Formula

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

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

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

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Examples
1. Calculate Population Skewness, Population Kurtosis from the following data
`85,96,76,108,85,80,100,85,70,95`

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

`=(85+96+76+108+85+80+100+85+70+95)/10`

`=880/10`

`=88`

`x`	`(x - bar x)` `=(x-88)`	`(x - bar x)^2` `=(x-88)^2`	`(x - bar x)^3` `=(x-88)^3`	`(x - bar x)^4` `=(x-88)^4`
85	-3	9	-27	81
96	8	64	512	4096
76	-12	144	-1728	20736
108	20	400	8000	160000
85	-3	9	-27	81
80	-8	64	-512	4096
100	12	144	1728	20736
85	-3	9	-27	81
70	-18	324	-5832	104976
95	7	49	343	2401
---	---	---	---	---
`880`	`0`	`1216`	`2430`	`317284`

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

`=sqrt(1216/10)`

`=sqrt(121.6)`

`=11.0272`

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

`=2430/(10*(11.0272)^3)`

`=2430/(10*1340.9123)`

`=0.1812`

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

`=317284/(10*(11.0272)^4)`

`=317284/(10*14786.56)`

`=2.1458`

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2. Calculate Population Skewness, Population Kurtosis from the following data
`69,66,67,69,64,63,65,68,72`

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

`=(69+66+67+69+64+63+65+68+72)/9`

`=603/9`

`=67`

`x`	`(x - bar x)` `=(x-67)`	`(x - bar x)^2` `=(x-67)^2`	`(x - bar x)^3` `=(x-67)^3`	`(x - bar x)^4` `=(x-67)^4`
69	2	4	8	16
66	-1	1	-1	1
67	0	0	0	0
69	2	4	8	16
64	-3	9	-27	81
63	-4	16	-64	256
65	-2	4	-8	16
68	1	1	1	1
72	5	25	125	625
---	---	---	---	---
`603`	`0`	`64`	`42`	`1012`

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

`=sqrt(64/9)`

`=sqrt(7.1111)`

`=2.6667`

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

`=42/(9*(2.6667)^3)`

`=42/(9*18.963)`

`=0.2461`

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

`=1012/(9*(2.6667)^4)`

`=1012/(9*50.5679)`

`=2.2236`

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