Formula

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

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

3. Kurtosis `= (sum(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:
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)^2` `(6)=(3)xx(5)`	`f*(x-bar x)^3` `(7)=(5)xx(6)`
2 - 4	3 `3=(2+4)/2`	3	9 `9=3xx3` `(4)=(2)xx(3)`	-2.2 `|3-5.2|=-2.2` `|x - 5.2|`	14.52 `14.52=3xx-2.2xx-2.2` `(6)=(2)xx(5)`	-31.944 `-31.944=14.52xx-2.2` `(7)=(2)xx(6)`
4 - 6	5 `5=(4+6)/2`	4	20 `20=4xx5` `(4)=(2)xx(3)`	-0.2 `|5-5.2|=-0.2` `|x - 5.2|`	0.16 `0.16=4xx-0.2xx-0.2` `(6)=(2)xx(5)`	-0.032 `-0.032=0.16xx-0.2` `(7)=(2)xx(6)`
6 - 8	7 `7=(6+8)/2`	2	14 `14=2xx7` `(4)=(2)xx(3)`	1.8 `|7-5.2|=1.8` `|x - 5.2|`	6.48 `6.48=2xx1.8xx1.8` `(6)=(2)xx(5)`	11.664 `11.664=6.48xx1.8` `(7)=(2)xx(6)`
8 - 10	9 `9=(8+10)/2`	1	9 `9=1xx9` `(4)=(2)xx(3)`	3.8 `|9-5.2|=3.8` `|x - 5.2|`	14.44 `14.44=1xx3.8xx3.8` `(6)=(2)xx(5)`	54.872 `54.872=14.44xx3.8` `(7)=(2)xx(6)`
---	---	---	---	---	---	---
--	--	`n=10`	`sum f*x=52`	`--	`=35.6`	`=34.56`

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

`=sqrt(35.6/9)`

`=sqrt(3.9556)`

`=1.9889`

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Skewness `= (sum(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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Kurtosis `= (sum(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:
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)^2` `(6)=(3)xx(5)`	`f*(x-bar x)^3` `(7)=(5)xx(6)`
0	1	0 `0=1xx0` `(4)=(2)xx(3)`	-2.2 `|0-2.2|=-2.2` `|x - 2.2|`	4.84 `4.84=1xx-2.2xx-2.2` `(6)=(2)xx(5)`	-10.648 `-10.648=4.84xx-2.2` `(7)=(2)xx(6)`
1	5	5 `5=5xx1` `(4)=(2)xx(3)`	-1.2 `|1-2.2|=-1.2` `|x - 2.2|`	7.2 `7.2=5xx-1.2xx-1.2` `(6)=(2)xx(5)`	-8.64 `-8.64=7.2xx-1.2` `(7)=(2)xx(6)`
2	10	20 `20=10xx2` `(4)=(2)xx(3)`	-0.2 `|2-2.2|=-0.2` `|x - 2.2|`	0.4 `0.4=10xx-0.2xx-0.2` `(6)=(2)xx(5)`	-0.08 `-0.08=0.4xx-0.2` `(7)=(2)xx(6)`
3	6	18 `18=6xx3` `(4)=(2)xx(3)`	0.8 `|3-2.2|=0.8` `|x - 2.2|`	3.84 `3.84=6xx0.8xx0.8` `(6)=(2)xx(5)`	3.072 `3.072=3.84xx0.8` `(7)=(2)xx(6)`
4	3	12 `12=3xx4` `(4)=(2)xx(3)`	1.8 `|4-2.2|=1.8` `|x - 2.2|`	9.72 `9.72=3xx1.8xx1.8` `(6)=(2)xx(5)`	17.496 `17.496=9.72xx1.8` `(7)=(2)xx(6)`
---	---	---	---	---	---
--	`n=25`	`sum f*x=55`	`--	`=26`	`=1.2`

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

`=sqrt(26/24)`

`=sqrt(1.0833)`

`=1.0408`

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Skewness `= (sum(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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Kurtosis `= (sum(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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