Relative Standard Deviation Calculator

RSD = (s ÷ x̄) × 100%

s

The standard deviation of the readings, in the same units as the data. Sample or population is your choice below, and it changes this number and therefore the answer

x̄

The mean of the readings. It must be strictly positive: a mean of zero puts a zero in the denominator, and a negative mean produces a ratio that means nothing as a measure of relative spread

RSD

The relative standard deviation, reported as a percentage. The value is dimensionless — the units of the standard deviation and of the mean cancel — so results from different instruments or different scales can be put side by side

n

How many readings went in, up to 200 here. Repeating a measurement more times tightens the standard deviation, so quoting the count is part of quoting the precision figure honestly

Use it whenever a spread has to be judged against the size of the thing being measured: repeat measurements of the same sample, the scatter of a calibration curve, the day-to-day variation of an assay, the noise on a sensor whose output drifts. The formula is the same one the coefficient of variation uses, and the two names travel together — which one you say depends on the field. Analytical chemistry, metrology and clinical laboratories report relative standard deviation, usually abbreviated RSD, and a percentage below about 5 is often taken as good repeatability, though the acceptable figure is set by the method and not by any universal rule. Social and economic work tends to say coefficient of variation for the same arithmetic, applied to incomes, prices or returns. Use whichever name your reader expects; the number is identical.

1. Eight values, sample standard deviation

   1. The mean is 40 / 8 = 5
   2. The sample standard deviation is 2.1381
   3. 2.1381 / 5 = 0.427618…, written as a percentage: 42.76%

   This list is deliberately spread out, and 42.76% says so: the typical distance from the mean is nearly half the mean itself. A single percentage carries the whole comparison — the same list measured in millimetres instead of units would give exactly the same number, because the units cancel in the division.

2. The same data under the population divisor

   1. Only the divisor changed: n instead of n − 1, giving a standard deviation of 2 rather than 2.1381
   2. The mean is unchanged at 5
   3. 2 / 5 = 0.4, so the relative standard deviation is 40.00%

   The two divisors differ by a factor of √(8/7) here, which moves the standard deviation by about 7% and the percentage from 42.76 to 40.00. Repeating a measurement five times is a sample of a larger process, so the n − 1 form is the usual choice; the population form is right when the list is genuinely the whole of what exists.

3. Five replicate readings of the same sample

   1. The five readings average 241 / 5 = 48.2
   2. Their sample standard deviation is 0.255
   3. 0.255 / 48.2 = 0.005289…, written as a percentage: 0.53%

   This is the shape of a real precision check: five repeats of one measurement, scattered by about a quarter of a unit around 48.2. The relative standard deviation of 0.53% is the figure that goes into the report — not the standard deviation of 0.255, which says nothing until you know the readings themselves are around 48. The same 0.255 around a mean of 1 would be 25.5%, and that is the whole reason the ratio is quoted instead.

4. All readings identical

   1. The mean is 10 and the standard deviation is 0 — neither reading differs from the other
   2. 0 / 10 = 0, so the relative standard deviation is 0.00%

   Zero is a real answer here rather than a failure: it says the measurement repeated exactly. That is possible with an instrument that reports to a fixed number of digits, and it is the one case where the relative standard deviation cannot distinguish very good precision from no measurement at all — the standard deviation of 0 is doing the talking.

The mean must be strictly positive, and that is a real restriction rather than a technicality. If the mean is zero the standard deviation is divided by zero and the answer is undefined; if the mean is negative the ratio is computable but meaningless, because a spread divided by a negative centre does not measure relative variability. Data that crosses zero — temperatures in Celsius, profit and loss, altitude around sea level — is therefore outside this tool's range, and so is any measurement on an interval scale rather than a ratio scale, since a percentage change depends on where zero sits. Two data sets can share a relative standard deviation and be completely different in shape: the figure uses only the mean and the standard deviation, so it cannot see skew, a heavy tail or a second cluster. A small number also is not automatically better — a relative standard deviation near zero can mean an instrument with coarse resolution that rounds everything to the same reading. There is no universal threshold for what counts as good: the acceptable figure comes from the method, the specification or the field's practice, which is why this page prints the number without a verdict. The list is capped at 200 values, and a token such as 1,500 is refused rather than guessed at, because a comma between digits is a decimal point in some countries and a thousands separator in others.

What is the relative standard deviation?

It is the standard deviation divided by the mean, reported as a percentage: RSD = (s ÷ x̄) × 100%. For readings of 2, 4, 4, 4, 5, 5, 7 and 9 the standard deviation is 2.1381 and the mean is 5, giving 42.76%. Dividing by the mean is what makes the figure comparable to the size of the measurement — a standard deviation of 0.255 is small next to a mean of 48.2 and gives 0.53%, while the same 0.255 next to a mean of 1 would be 25.5%.

Is it the same as the coefficient of variation?

Yes — the two names describe the same arithmetic, and the only difference is the multiplication by 100. Most software, including the reference used here, calls it the coefficient of variation and reports it as a plain ratio (0.4276 rather than 42.76%). Laboratories and metrology documents tend to call it the relative standard deviation and to write it as a percentage. So the value on this page is exactly 100 times the coefficient of variation for the same data, and the sample or population divisor is the only other thing that can make two tools disagree.

Is there a limit for what counts as good precision?

No universal one, and this page deliberately does not supply a threshold. A relative standard deviation below about 5% is commonly quoted as good repeatability in analytical chemistry, but that figure comes from the practice of a particular field and the requirements of a particular method — a geochemical assay and a clinical measurement can have very different acceptance limits. Treat a published threshold as a rule of thumb for that discipline, not as a property of the statistic. What you should do instead is compare against the precision the method or the instrument is specified to deliver.

My data includes negative numbers. What happens?

The calculation is refused, and that is by design. A relative standard deviation needs a strictly positive mean. If the mean is zero it would be dividing by zero; if the mean is negative, dividing a spread by it produces a negative percentage that says nothing useful about relative variability. The underlying reason is that this figure only makes sense for measurements on a ratio scale — one with a true zero, where 2 kg is meaningfully twice 1 kg. Celsius temperatures, profit and loss figures and altitudes above and below sea level are on interval scales, where zero is a convention, so no ratio of that kind is meaningful for them.

Should I use the sample or the population divisor?

Use n − 1, the sample form, whenever the readings are a sample of something larger — which includes the usual case of measuring one sample several times, since those repeats stand in for all the measurements you could have made. Use n, the population form, only when the list is the complete set of values that exist, such as every reading a fixed process produced in a batch you fully recorded. The difference is small for long lists and noticeable for short ones: with eight values the sample form gives 42.76% here against 40.00% for the population form, a gap of nearly three percentage points.

Can the relative standard deviation be zero?

Yes, whenever every reading is identical — the standard deviation is 0, so the answer is 0.00%. Two readings of 10 give a mean of 10 and a standard deviation of 0. That is the one case where a very small figure should be read with care rather than celebrated: an instrument that rounds its output can produce identical readings even when the underlying quantity is moving slightly, so a relative standard deviation of exactly zero may be measuring the resolution of the readout rather than the stability of the sample.

• Measures of Scale — NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.6 (defines the standard deviation this figure is built from, and states that squaring the distance from the mean gives greater weight to values far from it) — National Institute of Standards and Technology
• Normal Distribution — NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1 (lists the coefficient of variation σ / μ among the standard summaries of a distribution — the same quantity this page reports as a percentage; the handbook uses the coefficient-of-variation name) — National Institute of Standards and Technology
• GB/T 3358.1-2009, Statistics — Vocabulary and symbols, Part 1 (the Chinese national vocabulary standard covering the standard deviation and the measures of relative dispersion; the record page notes that no online full text is offered) — State Administration for Market Regulation, China