Formula

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

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

3. Kurtosis `= (sum f*(x - bar x)^4)/(n*S^4)`

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Examples
1. Calculate Population Skewness, Population Kurtosis from the following grouped data

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

Solution:
Skewness,Kurtosis :
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)`	`f*(x-bar x)^4` `(9)=(5)xx(8)`
2 - 4	3	3	9	-2.2	-6.6	14.52	-31.944	70.2768
4 - 6	5	4	20	-0.2	-0.8	0.16	-0.032	0.0064
6 - 8	7	2	14	1.8	3.6	6.48	11.664	20.9952
8 - 10	9	1	9	3.8	3.8	14.44	54.872	208.5136
---	---	---	---	---	---	---	---	---
--	--	`n=10`	`sum f*x=52`	--	`=0`	`=35.6`	`=34.56`	`=299.792`

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

`=sqrt(35.6/10)`

`=sqrt(3.56)`

`=1.8868`

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

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

`=34.56/(10*6.717)`

`=0.5145`

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Population Kurtosis `= (sum f*(x - bar x)^4)/(n*S^4)`

`=299.792/(10*(1.8868)^4)`

`=299.792/(10*12.6736)`

`=2.3655`

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2. Calculate Population Skewness, Population Kurtosis from the following grouped data

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

Solution:
Skewness,Kurtosis :
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)`	`f*(x-bar x)^4` `(9)=(5)xx(8)`
0	1	0	-2.2	-2.2	4.84	-10.648	23.4256
1	5	5	-1.2	-6	7.2	-8.64	10.368
2	10	20	-0.2	-2	0.4	-0.08	0.016
3	6	18	0.8	4.8	3.84	3.072	2.4576
4	3	12	1.8	5.4	9.72	17.496	31.4928
---	---	---	---	---	---	---	---
--	`n=25`	`sum f*x=55`	--	`=0`	`=26`	`=1.2`	`=67.76`

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

`=sqrt(26/25)`

`=sqrt(1.04)`

`=1.0198`

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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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Population Kurtosis `= (sum f*(x - bar x)^4)/(n*S^4)`

`=67.76/(25*(1.0198)^4)`

`=67.76/(25*1.0816)`

`=2.5059`

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