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Find Sample skewness {{61,5},{64,18},{67,42},{70,27},{73,8}}

Solution:
Your problem `->` Sample skewness {{61,5},{64,18},{67,42},{70,27},{73,8}}


Mean `bar x=(sum f x)/n`

`=6745/100`

`=67.45`

`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)`
`f*(x-bar x)^4`
`(8)=(5)xx(7)`
615 305 `305=5xx61`
`(4)=(2)xx(3)`
 -6.45 `|61-67.45|=-6.45`
`|x - 67.45|`
 208.0125 `208.0125=5xx-6.45xx-6.45`
`(6)=(2)xx(5)`
 -1341.6806 `-1341.6806=208.0125xx-6.45`
`(7)=(2)xx(6)`
 8653.84 `8653.84=-1341.6806xx-6.45`
`(8)=(2)xx(7)`
6418 1152 `1152=18xx64`
`(4)=(2)xx(3)`
 -3.45 `|64-67.45|=-3.45`
`|x - 67.45|`
 214.245 `214.245=18xx-3.45xx-3.45`
`(6)=(2)xx(5)`
 -739.1453 `-739.1453=214.245xx-3.45`
`(7)=(2)xx(6)`
 2550.0511 `2550.0511=-739.1453xx-3.45`
`(8)=(2)xx(7)`
6742 2814 `2814=42xx67`
`(4)=(2)xx(3)`
 -0.45 `|67-67.45|=-0.45`
`|x - 67.45|`
 8.505 `8.505=42xx-0.45xx-0.45`
`(6)=(2)xx(5)`
 -3.8273 `-3.8273=8.505xx-0.45`
`(7)=(2)xx(6)`
 1.7223 `1.7223=-3.8273xx-0.45`
`(8)=(2)xx(7)`
7027 1890 `1890=27xx70`
`(4)=(2)xx(3)`
 2.55 `|70-67.45|=2.55`
`|x - 67.45|`
 175.5675 `175.5675=27xx2.55xx2.55`
`(6)=(2)xx(5)`
 447.6971 `447.6971=175.5675xx2.55`
`(7)=(2)xx(6)`
 1141.6277 `1141.6277=447.6971xx2.55`
`(8)=(2)xx(7)`
738 584 `584=8xx73`
`(4)=(2)xx(3)`
 5.55 `|73-67.45|=5.55`
`|x - 67.45|`
 246.42 `246.42=8xx5.55xx5.55`
`(6)=(2)xx(5)`
 1367.631 `1367.631=246.42xx5.55`
`(7)=(2)xx(6)`
 7590.352 `7590.352=1367.631xx5.55`
`(8)=(2)xx(7)`
---------------------
--`n=100``sum f*x=6745``--`=852.75``=-269.325``=19937.5931`


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

`=sqrt(852.75/99)`

`=sqrt(8.6136)`

`=2.9349`



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

`=-269.325/(99*(2.9349)^3)`

`=-269.325/(99*25.2802)`

`=-0.1076`








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