How to Use the Coefficient of Variation Calculator
This calculator helps you find the population or sample coefficient of variation from either raw data values or summary data. To use the calculator:
- Select the Data Input type. If you have the individual data points, select the Raw Data option. However, if you already know the mean and standard deviation, choose the Summary Data option.
- Select the Data Type. If you’re working with raw data, you need to choose whether the values represent a sample or population. This is important because it dictates how the standard deviation is calculated. However, for the Summary Data option, just enter the given mean and standard deviation.
- Click Calculate.
The calculator will instantly return the coefficient of variation as both a ratio and a percentage, along with a clear, step-by-step solution.
Tip. When working with raw data, you can just copy-paste the values from Excel, Google Sheets, or text documents. Alternatively, you can enter the values manually, while separating them with commas, spaces, tabs, or line breaks.
What Is the Coefficient of Variation (CV)?
The coefficient of variation, or CV, is a statistical measure of relative variability that expresses the standard deviation as a percentage of the mean. While the standard deviation tells us how spread out the values are in the original units, the CV goes a step further by showing the size of the standard deviation relative to the mean. This makes CV useful when you want to compare variation between datasets with different units or different average values.
Coefficient of Variation Formula
Coefficient of variation is a ratio of the standard deviation to the mean of a dataset, which can either be expressed as a ratio or a percentage. The formula is CV = Standard Deviation/Mean as a ratio, or CV = (Standard Deviation/Mean) x 100% as a percentage.
When working with sample data, the coefficient of variation formula is often written as: CV = s / x̄ or CV = (s / x̄) × 100%. However, for population data, the formula is: CV = σ / μ or CV = (σ / μ) × 100%
Where:
- CV is the coefficient of variation
- s is the sample standard deviation.
- x̄ is the sample mean.
- σ is the population standard deviation.
- μ is the population mean.
How to Calculate the Coefficient of Variation
To calculate the coefficient of variation manually, follow these steps:
- Find the mean.
- Find the standard deviation.
- Divide the standard deviation by the mean to get the CV.
- Express the CV as a percentage
Example 1. Sample Data
A researcher records the following reaction times, in milliseconds, for a sample of students:
210, 225, 230, 240, 245
Find the coefficient of variation.
Solution
Since the data values come from a sample, we need to calculate the sample coefficient of variation. To find the sample coefficient of variation for the data by hand, follow these steps:
Step 1: Find the mean.
By definition, the sample mean formula is: x̄ = Σx / n
Using the data:
Σx = 210 + 225 + 230 + 240 + 245
= 1150
There are 5 observations. Thus, n = 5
Therefore: x̄ = 1150 / 5
= 230
You can also verify this value using the sample mean calculator.
Step 2: Find the standard deviation.
By definition, the sample standard deviation formula is: s = √[Σ(xi − x̄)² / (n − 1)]
Using x̄ = 230, the squared deviations are:
- (210 − 230)2 = 400
- (225 − 230)2 = 25
- (230 − 230)2 = 0
- (240 − 230)2 = 100
- (245 − 230)2 = 225
Therefore, Σ(xi − x̄)2 = 400 + 25 + 0 + 100 + 225
= 750
Since n = 5, we can find the sample standard deviation as follows:
s = √[750 / (5 − 1)]
= √187.5
= 13.693064
You can also verify the result using the sample standard deviation calculator.
Step 3: Divide the standard deviation by the mean.
By definition, CV = s / x̄
Substituting the values, CV = 13.693064 / 230
= 0.059535
Step 4: Express the CV as a percentage.
To express the CV as a percentage, we multiply the result in step 3 by 100%
Thus, CV = 0.059535 × 100%
= 5.953506%
This implies that the sample standard deviation is about 5.95% of the sample mean.
Example 2. Population Data
A small manufacturing company records the daily output, in units, for all 5 machines in a production line:
50, 55, 60, 65, 70
Find the coefficient of variation.
Solution
Since the data values represent all machines in the production line, we need to find the population coefficient of variation. To find the population CV manually, follow these steps:
Step 1: Find the mean.
By definition, the population mean formula is: μ = ΣX / N
Using the population data:
ΣX = 50 + 55 + 60 + 65 + 70
= 300
There are 5 values in the population. Thus, N = 5
Therefore, μ = 300 / 5
= 60
Step 2: Find the standard deviation.
By definition, the population standard deviation formula is: σ = √[Σ(Xi − μ)² / N]
Using μ = 60, the squared deviations are:
- (50 − 60)2 = 100
- (55 − 60)2 = 25
- (60 − 60)2 = 0
- (65 − 60)2 = 25
- (70 − 60)2 = 100
Therefore, Σ(xi − μ)2 = 100 + 25 + 0 + 25 + 100
= 250
Therefore, the population standard deviation, σ = √[250 / 5]
= √50
= 7.071068
You can also verify the above result using the population standard deviation calculator.
Step 3: Divide the standard deviation by the mean.
By definition, the population coefficient of variation as a ratio is: CV = σ / μ
Substituting the values, CV = 7.071068 / 60
= 0.117851
Step 4: Express the CV as a percentage.
To express the CV as a percentage, we multiply the result in Step 3 by 100%
Thus, CV= 0.117851 × 100%
= 11.785113%
This implies that the population standard deviation is about 11.79% of the population mean.
Why Use This Coefficient of Variation Calculator?
This calculator is designed to give you a quick answer while also showing how the coefficient of variation is calculated.
With the calculator, you can:
- Calculate the coefficient of variation from raw data or summary data
- Calculate sample or population CV from raw data
- Get the CV both as a ratio and percentage
- See the mean and standard deviation calculations when using raw data
- Follow a clear step-by-step solution
- Copy the solution for your notes
Frequently Asked Questions
This calculator finds the coefficient of variation from raw data or summary data. It returns the CV as both a ratio and a percentage and shows you how to find the CV manually, through a step-by-step solution approach.
The coefficient of variation is the ratio of the standard deviation to the mean. It measures relative variability.
For sample data, the coefficient of variation formula is: CV = s / x̄. However, for population data, the formula becomes: CV = σ / μ.
It can be expressed as either a ratio or a percentage. For example, CV = 0.25 is the same as CV = 25%.
A high coefficient of variation means the data have high variability relative to the mean. In many cases, this suggests less consistency.
A low coefficient of variation means the data values are more consistent relative to the mean.
