How do I calculate standard deviation and RSD for replicates?

The sample standard deviation is s = √[Σ(xᵢ − x̄)²/(n − 1)], dividing by n − 1 (the degrees of freedom), not n. The relative standard deviation (coefficient of variation) is RSD = s/x̄ × 100%. For 10.12, 10.15, 10.13, 10.18 and 10.14 the squared deviations add up to 0.00212, so s = √(0.00212/4) = 0.0230 and RSD = 0.0230/10.144 = 0.227%.

How is the confidence interval of the mean calculated?

μ = x̄ ± t·s/√n, where t is Student's t for the chosen confidence level and n − 1 degrees of freedom. For n = 5 at 95%, t = 2.776, so the example gives μ = 10.144 ± 2.776 × 0.0230/√5 = 10.144 ± 0.029 mL. More replicates shrink the interval, because t falls and √n grows. Report the mean rounded to the same decimal place as the uncertainty: 10.14 ± 0.03 mL.

Should I use Dixon's Q test, Grubbs' test or the 4d̄ rule?

The Q test compares Q = |suspect − nearest|/range with a critical value. It is quick but only tabulated for small n (3–10 here). Grubbs' test uses G = |suspect − mean|/s, so it draws on all the data. It is the test recommended by ISO 5725 and ASTM E178. The 4d̄ rule, common in Chinese textbooks, rejects a point that lies more than four average deviations from the mean of the others. It is simple but strict and can discard valid data. Test only one suspect value at a time, and never reject a point on statistics alone if you know of no experimental reason.

When do I use a t-test and when an F-test?

The F-test compares variances (precision): F = s²larger/s²smaller, and F below the critical value means the precisions do not differ significantly. t-tests compare means. Use t = |x̄ − μ|√n/s against a certified value. To compare two methods or analysts, use the pooled-variance t-test when the F-test passes and Welch's t-test when it does not. Use the paired t-test when each sample was measured by both methods. A difference is significant when |t| exceeds the critical value, or equivalently p < α.

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📏Lab Replicate Statistics - Mean, Standard Deviation, Confidence Interval, Q-Test and t-Test

This tool does the statistics of an analytical chemistry lab report in one place. Paste a set of replicate results, such as titration volumes. You get n, mean, median, sample standard deviation s, relative standard deviation (RSD, CV), standard error, range, average deviation d̄ and the confidence interval of the mean μ = x̄ ± t·s/√n at 90%, 95% and 99%. The t values come from an exact inverse of the incomplete beta function and match t tables. For five titrations of 10.12, 10.15, 10.13, 10.18 and 10.14 mL: x̄ = 10.144 mL, s = 0.0230 mL, RSD = 0.227% and t(0.05, 4) = 2.776. The 95% interval is 10.144 ± 0.029 mL, reported as 10.14 ± 0.03 mL. Enter a true or certified value to get the absolute and relative error and a t-test for systematic error. Outlier tests include Dixon's Q test (Rorabacher 1991 critical values, n = 3–10), Grubbs' test (critical value from the t distribution, two-sided or one-sided) and the 4d̄ rule, and you can drop the flagged point and recompute. In 0.167, 0.177, 0.181, 0.181, 0.204 the suspect 0.204 has Q = 0.023/0.037 = 0.62, below Q95 = 0.710, so it is retained, although the stricter 4d̄ rule would reject it. To compare data sets, the tool runs an F-test for precision, the pooled and Welch t-tests and a paired t-test, each with t, degrees of freedom, critical value and p value. It finishes with a copyable report and an accuracy-versus-precision target diagram.

Free · No sign-up · Runs in your browserLast updated: 2026-10-29

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How to use Lab Replicate Statistics

1Paste or type your replicate measurements and optionally set the unit, true / reference value and confidence level
2Read the descriptive statistics, confidence intervals, dot plot and outlier tests, and optionally remove an outlier and recompute
3Compare with a reference value or enter a second data set for F- and t-tests, then copy the report

FAQ

How do I calculate standard deviation and RSD for replicates?: The sample standard deviation is s = √[Σ(xᵢ − x̄)²/(n − 1)], dividing by n − 1 (the degrees of freedom), not n. The relative standard deviation (coefficient of variation) is RSD = s/x̄ × 100%. For 10.12, 10.15, 10.13, 10.18 and 10.14 the squared deviations add up to 0.00212, so s = √(0.00212/4) = 0.0230 and RSD = 0.0230/10.144 = 0.227%.
How is the confidence interval of the mean calculated?: μ = x̄ ± t·s/√n, where t is Student's t for the chosen confidence level and n − 1 degrees of freedom. For n = 5 at 95%, t = 2.776, so the example gives μ = 10.144 ± 2.776 × 0.0230/√5 = 10.144 ± 0.029 mL. More replicates shrink the interval, because t falls and √n grows. Report the mean rounded to the same decimal place as the uncertainty: 10.14 ± 0.03 mL.
Should I use Dixon's Q test, Grubbs' test or the 4d̄ rule?: The Q test compares Q = |suspect − nearest|/range with a critical value. It is quick but only tabulated for small n (3–10 here). Grubbs' test uses G = |suspect − mean|/s, so it draws on all the data. It is the test recommended by ISO 5725 and ASTM E178. The 4d̄ rule, common in Chinese textbooks, rejects a point that lies more than four average deviations from the mean of the others. It is simple but strict and can discard valid data. Test only one suspect value at a time, and never reject a point on statistics alone if you know of no experimental reason.
When do I use a t-test and when an F-test?: The F-test compares variances (precision): F = s²larger/s²smaller, and F below the critical value means the precisions do not differ significantly. t-tests compare means. Use t = |x̄ − μ|√n/s against a certified value. To compare two methods or analysts, use the pooled-variance t-test when the F-test passes and Welch's t-test when it does not. Use the paired t-test when each sample was measured by both methods. A difference is significant when |t| exceeds the critical value, or equivalently p < α.

Features

Mean, median, standard deviation, RSD, standard error, range, average and relative average deviation
90 / 95 / 99% confidence intervals with exact Student t values
Absolute and relative error plus a t-test of the mean against a known value
Dixon's Q test (Rorabacher table), Grubbs' test and the 4d̄ rule, with automatic recalculation
F-test, pooled-variance t-test, Welch's t-test and paired t-test with p values
Dot plot showing data, mean, confidence interval and true value
Report rounded to the uncertainty, ready to copy
Accuracy vs precision target diagram explaining systematic and random error

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