Repeated Observation Mean and Spread Calculator
Averages repeated readings with a numerically stable method and reports the sample or population standard deviation (clearly labelled), the standard error, residuals and range — removing nothing unless you exclude it with a reason.
Inputs
Mean, sample or population SD (clearly labelled), standard error and residuals for repeated readings — computed stably, with nothing removed unless you exclude it.
Enter the values and press Calculate.
Or load an example to see a worked result.
Before you rely on this: First-pass guide only. Verify safety-critical or regulated work against the relevant standards, your project requirements and a qualified professional.
How to use this calculator
- Paste your repeated readings of one quantity — one per line, or separated by spaces or commas — and optionally a unit label.
- Choose which spread to headline: the sample SD (n − 1) when the readings are a sample, or the population SD (n) when they are all the values there are.
- To leave a reading out, enter its number and your reason; it stays in the table, marked as excluded. An optional k·s rule flags large residuals without removing anything.
- If some readings are more precise than others, open “More options” and choose a weighted mode: enter each reading’s standard deviation σᵢ, or relative weights from your own procedure.
How it works
The mean and the sum of squared deviations M2 are computed with Welford's method, which stays accurate even for readings like 1,000,000,004 and 1,000,000,007, where the textbook sum-of-squares formula loses every digit.
Sample SD s = √(M2 ÷ (n − 1)); population SD σ = √(M2 ÷ n); standard error of the mean SEM = s ÷ √n. With a single reading, s and SEM are unavailable — not zero.
Residual = reading − mean. A small spread shows repeatability, not accuracy: a calibration offset or other systematic error moves every reading together and does not appear in the spread.
Weighted mode (optional): weighted mean = Σwx ÷ Σw, standard deviation of unit weight σ₀ = √(Σw r² ÷ (n − 1)) and its standard error σ₀ ÷ √Σw, using a stable incremental update. With w = 1 ÷ σᵢ² the tool also shows the standard error implied by your σᵢ alone (1 ÷ √Σw); σ₀ near 1 means the scatter agrees with the σᵢ you entered. Equal weights reproduce the ordinary results exactly.
Worked example
Five readings: 10, 12, 11, 13, 14. Mean = 12 and M2 = 10, so the sample SD s = √(10 ÷ 4) = 1.5811 and SEM = 1.5811 ÷ √5 = 0.7071. The largest residual is 2. The population SD would be √(10 ÷ 5) = 1.4142.
Tips
- More readings tighten the mean (the SEM falls with √n), but they cannot fix a systematic error — only a better procedure or calibration can.
Frequently asked questions
Does small spread mean accuracy?
No. Spread measures how well the readings agree with each other. A constant error — a wrong prism constant, an uncalibrated tape — shifts them all and leaves the spread unchanged.
SD or SEM?
The SD describes the scatter of individual readings; the SEM (s ÷ √n) describes how well the mean is determined, assuming independent readings. Report the one your question needs, and say which.
When should I weight the readings?
When some readings are genuinely more precise — taken with a better set-up, or each already the mean of several repetitions. The weights must come from your procedure; equal weights give the ordinary mean.
Can I remove an outlier?
Only with a reason you record — a known blunder, a disturbed set-up. The tool never removes readings by itself; an optional rule only flags them, and excluded readings stay visible in the table.
Is this official course material?
No. It is free study support mapped to surveying course levels — not official North Metropolitan TAFE content or advice. Always follow your lecturer and the official assessment brief, and check your own working.
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Cite this tool
Zerdly. (2026). Repeated Observation Mean and Spread Calculator. Retrieved from https://www.zerdly.com/tools/repeated-observation-mean-and-spread-calculator/
Tip: Enter any known values to calculate the remaining results.
Designed to run locally in your browser. Tool inputs are not intentionally stored by Zerdly unless stated otherwise.