Formula
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1. Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
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2. Skewness `= (sum(x - bar x)^3)/(n*S^3)`
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3. Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
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Examples
1. Calculate Population Skewness, Population Kurtosis from the following grouped data
| Class | Frequency |
| 2 - 4 | 3 |
| 4 - 6 | 4 |
| 6 - 8 | 2 |
| 8 - 10 | 1 |
Solution:
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=(2+4)/2` | 3 | 9 `9=3xx3`
`(4)=(2)xx(3)` | -2.2 `|3-5.2|=-2.2`
`|x - 5.2|` | 14.52 `14.52=3xx-2.2xx-2.2`
`(6)=(2)xx(5)` | -31.944 `-31.944=14.52xx-2.2`
`(7)=(2)xx(6)` |
| 4 - 6 | 5 `5=(4+6)/2` | 4 | 20 `20=4xx5`
`(4)=(2)xx(3)` | -0.2 `|5-5.2|=-0.2`
`|x - 5.2|` | 0.16 `0.16=4xx-0.2xx-0.2`
`(6)=(2)xx(5)` | -0.032 `-0.032=0.16xx-0.2`
`(7)=(2)xx(6)` |
| 6 - 8 | 7 `7=(6+8)/2` | 2 | 14 `14=2xx7`
`(4)=(2)xx(3)` | 1.8 `|7-5.2|=1.8`
`|x - 5.2|` | 6.48 `6.48=2xx1.8xx1.8`
`(6)=(2)xx(5)` | 11.664 `11.664=6.48xx1.8`
`(7)=(2)xx(6)` |
| 8 - 10 | 9 `9=(8+10)/2` | 1 | 9 `9=1xx9`
`(4)=(2)xx(3)` | 3.8 `|9-5.2|=3.8`
`|x - 5.2|` | 14.44 `14.44=1xx3.8xx3.8`
`(6)=(2)xx(5)` | 54.872 `54.872=14.44xx3.8`
`(7)=(2)xx(6)` |
| --- | --- | --- | --- | --- | --- | --- |
| -- | -- | `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`
Skewness `= (sum(x - bar x)^3)/(n*S^3)`
`=34.56/(10*(1.8868)^3)`
`=34.56/(10*6.717)`
`=0.5145`
Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
`=299.792/(10*(1.8868)^4)`
`=299.792/(10*12.6736)`
`=2.3655`
2. Calculate Population Skewness, Population Kurtosis from the following grouped data
Solution:
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 `0=1xx0`
`(4)=(2)xx(3)` | -2.2 `|0-2.2|=-2.2`
`|x - 2.2|` | 4.84 `4.84=1xx-2.2xx-2.2`
`(6)=(2)xx(5)` | -10.648 `-10.648=4.84xx-2.2`
`(7)=(2)xx(6)` |
| 1 | 5 | 5 `5=5xx1`
`(4)=(2)xx(3)` | -1.2 `|1-2.2|=-1.2`
`|x - 2.2|` | 7.2 `7.2=5xx-1.2xx-1.2`
`(6)=(2)xx(5)` | -8.64 `-8.64=7.2xx-1.2`
`(7)=(2)xx(6)` |
| 2 | 10 | 20 `20=10xx2`
`(4)=(2)xx(3)` | -0.2 `|2-2.2|=-0.2`
`|x - 2.2|` | 0.4 `0.4=10xx-0.2xx-0.2`
`(6)=(2)xx(5)` | -0.08 `-0.08=0.4xx-0.2`
`(7)=(2)xx(6)` |
| 3 | 6 | 18 `18=6xx3`
`(4)=(2)xx(3)` | 0.8 `|3-2.2|=0.8`
`|x - 2.2|` | 3.84 `3.84=6xx0.8xx0.8`
`(6)=(2)xx(5)` | 3.072 `3.072=3.84xx0.8`
`(7)=(2)xx(6)` |
| 4 | 3 | 12 `12=3xx4`
`(4)=(2)xx(3)` | 1.8 `|4-2.2|=1.8`
`|x - 2.2|` | 9.72 `9.72=3xx1.8xx1.8`
`(6)=(2)xx(5)` | 17.496 `17.496=9.72xx1.8`
`(7)=(2)xx(6)` |
| --- | --- | --- | --- | --- | --- |
| -- | `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`
Skewness `= (sum(x - bar x)^3)/(n*S^3)`
`=1.2/(25*(1.0198)^3)`
`=1.2/(25*1.0606)`
`=0.0453`
Kurtosis `= (sum(x - bar x)^4)/(n*S^4)`
`=67.76/(25*(1.0198)^4)`
`=67.76/(25*1.0816)`
`=2.5059`
This material is intended as a summary. Use your textbook for detail explanation.
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