Formula
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1. Mean `bar x = (sum fx)/n`
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2. Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`
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3. Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`
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4. Coefficient of Variation (Population) `=sigma / bar x * 100 %`
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Examples
1. Calculate Population Variance `(sigma^2)`, Population Standard deviation `(sigma)`, Population Coefficient of Variation from the following grouped data
| Class | Frequency |
| 2 - 4 | 3 |
| 4 - 6 | 4 |
| 6 - 8 | 2 |
| 8 - 10 | 1 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | Mid value `(x)`
`(3)` | `f*x`
`(4)=(2)xx(3)` | `f*x^2=(f*x)xx(x)`
`(5)=(4)xx(3)` |
| 2-4 | 3 | 3 `3=(2+4)/2` | 9 `9=3xx3`
`(4)=(2)xx(3)` | 27 `27=9xx3`
`(5)=(4)xx(3)` |
| 4-6 | 4 | 5 `5=(4+6)/2` | 20 `20=4xx5`
`(4)=(2)xx(3)` | 100 `100=20xx5`
`(5)=(4)xx(3)` |
| 6-8 | 2 | 7 `7=(6+8)/2` | 14 `14=2xx7`
`(4)=(2)xx(3)` | 98 `98=14xx7`
`(5)=(4)xx(3)` |
| 8-10 | 1 | 9 `9=(8+10)/2` | 9 `9=1xx9`
`(4)=(2)xx(3)` | 81 `81=9xx9`
`(5)=(4)xx(3)` |
| --- | --- | --- | --- | --- |
| -- | `n = 10` | -- | `sum f*x=52` | `sum f*x^2=306` |
Mean `bar x = (sum fx)/n`
`=52/10`
`=5.2`
Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`
`=(306 - (52)^2/10)/10`
`=(306 - 270.4)/10`
`=35.6/10`
`=3.56`
Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`
`=sqrt((306 - (52)^2/10)/10)`
`=sqrt((306 - 270.4)/10)`
`=sqrt(35.6/10)`
`=sqrt(3.56)`
`=1.8868`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=1.8868/5.2 * 100 %`
`=36.28 %`
2. Calculate Population Variance `(sigma^2)`, Population Standard deviation `(sigma)`, Population Coefficient of Variation from the following grouped data
Solution:
`x`
`(1)` | Frequency `(f)`
`(2)` | `f*x`
`(3)=(2)xx(1)` | `f*x^2=(f*x)xx(x)`
`(4)=(3)xx(1)` |
| 0 | 1 | 0 `0=1xx0`
`(3)=(2)xx(1)` | 0 `0=0xx0`
`(4)=(3)xx(1)` |
| 1 | 5 | 5 `5=5xx1`
`(3)=(2)xx(1)` | 5 `5=5xx1`
`(4)=(3)xx(1)` |
| 2 | 10 | 20 `20=10xx2`
`(3)=(2)xx(1)` | 40 `40=20xx2`
`(4)=(3)xx(1)` |
| 3 | 6 | 18 `18=6xx3`
`(3)=(2)xx(1)` | 54 `54=18xx3`
`(4)=(3)xx(1)` |
| 4 | 3 | 12 `12=3xx4`
`(3)=(2)xx(1)` | 48 `48=12xx4`
`(4)=(3)xx(1)` |
| --- | --- | --- | --- |
| `n=25` | `sum f*x=55` | `sum f*x^2=147` |
Mean `bar x = (sum fx)/n`
`=55/25`
`=2.2`
Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`
`=(147 - (55)^2/25)/25`
`=(147 - 121)/25`
`=26/25`
`=1.04`
Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`
`=sqrt((147 - (55)^2/25)/25)`
`=sqrt((147 - 121)/25)`
`=sqrt(26/25)`
`=sqrt(1.04)`
`=1.0198`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=1.0198/2.2 * 100 %`
`=46.35 %`
This material is intended as a summary. Use your textbook for detail explanation.
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