Mean, SD and %RSD calculator

Mean, sample standard deviation, %RSD and a confidence interval on the mean, from replicate results pasted straight out of a spreadsheet — with the t value and degrees of freedom shown, so the arithmetic can be reproduced by hand.

The formula

x̄ = Σxᵢ / n s = √( Σ(xᵢ − x̄)² / (n − 1) ) %RSD = 100 s / x̄ CI = x̄ ± t(α/2, n−1) · s / √n

n: number of replicate results
x̄: mean of the replicates
s: sample standard deviation, with the n − 1 denominator that every validation guideline means
%RSD: relative standard deviation — the SD as a percent of the mean, so it can be compared across concentration levels
t(α/2, n−1): two-sided Student's t critical value for the chosen confidence level and n − 1 degrees of freedom

Worked example

Six replicate assay results

Six replicates read 99.2, 100.1, 99.8, 100.4, 99.5 and 100.0. Their mean is 99.83 and the sample standard deviation is 0.4320, which as a percentage of the mean is a %RSD of 0.4328%. Every figure here is shown to four significant figures; the calculator carries full precision throughout.

The standard error is 0.1764 — the standard deviation divided by the square root of six. With five degrees of freedom, the two-sided t value at 95% is 2.571, giving a margin of error of 0.4535 and a 95% confidence interval of 99.38 to 100.3.

That interval says the mean is known to about ±0.45, not that individual results fall in that band — two of the six replicates lie outside it, as the plot above shows. It is the most commonly misread number on a validation report, and the distinction is the whole reason the standard error and the standard deviation are shown side by side.

Common pitfalls

The interval is on the mean, not on an individual result
A 95% confidence interval on the mean says where the true mean probably sits, given this many replicates. It is not the range a future single result will fall in — that is a prediction interval, and it is much wider. Reporting a confidence interval as though it bounded individual results understates the spread of the method, often badly.
%RSD is not the standard deviation
The standard deviation is in the units of the measurement, so it cannot be compared between a 1 ppm level and a 100 ppm level. %RSD divides it by the mean, which is exactly what makes precision comparable across concentration levels. Quoting one where a method asks for the other is a common report error.
n − 1, not n
The population form of the standard deviation, dividing by n, understates the spread of a sample. Every validation guideline means the sample form. A spreadsheet offers both — STDEV.S and STDEV.P in Excel — and picking the wrong one produces a number that is plausibly close and consistently too small.
The t value here is conservative between table rows
Critical values come from a published table. For a degrees of freedom that is not a tabulated row, this calculator steps down to the nearest lower row, which gives a slightly larger t and a slightly wider interval. Software that interpolates will report a marginally narrower interval for the same data; neither is wrong, and the value used is shown so the difference is never a mystery.
%RSD needs a mean well away from zero
Dividing by the mean is only meaningful for data on a ratio scale. For results that straddle zero — a bias, a difference between two methods — the ratio is unstable and can be enormous or negative for data that is perfectly well behaved. This calculator refuses rather than reporting one.
Decimal commas and thousands separators
Values are separated by commas, newlines, tabs and spaces, which means a decimal comma cannot also be a decimal point: “99,2” is read as two results, 99 and 2, and quietly turns one replicate into two. Use a dot. Numbers grouped like “1,234,567” — the way a spreadsheet formats a peak area — are rejected outright and listed as unread rather than guessed at, because reading them as either one value or three would invent a replicate either way. Safest of all: one value per line.
There is no universal pass mark
The ≤2% repeatability figure quoted from ICH Q2 is specific to a method and a matrix, and a trace-level assay that met 2% would be remarkable. Acceptance criteria belong to the method, not to the arithmetic, which is why this page reports numbers and leaves the verdict to your protocol.

Turn this into a validation study

The validation suite runs the same statistics, guided section by section, and ends in an auditor-ready PDF report. Sign in free and start one in minutes.