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

Solution:
Skewness :
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`
85	-3	9	-27
96	8	64	512
76	-12	144	-1728
108	20	400	8000
85	-3	9	-27
80	-8	64	-512
100	12	144	1728
85	-3	9	-27
70	-18	324	-5832
95	7	49	343
---	---	---	---
`880`	`0`	`1216`	`2430`

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

`=sqrt(1216/9)`

`=sqrt(135.1111)`

`=11.6237`

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

`=2430/(9*(11.6237)^3)`

`=2430/(9*1570.4951)`

`=0.1719`

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2. Calculate Sample Skewness from the following data
`10,50,30,20,10,20,70,30`

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

`=(10+50+30+20+10+20+70+30)/8`

`=240/8`

`=30`

`x`	`(x - bar x)` `=(x-30)`	`(x - bar x)^2` `=(x-30)^2`	`(x - bar x)^3` `=(x-30)^3`
10	-20	400	-8000
50	20	400	8000
30	0	0	0
20	-10	100	-1000
10	-20	400	-8000
20	-10	100	-1000
70	40	1600	64000
30	0	0	0
---	---	---	---
`240`	`0`	`3000`	`54000`

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

`=sqrt(3000/7)`

`=sqrt(428.5714)`

`=20.702`

---

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

`=54000/(7*(20.702)^3)`

`=54000/(7*8872.2715)`

`=0.8695`

---

3. Calculate Sample Skewness from the following data
`10,50,30,20,10,20,70,30`

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

`=(10+50+30+20+10+20+70+30)/8`

`=240/8`

`=30`

`x`	`(x - bar x)` `=(x-30)`	`(x - bar x)^2` `=(x-30)^2`	`(x - bar x)^3` `=(x-30)^3`
10	-20	400	-8000
50	20	400	8000
30	0	0	0
20	-10	100	-1000
10	-20	400	-8000
20	-10	100	-1000
70	40	1600	64000
30	0	0	0
---	---	---	---
`240`	`0`	`3000`	`54000`

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

`=sqrt(3000/7)`

`=sqrt(428.5714)`

`=20.702`

---

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

`=54000/(7*(20.702)^3)`

`=54000/(7*8872.2715)`

`=0.8695`

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