Coefficient of Variation Calculator

How to Use the Coefficient of Variation Calculator

This coefficient of variation calculator helps you calculate the sample and population coefficient of variation for either raw or summary data. The resulting coefficient of variation (CV) tells you how large the standard deviation is compared to the mean. In simple terms, it tells you how much relative variability exists in your data.

To use the calculator:

1. Choose Raw Data if you have raw data, either from a population or a sample, or Summary Data if you have the summary values (standard deviation and mean), either from a population or a sample
2. Select Sample CV if you want to calculate the coefficient of variation from sample data, or Population CV if you want to find the coefficient of variation from population data.
3. For raw data values, you can separate them using commas, spaces, tabs, or line breaks. You can also paste values directly from Excel, Google Sheets, or text documents. However, for summary data, enter the standard deviation and the mean.
4. Click Calculate.

The calculator will instantly return the coefficient of variation both as a ratio and as a percentage. It will also provide a clear, step-by-step solution, showing exactly how the coefficient of variation was computed for your data.

Example 1: Raw Data

A quality control officer records the weights, in grams, of a sample of 5 packaged items:

48, 52, 54, 56, 60

Find the sample coefficient of variation.

Solution

From the question, we’re given raw data values, which are from a sample. Thus, to find the coefficient of variation using the calculator:

1. Select the Raw Data option.
2. Select Sample CV (since we want to compute the coefficient of variation from sample data).
3. Copy and paste the data values in the data input field
4. Click Calculate.

The calculator will instantly return the coefficient of variation both as a decimal and a percentage as follows:

• CV = 0.082817
• CV% = 8.281733%

A coefficient of variation of 8.28% suggests that the sample standard deviation is about 8.28% of the sample mean. Practically, since the CV is relatively low, we can conclude that the weights are fairly consistent relative to their average weight.

Example 2: Summary Data

A researcher compares the consistency of scores in a training program. A sample of participants has a sample mean score of 72 and a sample standard deviation of 9.6. Find the coefficient of variation.

Solution

From the question, we know that:

• Sample mean, x̄ = 72
• Sample standard deviation, s = 9.6

With the calculator, you don’t have to memorize the coefficient of variation formula. Just follow these steps:

1. Switch to the Summary Data option.
2. Select the Sample CV option since we have summary data from a sample.
3. Enter the sample mean as 72 and the sample standard deviation as 9.6
4. Click Calculate.

The calculator will return the coefficient of variation both as a decimal and a percentage as follows:

• CV = 0.133333
• CV% = 13.333333%

A coefficient of variation of about 13.33% suggest that the sample standard deviation is about 13.33% of the sample mean. In other words, the scores show moderate relative variability around the mean.

What Is the Coefficient of Variation?

The coefficient of variation (CV) is a 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

From the definition, the coefficient of variation is a ratio of the standard deviation to the mean of a dataset. It is often expressed either as a ratio or a percentage. Therefore, the general coefficient of variation formula is:

CV = Standard Deviation/Mean

As a percentage, the formula becomes: CV = (Standard Deviation/Mean) x 100

From the general CV formula, we can write the population CV and Sample CV formulas as follows:

• Sample Coefficient of Variation Formula, CV = s / x̄ or CV% = (s / x̄) × 100%
• Population Coefficient of Variation Formula, CV = σ / μ or CV% = (σ / μ) × 100%

Where:

• CV is the coefficient of variation as a ratio.
• CV% is the coefficient of variation as a percentage.
• 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:

1. Find the mean.
2. Find the standard deviation.
3. Divide the standard deviation by the mean.
4. Multiply by 100% if you want the CV as a percentage.

Example 1

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 use the sample coefficient of variation. To find the CV manually, follow these steps:

Step 1: Find the mean.

By definition, the sample mean formula is: x̄ = Σx / n

From the dataset:

Σx = 210 + 225 + 230 + 240 + 245

= 1150

There are 5 observations. Thus, n = 5

Therefore: x̄ = 1150 / 5

= 230

Thus, the sample mean, x̄ = 230.

Want to find this value quickly and see the steps? Use 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

Thus, the sample standard deviation, s = 13.693064. Alternatively, you can quickly find this standard deviation using the sample standard deviation calculator.

Step 3: Divide the standard deviation by the mean.

By definition, the sample coefficient of variation as a ratio is:

CV = s / x̄

Substituting the values gives:

CV = 13.693064 / 230

= 0.059535

Step 4: Multiply by 100% to express the CV as a percentage.

CV% = 0.059535 × 100%

= 5.953506%

Therefore, the sample coefficient of variation is 5.953506%. This suggests that the sample standard deviation is about 5.95% of the sample mean.

Example 2

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 use the population coefficient of variation. To find the CV manually, follow these steps:

Step 1: Find the mean.

By definition, the population mean formula is: μ = Σx / N

From the dataset:

Σx = 50 + 55 + 60 + 65 + 70

= 300

There are 5 values in the population. Thus, N = 5

Therefore, μ = 300 / 5

= 60

Thus, the population mean, μ = 60. Want a tool that computes the population mean quickly and shows the steps? Use the population mean calculator.

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

Since N = 5, we can find the population standard deviation as follows:

σ = √[250 / 5]

= √50

= 7.071068

Thus, the population standard deviation, σ = 7.071068.

Want to quickly find the population standard deviation without manual computations? Our population standard deviation calculator might be useful.

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 into the formula and solving gives:

CV = 7.071068 / 60

= 0.117851

Step 4: Multiply by 100% to express the CV as a percentage.

CV% = 0.117851 × 100%

= 11.785113%

Therefore, the population coefficient of variation is 11.785113%. This suggests 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 be useful for both quick answers and learning.

It allows you to:

• Calculate CV from raw data
• Calculate CV from summary data
• Choose a sample CV or a population CV
• Get CV as a ratio
• Get CV as a percentage
• See the mean and standard deviation used in the calculation
• Review a clear step-by-step solution
• Copy the solution for later use

Frequently Asked Questions