1. Calculate Population Variance `(sigma^2)` 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
1	5	5	5
2	10	20	40
3	6	18	54
4	3	12	48
---	---	---	---
	`n=25`	`sum f*x=55`	`sum f*x^2=147`

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

`=55/25`

`=2.2`

---

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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2. Calculate Population Variance `(sigma^2)` from the following grouped data

X	Frequency
10	3
11	12
12	18
13	12
14	3

Solution:

`x` `(1)`	Frequency `(f)` `(2)`	`f*x` `(3)=(2)xx(1)`	`f*x^2=(f*x)xx(x)` `(4)=(3)xx(1)`
10	3	30	300
11	12	132	1452
12	18	216	2592
13	12	156	2028
14	3	42	588
---	---	---	---
	`n=48`	`sum f*x=576`	`sum f*x^2=6960`

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

`=576/48`

`=12`

---

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

`=(6960 - (576)^2/48)/48`

`=(6960 - 6912)/48`

`=48/48`

`=1`

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3. Calculate Population Variance `(sigma^2)` 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	9	27
4-6	4	5	20	100
6-8	2	7	14	98
8-10	1	9	9	81
---	---	---	---	---
--	`n = 10`	--	`sum f*x=52`	`sum f*x^2=306`

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

`=52/10`

`=5.2`

---

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`

---

4. Calculate Population Variance `(sigma^2)` from the following grouped data

Class	Frequency
0 - 2	5
2 - 4	16
4 - 6	13
6 - 8	7
8 - 10	5
10 - 12	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)`
0-2	5	1	5	5
2-4	16	3	48	144
4-6	13	5	65	325
6-8	7	7	49	343
8-10	5	9	45	405
10-12	4	11	44	484
---	---	---	---	---
--	`n = 50`	--	`sum f*x=256`	`sum f*x^2=1706`

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

`=256/50`

`=5.12`

---

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

`=(1706 - (256)^2/50)/50`

`=(1706 - 1310.72)/50`

`=395.28/50`

`=7.9056`

---

5. Calculate Population Variance `(sigma^2)` from the following grouped data

Class	Frequency
10 - 20	15
20 - 30	25
30 - 40	20
40 - 50	12
50 - 60	8
60 - 70	5
70 - 80	3

Solution:

Class `(1)`	Frequency `(f)` `(2)`	Mid value `(x)` `(3)`	`d=(x-A)/h=(x-45)/10` `A=45,h=10` `(4)`	`f*d` `(5)=(2)xx(4)`	`f*d^2` `(6)=(5)xx(4)`
10 - 20	15	15	-3	-45	135
20 - 30	25	25	-2	-50	100
30 - 40	20	35	-1	-20	20
40 - 50	12	45=A	0	0	0
50 - 60	8	55	1	8	8
60 - 70	5	65	2	10	20
70 - 80	3	75	3	9	27
---	---	---	---	---	---
	`n = 88`	-----	-----	`sum f*d=-88`	`sum f*d^2=310`

Mean `bar x = A + (sum fd)/n * h`

`=45 + -88/88 * 10`

`=45 + -1 * 10`

`=45 + -10`

`=35`

---

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

`=((310 - (-88)^2/88)/88) * 10^2`

`=((310 - 88)/88) * 100`

`=222/88 * 100`

`=2.5227 * 100`

`=252.2727`

---

6. Calculate Population Variance `(sigma^2)` from the following grouped data

Class	Frequency
20 - 25	110
25 - 30	170
30 - 35	80
35 - 40	45
40 - 45	40
45 - 50	35

Solution:

Class `(1)`	Frequency `(f)` `(2)`	Mid value `(x)` `(3)`	`d=(x-A)/h=(x-37.5)/5` `A=37.5,h=5` `(4)`	`f*d` `(5)=(2)xx(4)`	`f*d^2` `(6)=(5)xx(4)`
20 - 25	110	22.5	-3	-330	990
25 - 30	170	27.5	-2	-340	680
30 - 35	80	32.5	-1	-80	80
35 - 40	45	37.5=A	0	0	0
40 - 45	40	42.5	1	40	40
45 - 50	35	47.5	2	70	140
---	---	---	---	---	---
	`n = 480`	-----	-----	`sum f*d=-640`	`sum f*d^2=1930`

Mean `bar x = A + (sum fd)/n * h`

`=37.5 + -640/480 * 5`

`=37.5 + -1.3333 * 5`

`=37.5 + -6.6667`

`=30.8333`

---

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

`=((1930 - (-640)^2/480)/480) * 5^2`

`=((1930 - 853.3333)/480) * 25`

`=1076.6667/480 * 25`

`=2.2431 * 25`

`=56.0764`

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