1. Calculate Population Skewness from the following grouped data
Solution:
Skewness :
Mean `bar x=(sum f x)/n`
`=55/25`
`=2.2`
`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)` |
| 0 | 1 | 0 | -2.2 | 4.84 | -10.648 |
| 1 | 5 | 5 | -1.2 | 7.2 | -8.64 |
| 2 | 10 | 20 | -0.2 | 0.4 | -0.08 |
| 3 | 6 | 18 | 0.8 | 3.84 | 3.072 |
| 4 | 3 | 12 | 1.8 | 9.72 | 17.496 |
| --- | --- | --- | --- | --- | --- |
| -- | `n=25` | `sum f*x=55` | `-- | `=26` | `=1.2` |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(26/25)`
`=sqrt(1.04)`
`=1.0198`
Population Skewness `= (sum(x - bar x)^3)/(n*S^3)`
`=1.2/(25*(1.0198)^3)`
`=1.2/(25*1.0606)`
`=0.0453`
2. Calculate Population Skewness from the following grouped data
| X | Frequency |
| 10 | 3 |
| 11 | 12 |
| 12 | 18 |
| 13 | 12 |
| 14 | 3 |
Solution:
Skewness :
Mean `bar x=(sum f x)/n`
`=576/48`
`=12`
`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)` |
| 10 | 3 | 30 | -2 | 12 | -24 |
| 11 | 12 | 132 | -1 | 12 | -12 |
| 12 | 18 | 216 | 0 | 0 | 0 |
| 13 | 12 | 156 | 1 | 12 | 12 |
| 14 | 3 | 42 | 2 | 12 | 24 |
| --- | --- | --- | --- | --- | --- |
| -- | `n=48` | `sum f*x=576` | `-- | `=48` | `=0` |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(48/48)`
`=sqrt(1)`
`=1`
Population Skewness `= (sum(x - bar x)^3)/(n*S^3)`
`=0/(48*(1)^3)`
`=0/(48*1)`
`=0`
3. Calculate Population Skewness from the following grouped data
| Class | Frequency |
| 2 - 4 | 3 |
| 4 - 6 | 4 |
| 6 - 8 | 2 |
| 8 - 10 | 1 |
Solution:
Skewness :
Mean `bar x=(sum f x)/(sum f)`
`=52/10`
`=5.2`
Class
`(1)` | Mid value (`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)` |
| 2 - 4 | 3 | 3 | 9 | -2.2 | 14.52 | -31.944 |
| 4 - 6 | 5 | 4 | 20 | -0.2 | 0.16 | -0.032 |
| 6 - 8 | 7 | 2 | 14 | 1.8 | 6.48 | 11.664 |
| 8 - 10 | 9 | 1 | 9 | 3.8 | 14.44 | 54.872 |
| --- | --- | --- | --- | --- | --- | --- |
| -- | -- | `n=10` | `sum f*x=52` | `-- | `=35.6` | `=34.56` |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(35.6/10)`
`=sqrt(3.56)`
`=1.8868`
Population Skewness `= (sum(x - bar x)^3)/(n*S^3)`
`=34.56/(10*(1.8868)^3)`
`=34.56/(10*6.717)`
`=0.5145`
4. Calculate Population Skewness from the following grouped data
| Class | Frequency |
| 0 - 2 | 5 |
| 2 - 4 | 16 |
| 4 - 6 | 13 |
| 6 - 8 | 7 |
| 8 - 10 | 5 |
| 10 - 12 | 4 |
Solution:
Skewness :
Mean `bar x=(sum f x)/(sum f)`
`=256/50`
`=5.12`
Class
`(1)` | Mid value (`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)` |
| 0 - 2 | 1 | 5 | 5 | -4.12 | 84.872 | -349.6726 |
| 2 - 4 | 3 | 16 | 48 | -2.12 | 71.9104 | -152.45 |
| 4 - 6 | 5 | 13 | 65 | -0.12 | 0.1872 | -0.0225 |
| 6 - 8 | 7 | 7 | 49 | 1.88 | 24.7408 | 46.5127 |
| 8 - 10 | 9 | 5 | 45 | 3.88 | 75.272 | 292.0554 |
| 10 - 12 | 11 | 4 | 44 | 5.88 | 138.2976 | 813.1899 |
| --- | --- | --- | --- | --- | --- | --- |
| -- | -- | `n=50` | `sum f*x=256` | `-- | `=395.28` | `=649.6128` |
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(395.28/50)`
`=sqrt(7.9056)`
`=2.8117`
Population Skewness `= (sum(x - bar x)^3)/(n*S^3)`
`=649.6128/(50*(2.8117)^3)`
`=649.6128/(50*22.2281)`
`=0.5845`
This material is intended as a summary. Use your textbook for detail explanation.
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