Coefficient of Variation Calculator

CV = (s ÷ x̄) × 100%

s

The standard deviation of the list, in the original units. It is printed beside the answer, because without it the percentage cannot be checked

x̄

The mean of the same list. Dividing by it is the whole idea: the same spread is a small variation in a large quantity and a large variation in a small one. It must be greater than zero

s ÷ x̄

The ratio itself, dimensionless — the units cancel. That is what lets a spread in kilograms be compared with a spread in millimetres

× 100

Turn the ratio into a percentage. The coefficient of variation is written both ways in textbooks, as 0.4276 and as 42.76%; this page prints the percentage, and the worked examples below are all in percent so the two never disagree

n

How many values the list holds, up to 200 here. Whether the standard deviation used n − 1 or n is the sample-or-population choice on the panel, and it moves the answer

Use it whenever two spreads have to be compared and the quantities are not in the same units or not of the same size, which is most of the time: the variability of a laboratory assay across runs, the consistency of a filling machine against its target, the risk per unit of return in a set of measurements. It is also the honest answer to "is this spread large?", because a standard deviation alone cannot be large or small — only large or small relative to the mean it belongs to. Do not use it on data whose mean is near zero or negative. The ratio is undefined at zero and meaningless below it: the same spread would report a negative percentage, which reads as a very consistent measurement when it is nothing of the kind. Temperatures on the Celsius scale, gains and losses, and deviations from a baseline are all excluded for this reason, not because the arithmetic is hard but because the answer would be a number with no interpretation. Note also that there is no band or grade on this page — see the notes below for why.

1. The default list, as a sample

   1. Mean: 40 ÷ 8 = 5
   2. Sample standard deviation: 2.1381
   3. Ratio: 2.1381 ÷ 5 = 0.427618…
   4. As a percentage: 42.76%

   Read the last step carefully, because it is where this page goes wrong if it goes wrong anywhere: the ratio is 0.4276 and the answer shown is 42.76%. A page that forgot to multiply by 100 would print 0.43%, which looks like a perfectly reasonable coefficient of variation — small, plausible, and wrong by a factor of a hundred. The two fixed values printed beside it are the check: 2.1381 over 5 really is about 0.43, so about 43%.

2. The same list as a population

   1. Mean: still 5
   2. Population standard deviation: 2
   3. Ratio: 2 ÷ 5 = 0.4
   4. As a percentage: 40.00%

   42.76% and 40.00% from one list, differing only in the divisor used for the standard deviation. Dividing by the mean compresses the difference rather than removing it, which is worth remembering when comparing a figure from this page with one from a spreadsheet: if the two disagree slightly, the sample-or-population choice is the first thing to check, and the standard deviation printed on the panel will say which one was used.

3. No spread at all

   1. Mean: 20 ÷ 2 = 10
   2. Standard deviation: 0, since both values are identical
   3. Ratio: 0 ÷ 10 = 0
   4. As a percentage: 0.00%

   A coefficient of variation of zero is a real answer, not a failure: every reading is identical, so there is no variation relative to the mean. It is the one case where a zero is informative. It is also why the condition on this page is a positive mean rather than a non-zero standard deviation — zero spread over a positive mean is a perfectly well-defined 0%, while any spread at all over a mean of zero is undefined.

The mean must be positive, and that excludes more data than people expect: temperatures on a Celsius or Fahrenheit scale, gains and losses, and any measurement expressed as a deviation from a baseline all fail the condition, because their means can sit at or below zero. The reason is not a fussy guard but the ratio itself — divide the same spread by a negative mean and the answer comes out negative, which reads as a very consistent measurement when it is the opposite. There is also deliberately no band or grade on this page. Thresholds like "under 10% is low variability" circulate widely and are worth nothing here, because none of them can be traced to a source and they change with the field: analytical chemistry treats 5% as a line, survey research is untroubled by 30%. Publishing one would mean inventing a cut-off and presenting it as a standard. A third caveat is that the coefficient of variation is a ratio of two estimated quantities, so with a small sample it is unstable — the mean in the denominator is itself uncertain, and the percentage inherits that. Finally, the list is capped at 200 values, and a token like 1,500 is refused rather than guessed at; write 1500 or 1.5.

What is the coefficient of variation?

It is the standard deviation divided by the mean, usually written as a percentage: a spread expressed relative to the size of the quantity it belongs to. A standard deviation of 8 kg means one thing around a mean of 62 kg and quite another around a mean of 800 kg, and dividing turns both into a number that can be compared — 12.9% against 1%. Because the units cancel, the result is dimensionless.

What is the CV formula?

CV = (standard deviation ÷ mean) × 100%. Take the list 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the sample standard deviation is 2.1381, so the ratio is 0.4276 and the coefficient of variation is 42.76%. The panel prints the standard deviation and the mean beside the answer so the division can be checked without working them out again.

Why can't the mean be zero or negative?

Because the answer would be undefined at zero and misleading below it. Dividing by a mean of zero has no result at all, and dividing the same spread by a negative mean gives a negative percentage — which reads as a very consistent measurement when it is nothing of the kind. That rules out temperatures on the Celsius scale, gains and losses, and any reading expressed as a deviation from a baseline. Those data can still be described by a standard deviation; they just cannot be described by this ratio.

What counts as a high coefficient of variation?

There is no general answer, and this page deliberately gives no bands. Rules of thumb such as "under 10% is low variability" circulate widely, but they cannot be traced to a source and they change with the field — analytical chemistry commonly sets the line at 5% for a validated assay, while a survey researcher may not blink at 30%. The right comparison is against other measurements of the same kind, where the question becomes whether this run is more variable than the runs before it.

Is it the same as the relative standard deviation?

Yes — relative standard deviation, RSD, and coefficient of variation are three names for the same quantity. Laboratories tend to say RSD and write it as a percentage, statistics texts tend to say coefficient of variation and sometimes write it as the bare ratio 0.4276 instead of 42.76%. That last difference is only a factor of a hundred and a percent sign, and it is a common source of confusion when a figure from one source is compared with a figure from another.

Can a coefficient of variation be negative?

Not from this page, and a negative one anywhere is a sign something went wrong. The standard deviation is never negative, so the sign of the ratio can only come from the mean — and a negative mean is exactly the case this page refuses to answer. A negative coefficient of variation is therefore not a very consistent measurement; it is either a misread decimal place or a mean that should not have been used as a denominator.

• Normal Distribution — e-Handbook of Statistical Methods, section 1.3.6.6.1 (which lists the coefficient of variation as σ / μ) — National Institute of Standards and Technology (NIST)
• Measures of the Spread of the Data — Introductory Statistics 2e, section 2.7 — OpenStax, Rice University