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Find Population skewness {85,96,76,108,85,80,100,85,70,95} [ Calculator, Method and examples ]

Solution:
Your problem `->` Population skewness {85,96,76,108,85,80,100,85,70,95}


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`
`(x - bar x)^4`
`= (x - 88)^4`
85 -3 `(85-88)=-3`
`(x - 88)`
 9 `(85-88)^2=9`
`(x - 88)^2`
 -27 `(85-88)^3=-27`
`(x - 88)^3`
 81 `(85-88)^4=81`
`(x - 88)^4`
96 8 `(96-88)=8`
`(x - 88)`
 64 `(96-88)^2=64`
`(x - 88)^2`
 512 `(96-88)^3=512`
`(x - 88)^3`
 4096 `(96-88)^4=4096`
`(x - 88)^4`
76 -12 `(76-88)=-12`
`(x - 88)`
 144 `(76-88)^2=144`
`(x - 88)^2`
 -1728 `(76-88)^3=-1728`
`(x - 88)^3`
 20736 `(76-88)^4=20736`
`(x - 88)^4`
108 20 `(108-88)=20`
`(x - 88)`
 400 `(108-88)^2=400`
`(x - 88)^2`
 8000 `(108-88)^3=8000`
`(x - 88)^3`
 160000 `(108-88)^4=160000`
`(x - 88)^4`
85 -3 `(85-88)=-3`
`(x - 88)`
 9 `(85-88)^2=9`
`(x - 88)^2`
 -27 `(85-88)^3=-27`
`(x - 88)^3`
 81 `(85-88)^4=81`
`(x - 88)^4`
80 -8 `(80-88)=-8`
`(x - 88)`
 64 `(80-88)^2=64`
`(x - 88)^2`
 -512 `(80-88)^3=-512`
`(x - 88)^3`
 4096 `(80-88)^4=4096`
`(x - 88)^4`
100 12 `(100-88)=12`
`(x - 88)`
 144 `(100-88)^2=144`
`(x - 88)^2`
 1728 `(100-88)^3=1728`
`(x - 88)^3`
 20736 `(100-88)^4=20736`
`(x - 88)^4`
85 -3 `(85-88)=-3`
`(x - 88)`
 9 `(85-88)^2=9`
`(x - 88)^2`
 -27 `(85-88)^3=-27`
`(x - 88)^3`
 81 `(85-88)^4=81`
`(x - 88)^4`
70 -18 `(70-88)=-18`
`(x - 88)`
 324 `(70-88)^2=324`
`(x - 88)^2`
 -5832 `(70-88)^3=-5832`
`(x - 88)^3`
 104976 `(70-88)^4=104976`
`(x - 88)^4`
95 7 `(95-88)=7`
`(x - 88)`
 49 `(95-88)^2=49`
`(x - 88)^2`
 343 `(95-88)^3=343`
`(x - 88)^3`
 2401 `(95-88)^4=2401`
`(x - 88)^4`
---------------
880012162430317284


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`








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