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Find Standard Deviation and Coefficient of Variation for Mixed Data
1. Calculate Coefficient of variation from the follwing 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)`
`cf`
`(6)`
2
1
2
`2=2`
2
`2=1xx2`
`(4)=(2)xx(3)`
4
`4=2xx2`
`(5)=(4)xx(3)`
1
`1=0+1`
`(6)=`Previous `(6)+(2)`
3
2
3
`3=3`
6
`6=2xx3`
`(4)=(2)xx(3)`
18
`18=6xx3`
`(5)=(4)xx(3)`
3
`3=1+2`
`(6)=`Previous `(6)+(2)`
4
2
4
`4=4`
8
`8=2xx4`
`(4)=(2)xx(3)`
32
`32=8xx4`
`(5)=(4)xx(3)`
5
`5=3+2`
`(6)=`Previous `(6)+(2)`
5 - 9
8
7
`7=(5+9)/2`
56
`56=8xx7`
`(4)=(2)xx(3)`
392
`392=56xx7`
`(5)=(4)xx(3)`
13
`13=5+8`
`(6)=`Previous `(6)+(2)`
10 - 14
15
12
`12=(10+14)/2`
180
`180=15xx12`
`(4)=(2)xx(3)`
2160
`2160=180xx12`
`(5)=(4)xx(3)`
28
`28=13+15`
`(6)=`Previous `(6)+(2)`
15 - 19
8
17
`17=(15+19)/2`
136
`136=8xx17`
`(4)=(2)xx(3)`
2312
`2312=136xx17`
`(5)=(4)xx(3)`
36
`36=28+8`
`(6)=`Previous `(6)+(2)`
20 - 29
4
24.5
`24.5=(20+29)/2`
98
`98=4xx24.5`
`(4)=(2)xx(3)`
2401
`2401=98xx24.5`
`(5)=(4)xx(3)`
40
`40=36+4`
`(6)=`Previous `(6)+(2)`
---
---
---
---
---
---
`n = 40`
-----
`sum f*x=486`
`sum f*x^2=7319`
-----
Mean `bar x = (sum fx)/n`
`=486/40`
`=12.15`
To find Median Class
= value of `(n/2)^(th)` observation
= value of `(40/2)^(th)` observation
= value of `20^(th)` observation
From the column of cumulative frequency `cf`, we find that the `20^(th)` observation lies in the class `10 - 14`.
`:.` The median class is `9.5 - 14.5`.
Now,
`:. L = `lower boundary point of median class `=9.5`
`:. n = `Total frequency `=40`
`:. cf = `Cumulative frequency of the class preceding the median class `=13`
`:. f = `Frequency of the median class `=15`
`:. c = `class length of median class `=5`
Median `M = L + (n/2 - cf)/f * c`
`=9.5 + (20 - 13)/15 * 5`
`=9.5 + (7)/15 * 5`
`=9.5 + 2.33`
`=11.83`
`sigma = sqrt((sum f*x^2)/n - ((sum f*x)/n)^2)`
`=sqrt(7319/40 - (486/40)^2)`
`=sqrt(7319/40 - 236196/1600)`
`=sqrt(182.98 - 147.62)`
`=sqrt(35.35)`
`=5.95`
Co-efficient of Variation `=sigma / bar x * 100 %`
`=5.95/12.15 * 100 %`
`=48.94 %`