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Home > Statistical Methods calculators > Sample Variance, Standard deviation and coefficient of variation for grouped data example
Sample Variance, Standard deviation and coefficient of variation for grouped data Formula & Example ( Enter your problem )
( Enter your problem )
  1. Formula & Example
  2. Sample Variance Example
  3. Sample Standard deviation Example
  4. Sample coefficient of variation Example
 		Other related methods
  1. Mean, Median and Mode
  2. Quartile, Decile, Percentile, Octile, Quintile
  3. Population Variance, Standard deviation and coefficient of variation
  4. Sample Variance, Standard deviation and coefficient of variation
  5. Population Skewness, Kurtosis
  6. Sample Skewness, Kurtosis
  7. Geometric mean, Harmonic mean
  8. Mean deviation, Quartile deviation, Decile deviation, Percentile deviation
  9. Five number summary
  10. Box and Whisker Plots
  11. Mode using Grouping Method
  12. Less than type Cumulative frequency table
  13. More than type Cumulative frequency table
  14. Class and their frequency table
3. Population Variance, Standard deviation and coefficient of variation
(Previous method)		2. Sample Variance Example
(Next example)

1. Formula & Example


Formula
1. Mean `bar x = (sum fx)/n`
2. Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`
3. Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`
4. Coefficient of Variation (Sample) `=S / bar x * 100 %`
Examples
1. Calculate Sample Variance `(S^2)`, Sample Standard deviation `(S)`, Sample Coefficient of Variation from the following grouped data
ClassFrequency
2 - 43
4 - 64
6 - 82
8 - 101


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)`
2-43 3 `3=(2+4)/2` 9 `9=3xx3`
`(4)=(2)xx(3)`		 27 `27=9xx3`
`(5)=(4)xx(3)`
4-64 5 `5=(4+6)/2` 20 `20=4xx5`
`(4)=(2)xx(3)`		 100 `100=20xx5`
`(5)=(4)xx(3)`
6-82 7 `7=(6+8)/2` 14 `14=2xx7`
`(4)=(2)xx(3)`		 98 `98=14xx7`
`(5)=(4)xx(3)`
8-101 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`


Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`

`=(306 - (52)^2/10)/9`

`=(306 - 270.4)/9`

`=35.6/9`

`=3.9556`


Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`

`=sqrt((306 - (52)^2/10)/9)`

`=sqrt((306 - 270.4)/9)`

`=sqrt(35.6/9)`

`=sqrt(3.9556)`

`=1.9889`


Coefficient of Variation (Sample) `=S / bar x * 100 %`

`=1.9889/5.2 * 100 %`

`=38.25 %`
2. Calculate Sample Variance `(S^2)`, Sample Standard deviation `(S)`, Sample Coefficient of Variation from the following grouped data
XFrequency
01
15
210
36
43


Solution:
`x`
`(1)`		Frequency `(f)`
`(2)`		`f*x`
`(3)=(2)xx(1)`		`f*x^2=(f*x)xx(x)`
`(4)=(3)xx(1)`
01 0 `0=1xx0`
`(3)=(2)xx(1)`		 0 `0=0xx0`
`(4)=(3)xx(1)`
15 5 `5=5xx1`
`(3)=(2)xx(1)`		 5 `5=5xx1`
`(4)=(3)xx(1)`
210 20 `20=10xx2`
`(3)=(2)xx(1)`		 40 `40=20xx2`
`(4)=(3)xx(1)`
36 18 `18=6xx3`
`(3)=(2)xx(1)`		 54 `54=18xx3`
`(4)=(3)xx(1)`
43 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`


Sample Variance `S^2 = (sum f*x^2 - (sum f*x)^2/n)/(n-1)`

`=(147 - (55)^2/25)/24`

`=(147 - 121)/24`

`=26/24`

`=1.0833`


Sample Standard deviation `S = sqrt((sum f*x^2 - (sum f*x)^2/n)/(n-1))`

`=sqrt((147 - (55)^2/25)/24)`

`=sqrt((147 - 121)/24)`

`=sqrt(26/24)`

`=sqrt(1.0833)`

`=1.0408`


Coefficient of Variation (Sample) `=S / bar x * 100 %`

`=1.0408/2.2 * 100 %`

`=47.31 %`

This material is intended as a summary. Use your textbook for detail explanation.
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3. Population Variance, Standard deviation and coefficient of variation
(Previous method)		2. Sample Variance Example
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