# Coefficient of variation calculator

This coefficient of variation calculator calculates both population coefficient of variation and the sample coefficient of variation, providing solutions.

## How to use the coefficient of variation calculator

1. Enter delimited data
• You can copy the data from Excel and paste it into the calculator.
• You can choose the data delimiter under 'more options'.
2. Press the 'calculate' button.
3. After generating the plot, you will find the solution and customization buttons located below the plot.
4. The coefficient of variation calculator offers the option to exclude outliers using Tukey's fences method for better clarity.

### What is the coefficient of variation?

The coefficient of variation, also called the relative standard deviation (RSD), is a standardized measure of dispersion that represents the ratio between the standard deviation and the mean.

#### Coefficient of variation formula

Use this formula when you have knowledge of the population's standard deviation

 CV = σ x̄

#### Sample coefficient of variation formula

Use this formula when you are only able to estimate the standard deviation.

 Sample CV = S x̄

### Which measurement is better coefficient of variation or standard deviation?

When should I use the coefficient of variation?
The coefficient of variation is better for comparing the level of variation between groups with different means.
It also lacks a unit, allowing for comparisons between groups with different units.

When should I use the standard deviation?
For non ratio scale variable, the coefficient of variation is meaningless.
Since the standard deviation contains a unit, it is more intuitive.
The standard deviation is useful for constructing confidence intervals for the mean.
When the data contains positive and negative numbers and the mean is close to zero, the CV may not provide an accurate measure.

### Coefficient of variation example

The weight of a group of cats is (kg): 5.96, 4.93, 4.04, 4.63, 6.12, 4.59, 5.73, 4.41, 4.28, 4.99 (Mean=4.97, SD=0.73, CV=0.147).

The weight of a group of cows is (kg): 802.07, 800.38, 799.26, 793.42, 795.41, 792.46, 806.14, 794.71, 793.21, 797.19 (Mean=797.42, SD=4.47, CV=0.0056).

If you compare the magnitudes of the standard deviations, you find that the standard deviation of the cows (4.47) is much larger than the standard deviation of the cats (0.73).
It looks like the weight of the cows has much higher variation than the weight of the cats, is it?
The cow is a larger animal, with an average weight of ~800kg compared to an average of 5kg for the cats, hence we expect the standard deviation to be larger.

We all can agree that despite the fact the standard deviation of the cows is larger than the standard deviation of the cats, it looks like the dispersion of the cats is much larger than the dispersion of the cows, and the coefficient of variation of the cats (0.147) is really larger than the coefficient of variation of the cows (0.0056).

### Example of Incorrect Usage of the Coefficient of Variation

The measured temperature in Celsius are:
101.98,95.39,98.78,104.40,105.19,100.95 (Mean=101.11, SD=3.64, CV=0.036)

When you convert the temperatures to Kelvin:
-171.17,-177.76,-174.37,-168.75,-167.96,-172.20 (Mean=-172.03, SD=3.64, CV=-0.021)

The coefficient of variation doesn't have a unit. Therefore, one would expect to obtain the same result for the same temperatures in different units. However, this is not the case. The reason for this discrepancy is that Kelvin is a ratio scale variable, where there is a meaningful interpretation of the average, allowing the use of the coefficient of variation (CV). On the other hand, Celsius is an interval scale variable, where the average lacks meaningful significance, and thus the CV should not be used.

You may notice that the standard deviations are the same for Celsius and Kelvin, but this is only because a temperature change of 1 degree Celsius is equal to a temperature change of 1 Kelvin.

We invented another ratio scale call Chiki, where Chiki = 2 * Kelvin:
-342.34 -355.52 -348.74 -337.50 -335.92 -344.40 (Mean=-344.07, SD=7.29, CV=-0.021)

In this case, since Kelvin and Chiki are ratio scale variables, we obtain the same CV. However, due to the fact that a temperature change of 1 degree Celsius is not equal to a temperature change of 1 Chiki, we don't obtain the same standard deviation.

#### Steps for calculating the coefficient of variation for the cats group

 x̄  = Σxi = 5.96+4.93+4.04+4.63+6.12+4.59+5.73+4.41+4.28+4.99 = 4.968 n 10
Sums of squares = Σ(xi - x̄)² = (5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)²+(5.96 - 4.968)² = 4.7968
 σ2  = Σ(xi - x̄)2 = 4.797 =  0.69262 n 10
 S2  = Σ(xi - x̄)2 = 4.797 =  0.73012 n-1 9
 CV = σ = 0.693 = 0.1394 x̄ 4.968
 Sample CV= S = 0.73 = 0.147 x̄ 4.968