Population Variance, Standard deviation and coefficient of variation for mixed data Formula & Example

Formula

1. Mean

ˉx=∑fxn $bar x = (sum fx)/n$

2. Population Variance

σ2=∑f⋅x2-(∑f⋅x)2nn $sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n$

3. Population Standard deviation

σ=√∑f⋅x2-(∑f⋅x)2nn $sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)$

4. Coefficient of Variation (Population)

=σˉx⋅100% $=sigma / bar x * 100 %$

Examples
1. Calculate Population Variance (σ2) $(sigma^2)$ , Population Standard deviation (σ) $(sigma)$ , Population Coefficient of Variation from the following mixed data
Class Frequency
1 3
2 4
5 10
6 - 10 23
10 - 20 20
20 - 30 20
30 - 50 15
50 - 70 3
70 - 100 2

Solution:

Class (1) $(1)$	Frequency (f) $(f)$ (2) $(2)$	Mid value (x) $(x)$ (3) $(3)$	f⋅x $f*x$ (4)=(2)×(3) $(4)=(2)xx(3)$	f⋅x2=(f⋅x)×(x) $fx^2=(fx)xx(x)$ (5)=(4)×(3) $(5)=(4)xx(3)$
1	3	1 1=1 $1=1$	3 3=3×1 $3=3xx1$ (4)=(2)×(3) $(4)=(2)xx(3)$	3 3=3×1 $3=3xx1$ (5)=(4)×(3) $(5)=(4)xx(3)$
2	4	2 2=2 $2=2$	8 8=4×2 $8=4xx2$ (4)=(2)×(3) $(4)=(2)xx(3)$	16 16=8×2 $16=8xx2$ (5)=(4)×(3) $(5)=(4)xx(3)$
5	10	5 5=5 $5=5$	50 50=10×5 $50=10xx5$ (4)=(2)×(3) $(4)=(2)xx(3)$	250 250=50×5 $250=50xx5$ (5)=(4)×(3) $(5)=(4)xx(3)$
6 - 10	23	8 8=6+102 $8=(6+10)/2$	184 184=23×8 $184=23xx8$ (4)=(2)×(3) $(4)=(2)xx(3)$	1472 1472=184×8 $1472=184xx8$ (5)=(4)×(3) $(5)=(4)xx(3)$
10 - 20	20	15 15=10+202 $15=(10+20)/2$	300 300=20×15 $300=20xx15$ (4)=(2)×(3) $(4)=(2)xx(3)$	4500 4500=300×15 $4500=300xx15$ (5)=(4)×(3) $(5)=(4)xx(3)$
20 - 30	20	25 25=20+302 $25=(20+30)/2$	500 500=20×25 $500=20xx25$ (4)=(2)×(3) $(4)=(2)xx(3)$	12500 12500=500×25 $12500=500xx25$ (5)=(4)×(3) $(5)=(4)xx(3)$
30 - 50	15	40 40=30+502 $40=(30+50)/2$	600 600=15×40 $600=15xx40$ (4)=(2)×(3) $(4)=(2)xx(3)$	24000 24000=600×40 $24000=600xx40$ (5)=(4)×(3) $(5)=(4)xx(3)$
50 - 70	3	60 60=50+702 $60=(50+70)/2$	180 180=3×60 $180=3xx60$ (4)=(2)×(3) $(4)=(2)xx(3)$	10800 10800=180×60 $10800=180xx60$ (5)=(4)×(3) $(5)=(4)xx(3)$
70 - 100	2	85 85=70+1002 $85=(70+100)/2$	170 170=2×85 $170=2xx85$ (4)=(2)×(3) $(4)=(2)xx(3)$	14450 14450=170×85 $14450=170xx85$ (5)=(4)×(3) $(5)=(4)xx(3)$
---	---	---	---	---
	n=100 $n = 100$	-----	∑f⋅x=1995 $sum f*x=1995$	∑f⋅x2=67991 $sum f*x^2=67991$

Mean

ˉx=∑fxn $bar x = (sum fx)/n$

=1995100 $=1995/100$

=19.95 $=19.95$

Population Variance

σ2=∑f⋅x2-(∑f⋅x)2nn $sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n$

=67991-(1995)2100100 $=(67991 - (1995)^2/100)/100$

=67991-39800.25100 $=(67991 - 39800.25)/100$

=28190.75100 $=28190.75/100$

=281.9075 $=281.9075$

Population Standard deviation

σ=√∑f⋅x2-(∑f⋅x)2nn $sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)$

=√67991-(1995)2100100 $=sqrt((67991 - (1995)^2/100)/100)$

=√67991-39800.25100 $=sqrt((67991 - 39800.25)/100)$

=√28190.75100 $=sqrt(28190.75/100)$

=√281.9075 $=sqrt(281.9075)$

=16.7901 $=16.7901$

Coefficient of Variation (Population)

=σˉx⋅100% $=sigma / bar x * 100 %$

=16.790119.95⋅100% $=16.7901/19.95 * 100 %$

=84.16% $=84.16 %$

2. Calculate Population Variance (σ2) $(sigma^2)$ , Population Standard deviation (σ) $(sigma)$ , Population Coefficient of Variation from the following 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) $(1)$	Frequency (f) $(f)$ (2) $(2)$	Mid value (x) $(x)$ (3) $(3)$	f⋅x $f*x$ (4)=(2)×(3) $(4)=(2)xx(3)$	f⋅x2=(f⋅x)×(x) $fx^2=(fx)xx(x)$ (5)=(4)×(3) $(5)=(4)xx(3)$
2	1	2	2	4
3	2	3	6	18
4	2	4	8	32
5 - 9	8	7	56	392
10 - 14	15	12	180	2160
15 - 19	8	17	136	2312
20 - 29	4	24.5	98	2401
---	---	---	---	---
	n=40 $n = 40$	-----	∑f⋅x=486 $sum f*x=486$	∑f⋅x2=7319 $sum f*x^2=7319$

Mean

ˉx=∑fxn $bar x = (sum fx)/n$

=48640 $=486/40$

=12.15 $=12.15$

Population Variance

σ2=∑f⋅x2-(∑f⋅x)2nn $sigma^2 = (sum f*x^2 - (sum f*x)^2/n)/n$

=7319-(486)24040 $=(7319 - (486)^2/40)/40$

=7319-5904.940 $=(7319 - 5904.9)/40$

=1414.140 $=1414.1/40$

=35.3525 $=35.3525$

Population Standard deviation

σ=√∑f⋅x2-(∑f⋅x)2nn $sigma = sqrt((sum f*x^2 - (sum f*x)^2/n)/n)$

=√7319-(486)24040 $=sqrt((7319 - (486)^2/40)/40)$

=√7319-5904.940 $=sqrt((7319 - 5904.9)/40)$

=√1414.140 $=sqrt(1414.1/40)$

=√35.3525 $=sqrt(35.3525)$

=5.9458 $=5.9458$

Coefficient of Variation (Population)

=σˉx⋅100% $=sigma / bar x * 100 %$

=5.945812.15⋅100% $=5.9458/12.15 * 100 %$

=48.94% $=48.94 %$

This material is intended as a summary. Use your textbook for detail explanation.
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1. Formula & Example