Formula
1. Mean ˉx=∑fxn
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2. Population Variance σ2=∑f⋅x2-(∑f⋅x)2nn
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3. Population Standard deviation σ=√∑f⋅x2-(∑f⋅x)2nn
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4. Coefficient of Variation (Population) =σˉx⋅100%
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Examples
1. Calculate Population Variance (σ2), Population Standard deviation (σ), 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) | Frequency (f) (2) | Mid value (x) (3) | f⋅x (4)=(2)×(3) | f⋅x2=(f⋅x)×(x) (5)=(4)×(3) |
1 | 3 | 1 1=1 | 3 3=3×1 (4)=(2)×(3) | 3 3=3×1 (5)=(4)×(3) |
2 | 4 | 2 2=2 | 8 8=4×2 (4)=(2)×(3) | 16 16=8×2 (5)=(4)×(3) |
5 | 10 | 5 5=5 | 50 50=10×5 (4)=(2)×(3) | 250 250=50×5 (5)=(4)×(3) |
6 - 10 | 23 | 8 8=6+102 | 184 184=23×8 (4)=(2)×(3) | 1472 1472=184×8 (5)=(4)×(3) |
10 - 20 | 20 | 15 15=10+202 | 300 300=20×15 (4)=(2)×(3) | 4500 4500=300×15 (5)=(4)×(3) |
20 - 30 | 20 | 25 25=20+302 | 500 500=20×25 (4)=(2)×(3) | 12500 12500=500×25 (5)=(4)×(3) |
30 - 50 | 15 | 40 40=30+502 | 600 600=15×40 (4)=(2)×(3) | 24000 24000=600×40 (5)=(4)×(3) |
50 - 70 | 3 | 60 60=50+702 | 180 180=3×60 (4)=(2)×(3) | 10800 10800=180×60 (5)=(4)×(3) |
70 - 100 | 2 | 85 85=70+1002 | 170 170=2×85 (4)=(2)×(3) | 14450 14450=170×85 (5)=(4)×(3) |
--- | --- | --- | --- | --- |
| n=100 | ----- | ∑f⋅x=1995 | ∑f⋅x2=67991 |
Mean
ˉx=∑fxn=1995100=19.95
Population Variance
σ2=∑f⋅x2-(∑f⋅x)2nn=67991-(1995)2100100=67991-39800.25100=28190.75100=281.9075
Population Standard deviation
σ=√∑f⋅x2-(∑f⋅x)2nn=√67991-(1995)2100100=√67991-39800.25100=√28190.75100=√281.9075=16.7901
Coefficient of Variation (Population)
=σˉx⋅100%=16.790119.95⋅100%=84.16%
2. Calculate Population Variance (σ2), Population Standard deviation (σ), 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) | Frequency (f) (2) | Mid value (x) (3) | f⋅x (4)=(2)×(3) | f⋅x2=(f⋅x)×(x) (5)=(4)×(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 | ----- | ∑f⋅x=486 | ∑f⋅x2=7319 |
Mean
ˉx=∑fxn=48640=12.15
Population Variance
σ2=∑f⋅x2-(∑f⋅x)2nn=7319-(486)24040=7319-5904.940=1414.140=35.3525
Population Standard deviation
σ=√∑f⋅x2-(∑f⋅x)2nn=√7319-(486)24040=√7319-5904.940=√1414.140=√35.3525=5.9458
Coefficient of Variation (Population)
=σˉx⋅100%=5.945812.15⋅100%=48.94%
This material is intended as a summary. Use your textbook for detail explanation.
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