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Home > Statistical Methods calculators > Sample Variance, Standard deviation and coefficient of variation for ungrouped data example
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Sample Variance, Standard deviation and coefficient of variation for ungrouped data Formula & Example
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- Formula & Example
- Sample Variance Example
- Sample Standard deviation Example
- Sample 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. Sample Variance S^2 = (sum x^2 - (sum x)^2/n)/(n-1)
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3. Sample Standard deviation S = sqrt((sum x^2 - (sum 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 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
Sample Variance S^2 = (sum x^2 - (sum x)^2/n)/(n-1)
=(582 - (58)^2/9)/8
=(582 - 373.7778)/8
=208.2222/8
=26.0278
Sample Standard deviation S = sqrt((sum x^2 - (sum x)^2/n)/(n-1))
=sqrt((582 - (58)^2/9)/8)
=sqrt((582 - 373.7778)/8)
=sqrt(208.2222/8)
=sqrt(26.0278)
=5.1017
Coefficient of Variation (Sample) =S / bar x * 100 %
=5.1017/6.4444 * 100 %
=79.16 %
2. Calculate Sample Variance (S^2), Sample Standard deviation (S), Sample 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
Sample Variance S^2 = (sum (x - bar x)^2)/(n-1)
=1216/9
=135.1111
Sample Standard deviation S = sqrt((sum (x - bar x)^2)/(n-1))
=sqrt(1216/9)
=sqrt(135.1111)
=11.6237
Coefficient of Variation (Sample) =S / bar x * 100 %
=11.6237/88 * 100 %
=13.21 %
This material is intended as a summary. Use your textbook for detail explanation.
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