1. Calculate Population Skewness from the following grouped data

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

Solution:
Skewness :
Mean `bar x=(sum f x)/n`

`=55/25`

`=2.2`

`x` `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`(x-bar x)` `(5)`	`f*(x-bar x)` `(6)=(3)xx(5)`	`f*(x-bar x)^2` `(7)=(5)xx(6)`	`f*(x-bar x)^3` `(8)=(5)xx(7)`
0	1	0	-2.2	-2.2	4.84	-10.648
1	5	5	-1.2	-6	7.2	-8.64
2	10	20	-0.2	-2	0.4	-0.08
3	6	18	0.8	4.8	3.84	3.072
4	3	12	1.8	5.4	9.72	17.496
---	---	---	---	---	---	---
--	`n=25`	`sum f*x=55`	--	`=0`	`=26`	`=1.2`

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

`=sqrt(26/25)`

`=sqrt(1.04)`

`=1.0198`

---

Population Skewness `= (sum f*(x - bar x)^3)/(n*S^3)`

`=1.2/(25*(1.0198)^3)`

`=1.2/(25*1.0606)`

`=0.0453`

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2. Calculate Population Skewness from the following grouped data

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

Solution:
Skewness :
Mean `bar x=(sum f x)/n`

`=576/48`

`=12`

`x` `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`(x-bar x)` `(5)`	`f*(x-bar x)` `(6)=(3)xx(5)`	`f*(x-bar x)^2` `(7)=(5)xx(6)`	`f*(x-bar x)^3` `(8)=(5)xx(7)`
10	3	30	-2	-6	12	-24
11	12	132	-1	-12	12	-12
12	18	216	0	0	0	0
13	12	156	1	12	12	12
14	3	42	2	6	12	24
---	---	---	---	---	---	---
--	`n=48`	`sum f*x=576`	--	`=0`	`=48`	`=0`

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

`=sqrt(48/48)`

`=sqrt(1)`

`=1`

---

Population Skewness `= (sum f*(x - bar x)^3)/(n*S^3)`

`=0/(48*(1)^3)`

`=0/(48*1)`

`=0`

---

3. Calculate Population Skewness from the following grouped data

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

Solution:
Skewness :
Mean `bar x=(sum f x)/(sum f)`

`=52/10`

`=5.2`

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`(x-bar x)` `(5)`	`f*(x-bar x)` `(6)=(3)xx(5)`	`f*(x-bar x)^2` `(7)=(5)xx(6)`	`f*(x-bar x)^3` `(8)=(5)xx(7)`
2 - 4	3	3	9	-2.2	-6.6	14.52	-31.944
4 - 6	5	4	20	-0.2	-0.8	0.16	-0.032
6 - 8	7	2	14	1.8	3.6	6.48	11.664
8 - 10	9	1	9	3.8	3.8	14.44	54.872
---	---	---	---	---	---	---	---
--	--	`n=10`	`sum f*x=52`	--	`=0`	`=35.6`	`=34.56`

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

`=sqrt(35.6/10)`

`=sqrt(3.56)`

`=1.8868`

---

Population Skewness `= (sum f*(x - bar x)^3)/(n*S^3)`

`=34.56/(10*(1.8868)^3)`

`=34.56/(10*6.717)`

`=0.5145`

---

4. Calculate Population Skewness 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:
Skewness :
Mean `bar x=(sum f x)/(sum f)`

`=256/50`

`=5.12`

Class `(1)`	Mid value (`x`) `(2)`	`f` `(3)`	`f*x` `(4)=(2)xx(3)`	`(x-bar x)` `(5)`	`f*(x-bar x)` `(6)=(3)xx(5)`	`f*(x-bar x)^2` `(7)=(5)xx(6)`	`f*(x-bar x)^3` `(8)=(5)xx(7)`
0 - 2	1	5	5	-4.12	-20.6	84.872	-349.6726
2 - 4	3	16	48	-2.12	-33.92	71.9104	-152.45
4 - 6	5	13	65	-0.12	-1.56	0.1872	-0.0225
6 - 8	7	7	49	1.88	13.16	24.7408	46.5127
8 - 10	9	5	45	3.88	19.4	75.272	292.0554
10 - 12	11	4	44	5.88	23.52	138.2976	813.1899
---	---	---	---	---	---	---	---
--	--	`n=50`	`sum f*x=256`	--	`=0`	`=395.28`	`=649.6128`

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

`=sqrt(395.28/50)`

`=sqrt(7.9056)`

`=2.8117`

---

Population Skewness `= (sum f*(x - bar x)^3)/(n*S^3)`

`=649.6128/(50*(2.8117)^3)`

`=649.6128/(50*22.2281)`

`=0.5845`

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