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`

---

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

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

`=2430/(10*1340.9123)`

`=0.1812`

---

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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