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Method and examples
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Method
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Find Population Skewness, Kurtosis for ungrouped data
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Enter Data
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Standard deviation using Direct method or Assumed mean method
Method
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- `85,96,76,108,85,80,100,85,70,95`
- `3,13,11,15,5,4,2`
- `3,23,13,11,15,5,4,2`
- `69,66,67,69,64,63,65,68,72`
- `4,14,12,16,6,3,1,2,3`
- `73,70,71,73,68,67,69,72,76,71`
- `10,50,30,20,10,20,70,30`
- `21,23,19,17,12,15,15,17,17,19,23,23,21,23,25,25,21,19,19,19`
- `2,3,7,8,8,5,5,3,7,3,1,8,6,7,4,5,6,3,2,4,3,4,9,8,3`
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Decimal Place =
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log(x)/ln(x) Option for Geometric mean =
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Solution
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Solution provided by AtoZmath.com
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Population Skewness, Kurtosis for ungrouped data calculator
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 1. 85,96,76,108,85,80,100,85,70,95
2. 3,13,11,11,5,4,2
3. 3,23,13,11,15,3,5,4,2
4. 69,66,67,69,64,63,65,68,72
5. 4,14,12,16,6,3,1,2,3
6. 73,70,71,73,68,67,69,72,76,71
7. 10,50,30,20,10,20,70,30
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Example1. Calculate Population 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 |
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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