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Home > Statistical Methods calculators > Population Variance, Standard deviation and coefficient of variation for ungrouped data example
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Population Variance, Standard deviation and coefficient of variation for ungrouped data Formula & Example
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- Formula & Example
- Population Variance Example
- Population Standard deviation Example
- Population coefficient of variation Example
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Other related methods
- Mean, Median and Mode
- Quartile, Decile, Percentile, Octile, Quintile
- Population Variance, Standard deviation and coefficient of variation
- Sample Variance, Standard deviation and coefficient of variation
- Population Skewness, Kurtosis
- Sample Skewness, Kurtosis
- Geometric mean, Harmonic mean
- Mean deviation, Quartile deviation, Decile deviation, Percentile deviation
- Five number summary
- Box and Whisker Plots
- Construct an ungrouped frequency distribution table
- Construct a grouped frequency distribution table
- Maximum, Minimum
- Sum, Length
- Range, Mid Range
- Stem and leaf plot
- Ascending order, Descending order
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1. Formula & Example
Formula
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1. Mean `bar x = (sum x)/n`
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2. Population Variance `sigma^2 = (sum x^2 - (sum x)^2/n)/n`
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3. Population Standard deviation `sigma = sqrt((sum x^2 - (sum 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 data
3,13,11,15,5,4,2,3,2
Solution:
| `x` | `x^2` | | 3 | 9 `9=3xx3` | | 13 | 169 `169=13xx13` | | 11 | 121 `121=11xx11` | | 15 | 225 `225=15xx15` | | 5 | 25 `25=5xx5` | | 4 | 16 `16=4xx4` | | 2 | 4 `4=2xx2` | | 3 | 9 `9=3xx3` | | 2 | 4 `4=2xx2` | | --- | --- | | `sum x=58` | `sum x^2=582` |
Mean `bar x = (sum x)/n`
`=(3 + 13 + 11 + 15 + 5 + 4 + 2 + 3 + 2)/9`
`=58/9`
`=6.4444`
Population Variance `sigma^2 = (sum x^2 - (sum x)^2/n)/n`
`=(582 - (58)^2/9)/9`
`=(582 - 373.7778)/9`
`=208.2222/9`
`=23.1358`
Population Standard deviation `sigma = sqrt((sum x^2 - (sum x)^2/n)/n)`
`=sqrt((582 - (58)^2/9)/9)`
`=sqrt((582 - 373.7778)/9)`
`=sqrt(208.2222/9)`
`=sqrt(23.1358)`
`=4.81`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=4.81/6.4444 * 100 %`
`=74.64 %`
2. Calculate Population Variance `(sigma^2)`, Population Standard deviation `(sigma)`, Population Coefficient of Variation from the following data
85,96,76,108,85,80,100,85,70,95
Solution:
| `x` | `x - bar x = x - 88` | `(x - bar x)^2` | | 85 | -3 `-3=85-88` | 9 `9=-3xx-3` | | 96 | 8 `8=96-88` | 64 `64=8xx8` | | 76 | -12 `-12=76-88` | 144 `144=-12xx-12` | | 108 | 20 `20=108-88` | 400 `400=20xx20` | | 85 | -3 `-3=85-88` | 9 `9=-3xx-3` | | 80 | -8 `-8=80-88` | 64 `64=-8xx-8` | | 100 | 12 `12=100-88` | 144 `144=12xx12` | | 85 | -3 `-3=85-88` | 9 `9=-3xx-3` | | 70 | -18 `-18=70-88` | 324 `324=-18xx-18` | | 95 | 7 `7=95-88` | 49 `49=7xx7` | | --- | --- | --- | | `sum x=880` | `sum (x - bar x)=0` | `sum (x - bar x)^2=1216` |
Mean `bar x = (sum x)/n`
`=(85 + 96 + 76 + 108 + 85 + 80 + 100 + 85 + 70 + 95)/10`
`=880/10`
`=88`
Population Variance `sigma^2 = (sum (x - bar x)^2)/n`
`=1216/10`
`=121.6`
Population Standard deviation `sigma = sqrt((sum (x - bar x)^2)/n)`
`=sqrt(1216/10)`
`=sqrt(121.6)`
`=11.0272`
Coefficient of Variation (Population) `=sigma / bar x * 100 %`
`=11.0272/88 * 100 %`
`=12.53 %`
This material is intended as a summary. Use your textbook for detail explanation.
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