Formula
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1. Mean `bar x = (sum fx)/n`
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2. Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
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3. Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`
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4. Coefficient of Variation (Sample) `=S / bar x * 100 %`
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Examples
1. Calculate Sample Variance `(S^2)`, Sample Standard deviation `(S)`, Sample Coefficient of Variation from the following mixed data
| Class | Frequency |
| 1 | 3 |
| 2 | 4 |
| 5 | 10 |
| 6 - 10 | 23 |
| 10 - 20 | 20 |
| 20 - 30 | 20 |
| 30 - 50 | 15 |
| 50 - 70 | 3 |
| 70 - 100 | 2 |
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)` |
| 1 | 3 | 1 `1=1` | 3 `3=3xx1`
`(4)=(2)xx(3)` | 3 `3=3xx1`
`(5)=(4)xx(3)` |
| 2 | 4 | 2 `2=2` | 8 `8=4xx2`
`(4)=(2)xx(3)` | 16 `16=8xx2`
`(5)=(4)xx(3)` |
| 5 | 10 | 5 `5=5` | 50 `50=10xx5`
`(4)=(2)xx(3)` | 250 `250=50xx5`
`(5)=(4)xx(3)` |
| 6 - 10 | 23 | 8 `8=(6+10)/2` | 184 `184=23xx8`
`(4)=(2)xx(3)` | 1472 `1472=184xx8`
`(5)=(4)xx(3)` |
| 10 - 20 | 20 | 15 `15=(10+20)/2` | 300 `300=20xx15`
`(4)=(2)xx(3)` | 4500 `4500=300xx15`
`(5)=(4)xx(3)` |
| 20 - 30 | 20 | 25 `25=(20+30)/2` | 500 `500=20xx25`
`(4)=(2)xx(3)` | 12500 `12500=500xx25`
`(5)=(4)xx(3)` |
| 30 - 50 | 15 | 40 `40=(30+50)/2` | 600 `600=15xx40`
`(4)=(2)xx(3)` | 24000 `24000=600xx40`
`(5)=(4)xx(3)` |
| 50 - 70 | 3 | 60 `60=(50+70)/2` | 180 `180=3xx60`
`(4)=(2)xx(3)` | 10800 `10800=180xx60`
`(5)=(4)xx(3)` |
| 70 - 100 | 2 | 85 `85=(70+100)/2` | 170 `170=2xx85`
`(4)=(2)xx(3)` | 14450 `14450=170xx85`
`(5)=(4)xx(3)` |
| --- | --- | --- | --- | --- |
| `n = 100` | ----- | `sum f*x=1995` | `sum f*x^2=67991` |
Mean `bar x = (sum fx)/n`
`=1995/100`
`=19.95`
Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
`=(67991 - (1995)^2/100)/99`
`=(67991 - 39800.25)/99`
`=28190.75/99`
`=284.7551`
Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`
`=sqrt((67991 - (1995)^2/100)/99)`
`=sqrt((67991 - 39800.25)/99)`
`=sqrt(28190.75/99)`
`=sqrt(284.7551)`
`=16.8747`
Coefficient of Variation (Sample) `=S / bar x * 100 %`
`=16.8747/19.95 * 100 %`
`=84.58 %`
2. Calculate Sample Variance `(S^2)`, Sample Standard deviation `(S)`, Sample Coefficient of Variation from the following mixed data
| Class | Frequency |
| 2 | 1 |
| 3 | 2 |
| 4 | 2 |
| 5 - 9 | 8 |
| 10 - 14 | 15 |
| 15 - 19 | 8 |
| 20 - 29 | 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)` |
| 2 | 1 | 2 | 2 | 4 |
| 3 | 2 | 3 | 6 | 18 |
| 4 | 2 | 4 | 8 | 32 |
| 5 - 9 | 8 | 7 | 56 | 392 |
| 10 - 14 | 15 | 12 | 180 | 2160 |
| 15 - 19 | 8 | 17 | 136 | 2312 |
| 20 - 29 | 4 | 24.5 | 98 | 2401 |
| --- | --- | --- | --- | --- |
| `n = 40` | ----- | `sum f*x=486` | `sum f*x^2=7319` |
Mean `bar x = (sum fx)/n`
`=486/40`
`=12.15`
Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
`=(7319 - (486)^2/40)/39`
`=(7319 - 5904.9)/39`
`=1414.1/39`
`=36.259`
Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`
`=sqrt((7319 - (486)^2/40)/39)`
`=sqrt((7319 - 5904.9)/39)`
`=sqrt(1414.1/39)`
`=sqrt(36.259)`
`=6.0215`
Coefficient of Variation (Sample) `=S / bar x * 100 %`
`=6.0215/12.15 * 100 %`
`=49.56 %`
This material is intended as a summary. Use your textbook for detail explanation.
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