Reporting lab measurements as mean ± SD

The worked example: eight replicate determinations

Suppose an analyst measures the chloride concentration of one water sample eight times. The numbers are hypothetical, recorded to 0.1 mg/L: 48.7, 49.3, 48.9, 49.6, 48.4, 49.1, 49.0, 48.8.

x̄ = 391.8 / 8 = 48.975 mg/L s = √[0.955 / (8 − 1)] = √0.136429 = 0.36936 mg/L %RSD = 100 × 0.36936 / 48.975 = 0.754%

That is the arithmetic, and you can check it in the mean and standard deviation calculator. The sample SD (n − 1) is the right one here, because eight determinations are a sample of what the method could produce. Keep every digit through the calculation; rounding happens once, at the end.

Try it: RSD calculator

The eight chloride readings are already filled in below; replace them with your own replicates to get the mean, sample SD and %RSD in one step.

Open the full relative standard deviation calculator for more on %RSD and the coefficient of variation.

Round the SD first, then the mean to match

The standard deviation of eight readings is only a rough estimate of the method's true spread. With n = 8, a second batch of eight could easily give an SD 30% higher or lower. So the third and later significant figures of s are noise. The usual rule:

1. Round s to 1 or 2 significant figures. 2. Round x̄ to the same decimal place as the rounded s.

Here s = 0.36936 becomes 0.37 (two significant figures). Its last digit is in the hundredths place, so the mean is rounded to hundredths: 48.975 becomes 48.98. The result is 48.98 ± 0.37 mg/L. If your lab uses one significant figure for the SD, the same data gives 49.0 ± 0.4 mg/L. Both are defensible; what is not defensible is 48.975 ± 0.36936, or the mismatched 48.975 ± 0.4, whose mean claims a precision the SD says it does not have.

Notice the mean is quoted to 0.01 mg/L even though each reading was recorded to 0.1. That is fine: averaging eight readings pins the centre down better than any one of them. The size of the spread sets the last digit, not the resolution of the instrument.

Add %RSD, and always add n

The %RSD (relative standard deviation, the same thing as the coefficient of variation expressed as a percentage) puts precision on a scale that does not depend on the concentration. Here it is 0.754%, reported as 0.75%. Method validation criteria are usually written as a maximum %RSD, which is why labs report it alongside the SD. The RSD calculator gives it directly. Round %RSD to two significant figures for the same reason as the SD: it inherits the SD's uncertainty.

n belongs in every line. "48.98 ± 0.37 mg/L" from 3 replicates and from 30 replicates are very different statements, and a reader cannot recover n from the SD. A complete line looks like this:

Chloride: 48.98 ± 0.37 mg/L (mean ± SD, n = 8; RSD 0.75%)

When the ± should be a standard error or a confidence interval

Mean ± SD describes the scatter of individual determinations: the repeatability of the method. Often the reader wants something else, namely how well you know the concentration of this sample. That calls for the standard error of the mean or a 95% confidence interval.

The three lines look alike and mean different things. The SD will not shrink if you run more replicates; the SE and the confidence interval will. The t value of 2.365 is for 7 degrees of freedom; with small n it is noticeably bigger than 1.96. Label which one you used, every time. The standard error calculator and the confidence interval calculator work both out from the raw values.

Before you report: three checks

Look at the values, not only the summary. With eight numbers a quick sort shows whether one reading sits far from the rest. Here the extremes are 48.4 and 49.6, each within 1.7 SD of the mean, so nothing stands out. If a reading does stand out, test it (the outlier calculator covers the common rules) and record any exclusion with its reason. Quietly dropping the awkward replicate makes the SD look better than the method is.

Check the %RSD against the method's acceptance limit. If the limit is 2% and you got 0.75%, the replicate set passes; if you got 3%, report the result but flag it.

Keep units and the replicate type straight. "Eight injections of one extract" measures instrument repeatability. "Eight separate extractions" also captures sample preparation, and will usually have a larger SD. Say which kind of replicate the SD describes, because the two answer different questions about the method. The broader difference between describing spread and describing a mean is covered in standard deviation vs mean.