1. Calculate 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 |
| 1 | 5 | 5 | 5 |
| 2 | 10 | 20 | 40 |
| 3 | 6 | 18 | 54 |
| 4 | 3 | 12 | 48 |
| --- | --- | --- | --- |
| `n=25` | `sum f*x=55` | `sum f*x^2=147` |
Mean `bar x = (sum fx)/n`
`=55/25`
`=2.2`
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 %`
2. Calculate Population Coefficient of Variation from the following grouped data
| X | Frequency |
| 10 | 3 |
| 11 | 12 |
| 12 | 18 |
| 13 | 12 |
| 14 | 3 |
Solution:
`x`
`(1)` | Frequency `(f)`
`(2)` | `f*x`
`(3)=(2)xx(1)` | `f*x^2=(f*x)xx(x)`
`(4)=(3)xx(1)` |
| 10 | 3 | 30 | 300 |
| 11 | 12 | 132 | 1452 |
| 12 | 18 | 216 | 2592 |
| 13 | 12 | 156 | 2028 |
| 14 | 3 | 42 | 588 |
| --- | --- | --- | --- |
| `n=48` | `sum f*x=576` | `sum f*x^2=6960` |
Mean `bar x = (sum fx)/n`
`=576/48`
`=12`
Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`
`=sqrt((6960 - (576)^2/48)/48)`
`=sqrt((6960 - 6912)/48)`
`=sqrt(48/48)`
`=sqrt(1)`
`=1`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=1/12 * 100 %`
`=8.33 %`
3. Calculate 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 | 9 | 27 |
| 4-6 | 4 | 5 | 20 | 100 |
| 6-8 | 2 | 7 | 14 | 98 |
| 8-10 | 1 | 9 | 9 | 81 |
| --- | --- | --- | --- | --- |
| -- | `n = 10` | -- | `sum f*x=52` | `sum f*x^2=306` |
Mean `bar x = (sum fx)/n`
`=52/10`
`=5.2`
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 %`
4. Calculate Population Coefficient of Variation 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:
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)` |
| 0-2 | 5 | 1 | 5 | 5 |
| 2-4 | 16 | 3 | 48 | 144 |
| 4-6 | 13 | 5 | 65 | 325 |
| 6-8 | 7 | 7 | 49 | 343 |
| 8-10 | 5 | 9 | 45 | 405 |
| 10-12 | 4 | 11 | 44 | 484 |
| --- | --- | --- | --- | --- |
| -- | `n = 50` | -- | `sum f*x=256` | `sum f*x^2=1706` |
Mean `bar x = (sum fx)/n`
`=256/50`
`=5.12`
Population Standard deviation `sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)`
`=sqrt((1706 - (256)^2/50)/50)`
`=sqrt((1706 - 1310.72)/50)`
`=sqrt(395.28/50)`
`=sqrt(7.9056)`
`=2.8117`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=2.8117/5.12 * 100 %`
`=54.92 %`
5. Calculate Population Coefficient of Variation from the following grouped data
| Class | Frequency |
| 10 - 20 | 15 |
| 20 - 30 | 25 |
| 30 - 40 | 20 |
| 40 - 50 | 12 |
| 50 - 60 | 8 |
| 60 - 70 | 5 |
| 70 - 80 | 3 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | Mid value `(x)`
`(3)` | `d=(x-A)/h=(x-45)/10`
`A=45,h=10`
`(4)` | `f*d`
`(5)=(2)xx(4)` | `f*d^2`
`(6)=(5)xx(4)` |
| 10 - 20 | 15 | 15 | -3 | -45 | 135 |
| 20 - 30 | 25 | 25 | -2 | -50 | 100 |
| 30 - 40 | 20 | 35 | -1 | -20 | 20 |
| 40 - 50 | 12 | 45=A | 0 | 0 | 0 |
| 50 - 60 | 8 | 55 | 1 | 8 | 8 |
| 60 - 70 | 5 | 65 | 2 | 10 | 20 |
| 70 - 80 | 3 | 75 | 3 | 9 | 27 |
| --- | --- | --- | --- | --- | --- |
| `n = 88` | ----- | ----- | `sum f*d=-88` | `sum f*d^2=310` |
Mean `bar x = A + (sum fd)/n * h`
`=45 + -88/88 * 10`
`=45 + -1 * 10`
`=45 + -10`
`=35`
Population Standard deviation `sigma = sqrt((sum f*d^2 - (sum f*d)^2/n)/n) * h`
`=sqrt((310 - (-88)^2/88)/88) * 10`
`=sqrt((310 - 88)/88) * 10`
`=sqrt(222/88) * 10`
`=sqrt(2.5227) * 10`
`=1.5883 * 10`
`=15.8831`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=15.8831/35 * 100 %`
`=45.38 %`
6. Calculate Population Coefficient of Variation from the following grouped data
| Class | Frequency |
| 20 - 25 | 110 |
| 25 - 30 | 170 |
| 30 - 35 | 80 |
| 35 - 40 | 45 |
| 40 - 45 | 40 |
| 45 - 50 | 35 |
Solution:
Class
`(1)` | Frequency `(f)`
`(2)` | Mid value `(x)`
`(3)` | `d=(x-A)/h=(x-37.5)/5`
`A=37.5,h=5`
`(4)` | `f*d`
`(5)=(2)xx(4)` | `f*d^2`
`(6)=(5)xx(4)` |
| 20 - 25 | 110 | 22.5 | -3 | -330 | 990 |
| 25 - 30 | 170 | 27.5 | -2 | -340 | 680 |
| 30 - 35 | 80 | 32.5 | -1 | -80 | 80 |
| 35 - 40 | 45 | 37.5=A | 0 | 0 | 0 |
| 40 - 45 | 40 | 42.5 | 1 | 40 | 40 |
| 45 - 50 | 35 | 47.5 | 2 | 70 | 140 |
| --- | --- | --- | --- | --- | --- |
| `n = 480` | ----- | ----- | `sum f*d=-640` | `sum f*d^2=1930` |
Mean `bar x = A + (sum fd)/n * h`
`=37.5 + -640/480 * 5`
`=37.5 + -1.3333 * 5`
`=37.5 + -6.6667`
`=30.8333`
Population Standard deviation `sigma = sqrt((sum f*d^2 - (sum f*d)^2/n)/n) * h`
`=sqrt((1930 - (-640)^2/480)/480) * 5`
`=sqrt((1930 - 853.3333)/480) * 5`
`=sqrt(1076.6667/480) * 5`
`=sqrt(2.2431) * 5`
`=1.4977 * 5`
`=7.4884`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=7.4884/30.8333 * 100 %`
`=24.29 %`
This material is intended as a summary. Use your textbook for detail explanation.
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