Absolute Uncertainty Calculator
This absolute uncertainty calculator turns a set of repeated measurements into a single best estimate plus its absolute uncertainty, the standard way to report a result in a science lab. When you measure the same thing several times you rarely get exactly the same number, so no single reading is the whole truth. The accepted approach is to quote the mean of your readings as the best estimate, then state how far the readings spread as the absolute uncertainty, written with a plus-or-minus sign in the same unit as the measurement. This tool uses the half-range method taught in NZ secondary schools and first-year labs: the absolute uncertainty is the largest reading minus the smallest, divided by two, which brackets every reading you took. Enter between two and eight readings and the calculator returns the mean, the absolute uncertainty and the relative (percentage) uncertainty, then writes the result in the usual best-estimate-plus-or-minus form. Students use it to write up physics and chemistry practicals, and anyone repeating a measurement, timing laps, weighing samples, reading a ruler, can use it to say honestly how sure they are. For a large, carefully collected data set the standard deviation or standard error is a more refined measure of spread, but the half-range is the clear, defensible choice for the handful of readings a typical experiment produces.
Enter two or more readings in the same unit. Leave the rest blank. The unit shown is just a label; the maths is the same for any unit.
From 5 readings the mean is 24.1 and the range is 0.4, so the absolute uncertainty is half of that, 0.2. Report the measurement as 24.1 ± 0.2, an uncertainty of about 0.83%.
Uses the half-range method: absolute uncertainty = (largest reading − smallest reading) / 2. For large data sets the standard deviation or standard error gives a more refined spread. Estimate only.
How it works
The best estimate of a repeated measurement is the mean: add every reading and divide by how many you took. The absolute uncertainty is half the range, the difference between the largest and smallest reading divided by two. This works because the half-range is the distance from the middle of your readings to either extreme, so writing the result as mean plus or minus the half-range creates a band that contains every reading. The relative uncertainty is the absolute uncertainty divided by the mean, expressed as a percentage, which lets you compare the precision of measurements of very different sizes. By convention the absolute uncertainty is rounded to one significant figure and the best estimate is rounded to the same decimal place, so the two numbers line up when written together.
Worked example
You measure a length five times and get 24.1, 24.3, 23.9, 24.0 and 24.2 cm. The mean is (24.1 + 24.3 + 23.9 + 24.0 + 24.2) / 5 = 120.5 / 5 = 24.1 cm, your best estimate. The largest reading is 24.3 and the smallest is 23.9, so the range is 0.4 cm and the absolute uncertainty is 0.4 / 2 = 0.2 cm. The relative uncertainty is 0.2 / 24.1 = 0.0083, or 0.83%. You would report the length as 24.1 ± 0.2 cm. If a sixth reading of 24.6 came in, the range would widen to 0.7 cm and the absolute uncertainty would rise to about 0.35 cm, showing how a single outlier stretches the half-range.
Related calculators
- Measurement Uncertainty Calculator: value and uncertainty to a range.
- Relative Error Calculator: error as a fraction and percentage.
- Absolute Error Calculator: measured against true value.
- Percent Error Calculator: percentage error.
- Standard Error Calculator: standard error of the mean.