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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 %`