Formula

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

2. Skewness `= (sum f*(x - bar x)^3)/((n-1)*S^3)`

3. Kurtosis `= (sum f*(x - bar x)^4)/((n-1)*S^4)`

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Examples
1. Calculate Sample Skewness, Sample 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`

Sample Standard deviation `S = sqrt((sum f*(x - bar x)^2)/(n-1))`

`=sqrt(35.6/9)`

`=sqrt(3.9556)`

`=1.9889`

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Sample Skewness `= (sum f*(x - bar x)^3)/((n-1)*S^3)`

`=34.56/(9*(1.9889)^3)`

`=34.56/(9*7.867)`

`=0.4881`

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Sample Kurtosis `= (sum f*(x - bar x)^4)/((n-1)*S^4)`

`=299.792/(9*(1.9889)^4)`

`=299.792/(9*15.6464)`

`=2.1289`

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2. Calculate Sample Skewness, Sample 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`

Sample Standard deviation `S = sqrt((sum f*(x - bar x)^2)/(n-1))`

`=sqrt(26/24)`

`=sqrt(1.0833)`

`=1.0408`

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Sample Skewness `= (sum f*(x - bar x)^3)/((n-1)*S^3)`

`=1.2/(24*(1.0408)^3)`

`=1.2/(24*1.1276)`

`=0.0443`

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Sample Kurtosis `= (sum f*(x - bar x)^4)/((n-1)*S^4)`

`=67.76/(24*(1.0408)^4)`

`=67.76/(24*1.1736)`

`=2.4057`

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