Math Buddyv1.0

Coefficient of variation calculator

Type or scan a data set. MathBuddy finds the mean, the standard deviation and their ratio.

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What this page covers

  • The coefficient of variation of a sample, using the sample standard deviation.
  • The population version, when the data is the whole group you care about.
  • The result as a decimal or as a percentage (relative standard deviation, RSD).
  • Not covered here: the standard deviation on its own (use Standard deviation calculator) and the variance (use Variance calculator).

How to enter the problem

  • Type it or paste it. Write coefficient of variation or RSD, then the numbers separated by commas. Say population if the data is the whole group.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

A sample of five measurements is 4,6,8,10,12\displaystyle 4, 6, 8, 10, 12. Find the coefficient of variation.

Answer

Verified

The answer is

CV=108≈0.395CV = \frac{\sqrt{10}}{8} \approx 0.395

Explanation

  1. Find the mean
    xˉ=4+6+8+10+125=8\bar{x} = \frac{4 + 6 + 8 + 10 + 12}{5} = 8
  2. Square the deviations from the mean
    (−4)2+(−2)2+02+22+42=40(-4)^2 + (-2)^2 + 0^2 + 2^2 + 4^2 = 40
  3. Find the sample standard deviation
    Divide by n−1=4n - 1 = 4: s2=10s^2 = 10, so s=10s = \sqrt{10}.
  4. Divide by the mean
    CV=sxˉ=108≈0.395CV = \frac{s}{\bar{x}} = \frac{\sqrt{10}}{8} \approx 0.395, about 39.5%39.5\%.

Example 2

Problem

The 4 players on a team scored 70,80,90,80\displaystyle 70, 80, 90, 80. Treat the team as the whole population and find the coefficient of variation.

Answer

Verified

The answer is

CV=216≈0.088CV = \frac{\sqrt{2}}{16} \approx 0.088

Explanation

  1. Find the mean
    μ=70+80+90+804=80\mu = \frac{70 + 80 + 90 + 80}{4} = 80
  2. Find the population variance
    The squared deviations are 100,0,100,0100, 0, 100, 0. Divide their sum by n=4n = 4
    σ2=50\sigma^2 = 50
  3. Take the square root
    σ=50=52\sigma = \sqrt{50} = 5\sqrt{2}
  4. Divide by the mean
    CV=5280=216≈0.088CV = \frac{5\sqrt{2}}{80} = \frac{\sqrt{2}}{16} \approx 0.088, about 8.8%8.8\%.

Common mistakes

  • Using the population formula (divide by nn) for a sample. For 18,24,3018, 24, 30 that gives about 0.2040.204 instead of 0.250.25.
  • Dividing the variance by the mean instead of the standard deviation.
  • Forgetting to multiply by 100 when the answer is asked as a percentage, or multiplying twice.
  • Using it for data with a mean near zero or with negative values, where the ratio has no clear meaning.

Checks, assumptions and limits

  • Check the scale. Multiplying every value by the same positive number leaves the coefficient of variation unchanged, because the mean and the standard deviation scale together.
  • The coefficient of variation is meant for data on a ratio scale with a positive mean, such as lengths, weights or times. This page uses the sample standard deviation unless you say population.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Yes. The relative standard deviation is the coefficient of variation written as a percentage: multiply the decimal by 100.

Use the sample version (divide by n − 1) when the data is a sample from a larger group. Use the population version when the data is the whole group. Say which one when you type the problem.

Yes. The mean, the squared deviations and the standard deviation each get a numbered step before the final ratio.

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