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Home > Statistical Methods calculators > Population Variance, Standard deviation and coefficient of variation for grouped data example
Population Variance Example for grouped data ( Enter your problem )
( Enter your problem )
  1. Formula & Example
  2. Population Variance Example
  3. Population Standard deviation Example
  4. Population 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
1. Formula & Example
(Previous example)		3. Population Standard deviation Example
(Next example)

2. Population Variance Example


1. Calculate Population Variance `(sigma^2)` 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)`
0100
1555
2102040
361854
431248
------------
`n=25``sum f*x=55``sum f*x^2=147`


Mean `bar x = (sum fx)/n`

`=55/25`

`=2.2`


Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

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

`=(147 - 121)/25`

`=26/25`

`=1.04`

2. Calculate Population Variance `(sigma^2)` from the following grouped data
XFrequency
103
1112
1218
1312
143


Solution:
`x`
`(1)`		Frequency `(f)`
`(2)`		`f*x`
`(3)=(2)xx(1)`		`f*x^2=(f*x)xx(x)`
`(4)=(3)xx(1)`
10330300
11121321452
12182162592
13121562028
14342588
------------
`n=48``sum f*x=576``sum f*x^2=6960`


Mean `bar x = (sum fx)/n`

`=576/48`

`=12`


Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

`=(6960 - (576)^2/48)/48`

`=(6960 - 6912)/48`

`=48/48`

`=1`

3. Calculate Population Variance `(sigma^2)` 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)`
2-433927
4-64520100
6-8271498
8-1019981
---------------
--`n = 10`--`sum f*x=52``sum f*x^2=306`


Mean `bar x = (sum fx)/n`

`=52/10`

`=5.2`


Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

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

`=(306 - 270.4)/10`

`=35.6/10`

`=3.56`

4. Calculate Population Variance `(sigma^2)` from the following grouped data
ClassFrequency
0 - 25
2 - 416
4 - 613
6 - 87
8 - 105
10 - 124


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)`
0-25155
2-416348144
4-613565325
6-87749343
8-105945405
10-1241144484
---------------
--`n = 50`--`sum f*x=256``sum f*x^2=1706`


Mean `bar x = (sum fx)/n`

`=256/50`

`=5.12`


Population Variance `sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n`

`=(1706 - (256)^2/50)/50`

`=(1706 - 1310.72)/50`

`=395.28/50`

`=7.9056`

5. Calculate Population Variance `(sigma^2)` from the following grouped data
ClassFrequency
10 - 2015
20 - 3025
30 - 4020
40 - 5012
50 - 608
60 - 705
70 - 803


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 - 201515-3-45135
20 - 302525-2-50100
30 - 402035-1-2020
40 - 501245=A000
50 - 60855188
60 - 7056521020
70 - 803753927
------------------
`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 Variance `sigma^2 = ((sum f*d^2 - (sum f*d)^2/n)/n) * h^2`

`=((310 - (-88)^2/88)/88) * 10^2`

`=((310 - 88)/88) * 100`

`=222/88 * 100`

`=2.5227 * 100`

`=252.2727`

6. Calculate Population Variance `(sigma^2)` from the following grouped data
ClassFrequency
20 - 25110
25 - 30170
30 - 3580
35 - 4045
40 - 4540
45 - 5035


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 - 2511022.5-3-330990
25 - 3017027.5-2-340680
30 - 358032.5-1-8080
35 - 404537.5=A000
40 - 454042.514040
45 - 503547.5270140
------------------
`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 Variance `sigma^2 = ((sum f*d^2 - (sum f*d)^2/n)/n) * h^2`

`=((1930 - (-640)^2/480)/480) * 5^2`

`=((1930 - 853.3333)/480) * 25`

`=1076.6667/480 * 25`

`=2.2431 * 25`

`=56.0764`

This material is intended as a summary. Use your textbook for detail explanation.
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