variance `(sigma^2)` from the following data
`10,50,30,20,10,20,70,30`

Solution:

`x`	`x - bar x = x - 30`	`(x - bar x)^2`
10	-20	400
50	20	400
30	0	0
20	-10	100
10	-20	400
20	-10	100
70	40	1600
30	0	0
---	---	---
`sum x=240`	`sum (x - bar x)=0`	`sum (x - bar x)^2=3000`

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

`=(10+50+30+20+10+20+70+30)/8`

`=240/8`

`=30`

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

`=3000/8`

`=375`

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3. Calculate Population Variance `(sigma^2)` from the following data
`85,96,76,108,85,80,100,85,70,95`

Solution:

`x`	`x - bar x = x - 88`	`(x - bar x)^2`
85	-3	9
96	8	64
76	-12	144
108	20	400
85	-3	9
80	-8	64
100	12	144
85	-3	9
70	-18	324
95	7	49
---	---	---
`sum x=880`	`sum (x - bar x)=0`	`sum (x - bar x)^2=1216`

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

`=(85+96+76+108+85+80+100+85+70+95)/10`

`=880/10`

`=88`

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

`=1216/10`

`=121.6`

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