Last modified on January 2nd, 2025

chapter outline

 

Coefficient of Variation

The coefficient of variation (CV) is a measurement in statistics that shows the relative dispersion of data points with respect to its mean. 

Unlike standard deviation, which measures absolute variability, CV measures variability in decimal form or as a percentage. It is used to compare the variability of datasets with different units or scales, such as comparing financial returns or experimental results.

Mathematically, it is the ratio of the standard deviation to the mean. 

Formula

The formula to determine the coefficient of variation depends on whether the dataset represents an entire group or just a subset (sample coefficient of variation).

Population Coefficient of Variation vs Coefficient of Variation

Population Coefficient of Variation

If the dataset represents an entire group, it is called the population coefficient of variation. The formula to find the coefficient of variation for a population is:

CV = ${\dfrac{\sigma }{\mu }\times 100\%}$

Here,

  • σ = ${\sqrt{\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}}$ is the population standard deviation
  • μ is the mean for the population

Sample Coefficient of Variation

If the dataset represents only a subset of an entire group, it is called the sample coefficient of variation. The formula to find it is:

CV = ${\dfrac{s}{\overline{x}}\times 100\%}$

Here,

  • s = ${\sqrt{\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}}$ is the sample standard deviation
  • ${\overline{x}}$ is the mean for the sample

Steps To Find

Population Coefficient of Variation

Let us find the coefficient of variation for the dataset {4.5, 5.0, 4.8, 5.1, 4.9}

Finding the Mean (μ) 

μ = ${\dfrac{4.5+5.0+4.8+5.1+4.9}{5}}$ = ${\dfrac{24.3}{5}}$ = 4.86

Finding the Population Standard Deviation (σ)

As we know, σ = ${\sqrt{\dfrac{\sum \left( x_{i}-\mu \right) ^{2}}{N}}}$

Here, N = 5

Now, calculating the squared deviations, we get

(4.5 – 4.86)2 = 0.1296

(5.0 – 4.86)2 = 0.0196

(4.8 – 4.86)2 = 0.0036

(5.1 – 4.86)2 = 0.0576

(4.9 – 4.86)2 = 0.0016

Adding the squared deviations, we get

${\sum \left( x_{i}-\mu \right) ^{2}}$ = 0.1296 + 0.0196 + 0.0036 + 0.0576 + 0.0016 = 0.212

Thus, σ = ${\sqrt{\dfrac{0.212}{5}}}$ = 0.206

Calculating the Population Coefficient of Variation

CV = ${\dfrac{\sigma }{\mu }\times 100\%}$

= ${\dfrac{0.206}{4.86}\times 100\%}$

= 4.24%

Thus, the Coefficient of Variation is 4.24%

Sample Coefficient of Variation

Let us find the coefficient of variation for the sample {12, 14, 15, 11, 13, 12}

Finding the Mean (${\overline{x}}$)

${\overline{x}}$ = ${\dfrac{12+14+15+11+13+12}{6}}$ = ${\dfrac{77}{6}}$ = 12.83

Finding the Sample Standard Deviation (s)

As we know, s = ${\sqrt{\dfrac{\sum \left( x_{i}-\overline{x}\right) ^{2}}{n-1}}}$

Here, n = 6

Now, calculating the squared deviations, we get

(12 – 12.83)2 = 0.6889

(14 – 12.83)2 = 1.3689

(15 – 12.83)2 = 4.6889

(11 – 12.83)2 = 3.3489

(13 – 12.83)2 = 0.0289

(12 – 12.83)2 = 0.6889

Adding the squared deviations, we get

${\sum \left( x_{i}-\overline{x}\right) ^{2}}$ = 0.6889 + 1.3689 + 4.6889 + 3.3489 + 0.0289 + 0.6889 = 10.8134

Thus, s = ${\sqrt{\dfrac{10.8134}{6-1}}}$ = ${\sqrt{\dfrac{10.8134}{5}}}$ = 1.47

Calculating the Sample Coefficient of Variation

CV = ${\dfrac{s}{\overline{x}}\times 100\%}$

= ${\dfrac{1.47}{12.83}\times 100\%}$

= 11.45%

Thus, the Coefficient of Variation is 11.45%

Coefficient of Variation (CV) vs Standard Deviation (SD)

Both the coefficient of variation and the standard deviation are commonly used to measure the dispersion of values in a dataset. Although both metrics measure variability, their purposes and applications differ significantly.

BasisCoefficient of Variation (CV)Standard Deviation (SD)
Type of DispersionMeasures the relative variability as a percentage of the meanMeasures the absolute variability in a dataset
UnitsIt is unitlessIt has the same unit as the data
Impact of Small or Zero MeansCan get affected Does not get affected
UsesIt is used to compare the variation of different datasets with different units/scalesIt is used to measure the dispersion of data around the mean within a single dataset

Solved Examples

Find the population coefficient of variation for the dataset: {8, 10, 12, 14, 16}

Solution:

As we know, 
CV = ${\dfrac{\sigma }{\mu }\times 100\%}$
Here,
N = 5
μ = ${\dfrac{8+10+12+14+16}{5}}$ = ${\dfrac{60}{5}}$ = 12
Now, ${\sum \left( x_{i}-\mu \right) ^{2}}$ 
= (8 – 12)2 + (10 – 12)2 + (12 – 12)2 + (14 – 12)2 + (16 – 12)2 
= 16 + 4 + 0 + 4 + 16
= 40
Thus, σ = ${\sqrt{\dfrac{40}{5}}}$ = ${\sqrt{8}}$ = 2.83
Here, CV = ${\dfrac{2.83}{12}\times 100\%}$ ≈ 23.58%
Thus, the coefficient of variation for the dataset is approximately 23.58%

If two samples have the following data:
Sample 1: {5, 10, 15, 20, 25}
Sample 2: {10, 20, 30, 40, 50}
Which sample has greater relative variability?

Solution:

As we know, 
CV = ${\dfrac{s}{\overline{x}}\times 100\%}$
For sample 1, 
${\overline{x}}$ = ${\dfrac{5+10+15+20+25}{5}}$ = ${\dfrac{75}{5}}$ = 15
${\sum \left( x_{i}-\overline{x}\right) ^{2}}$ = (5 – 15)2 + (10 – 15)2 + (15 – 15)2 + (20 – 15)2 + (25 – 15)2 = 250
s = ${\sqrt{\dfrac{250}{5-1}}}$ = ${\sqrt{62.5}}$ = 7.91
Thus, CV1 = ${\dfrac{7.91}{15}\times 100\%}$ ≈ 52.73% …..(i)
For sample 2,
${\overline{x}}$ = ${\dfrac{10+20+30+40+50}{5}}$ = ${\dfrac{150}{5}}$ = 30
${\sum \left( x_{i}-\overline{x}\right) ^{2}}$ = (10 – 30)2 + (20 – 30)2 + (30 – 30)2 + (40 – 30)2 + (50 – 30)2 = 1000
s = ${\sqrt{\dfrac{1000}{5-1}}}$ = ${\sqrt{250}}$ = 15.81
Thus, CV2 = ${\dfrac{15.81}{30}\times 100\%}$ ≈ 52.7 …..(ii)
From (i) and (ii), we get
CV1 and CV2 are approximately equal
Thus, both samples have the same relative variability, with a CV of approximately 52.7%

Last modified on January 2nd, 2025