Coefficient of Variation: Meaning, Formula and Examples

Standard Deviation is an absolute measure of dispersion, which means it cannot be used to compare the variability of datasets expressed in different units. To make such comparisons possible, relative measures are used, the most common being the Coefficient of Variation.

The Coefficient of Variation, introduced by Karl Pearson is a relative measure that expresses the ratio of Standard Deviation to the Mean. It is useful for comparing datasets in terms of stability, homogeneity or consistency and for analyzing variability.

Formula

Coefficient of Variation is expressed as percentage and can be determined using the following:

Coefficient~of~Variation(C.V.)=\frac{\sigma}{\bar{X}}\times{100}

Where,

• C.V. = Coefficient of Variation
• σ = Standard Deviation
• \bar{X}   = Arithmetic Mean

Let's see few examples to understand Coefficient of Variation better.

Example 1

Series X: [11.5, 46, 80.5]. We need to find the Coefficient of Variance of series X.

Solution:

• Mean \bar{X}  of series X= \frac{\sum X}{N} = \frac{138}{3} = 46
• Standard Deviation σ_X = \sqrt{\frac{1}{N} \sum_{i=1}^N \left(X_i - \bar{X} \right)^2} = 25.5
• Coefficient of Variation of X = \frac{25.5}{46}\times{100}=55.43{\%}

C.V. of Series X = 55.43%.

Example 2

The number of points scored by two teams in a hockey match is given below. With the help of Coefficient of Variation, we need to determine which team is more consistent.

Solution:

Coefficient of Variation (Team A):

• Mean \bar{X}_A=\frac{\sum{X_A}}{N}=\frac{42}{6}=7
• Standard Deviation \sigma_A=\sqrt{\frac{\sum{x^2_A}}{N}}=\sqrt{\frac{252}{6}}=6.48
• Coefficient of Variation (Team A) = \frac{\sigma}{\bar{X}}\times{100}=\frac{6.48}{7}\times{100}
• C.V. (Team A) = 92.5%

Coefficient of Variation (Team B):

• Mean \bar{X}_B=\frac{\sum{X_B}}{N}=\frac{42}{6}=7
• Standard Deviation \sigma_B=\sqrt{\frac{\sum{x^2_B}}{N}}=\sqrt{\frac{118}{6}}=4.42
• Coefficient of Variation (Team B) = \frac{\sigma}{\bar{X}}\times{100}=\frac{4.39}{7}\times{100}
• C.V. (Team B) = 63.24%

Coefficient of Variation of Team B is less, i.e., 63.24% so Team B is more consistent.

Interpretation of C.V.

Ways to interpret Coefficient of Variation (C.V.):

1. Low C.V: Indicates that the data points are closely clustered around the mean showing low relative variability and more consistency.
2. High C.V: Suggests that the data has high relative variability compared to the mean, meaning the values are more spread out.
3. Cross Dataset Comparison: It allows you to compare the variability of two or more datasets even if they have different units or scales.

Coefficient of Variation in Python

Python implementation of coefficient of variation using Numpy:

import numpy as np

def calculate_coefficient_of_variation(data):
    
    mean = np.mean(data)
    std_dev = np.std(data)

    if mean == 0:
        return None  

    cv = (std_dev / mean) * 100 
    return cv

data = [30, 35, 40, 20, 50, 45, 35]
cv = calculate_coefficient_of_variation(data)
if cv is not None:
    print(f"Coefficient of Variation for data: {cv:.2f}%")
else:
    print("Cannot calculate CV for data: Mean is zero.")

Output:

Coefficient of Variation for data: 25.11%

What Is Considered a Good Coefficient of Variation?

In fields like finance or biology, a slightly higher C.V. might still be acceptable while in engineering or manufacturing, a lower CV is preferred for reliability.

1. Below 10%: Usually considered very good, meaning the data is highly consistent with little variation.
2. 10–20%: Acceptable in many real-world applications indicating moderate variability.
3. Above 20%: Often seen as high variation, which may signal instability or inconsistency in the dataset.

Applications

Here are some of the key applications of Coefficient of Variance.

1. Finance and Investment: Used to compare the risk-to-return ratio of different assets or portfolios.
2. Manufacturing and Quality Control: Helps measure consistency in production processes and detect variability in product quality.
3. Healthcare and Medicine: Applied to assess variability in clinical trials, patient measurements or lab test results.
4. Biology and Environmental Science: Used to compare variability in population studies, growth rates or environmental measurements.
5. Education and Research: Helps analyze student performance, survey results or experimental data for consistency and reliability.

Comparison with Other Measures

Comparison table among Standard Deviation, Variance and Coefficient of Variance.