Absolute Uncertainty Calculator

What the absolute uncertainty calculator does

Every measurement carries some uncertainty, and it can be expressed two ways: as a percentage of the measurement, or as an amount in the measurement's own units. This calculator converts the first into the second, taking a measured value and its relative uncertainty and returning the absolute uncertainty.

It answers the question, if my measurement is uncertain by some percentage, how much is that in actual units. Below is what the two kinds of uncertainty are and how they convert.

How to use it

1. Enter the measured value.
2. Enter the relative uncertainty as a percentage.
3. Press Calculate for the absolute uncertainty, or Reset to clear it.

Absolute and relative uncertainty

Absolute uncertainty is expressed in the same units as the measurement itself. If you measure a length as 50 metres give or take a metre, that one metre is the absolute uncertainty, and it tells you directly how much the true value might differ from what you recorded.

Relative uncertainty is that same doubt expressed as a fraction or percentage of the measurement. A one-metre uncertainty on 50 metres is a relative uncertainty of 2 percent. Being a pure ratio, it has no units, which makes it a clean way to describe how precise a measurement is regardless of what was measured. The two are just different clothes on the same underlying doubt.

How the conversion works

Going from relative to absolute is a matter of taking that percentage of the measured value:

Absolute uncertainty = measured value × (relative uncertainty ÷ 100)

The relative uncertainty is a percentage, so dividing by 100 turns it into a plain fraction, and multiplying by the measured value scales it into the measurement's units. The result is the actual size of the doubt, in the same units you measured in. To go the other way, from absolute to relative, you would instead divide the absolute uncertainty by the measured value.

Reading it as a range

The absolute uncertainty is what lets you write a measurement as a range rather than a single figure. A value of 50 with an absolute uncertainty of 1 becomes 50 plus or minus 1, meaning the true value most likely lies somewhere between 49 and 51.

This plus-or-minus form is how measurements are properly reported, because it carries the honesty of the measurement with it. A bare number pretends to a precision it does not have, while a number with its uncertainty tells the reader how much to trust the last digits. The absolute uncertainty is the half-width of that range on either side of the measured value.

Which one to use when

The two forms suit different jobs. Absolute uncertainty is what you want when you care about the actual spread of a single measurement, since it is in real units and feeds directly into a plus-or-minus range. It is also what you carry through calculations when combining measurements.

Relative uncertainty comes into its own for comparing precision across different measurements. A one-metre error means very different things on a 50-metre wall and on a journey of many kilometres, but expressed as a percentage, the precision of the two can be compared directly. So reach for absolute units when you want the concrete spread, and relative percentages when you want to judge how precise something is on its own terms.

A worked example

Suppose you measure a length as 50 metres with a relative uncertainty of 2 percent. Converting, the absolute uncertainty is 50 times 2 divided by 100 = 1 metre.

So the measurement is best written as 50 plus or minus 1 metre, with the true length most likely between 49 and 51 metres. That single metre is the concrete size of the doubt, which the 2 percent expressed only as a proportion. If the relative uncertainty had been 5 percent instead, the absolute uncertainty would be 2.5 metres, and the range would widen to between 47.5 and 52.5.

Entering your values

Enter the measured value and the relative uncertainty as a percentage, which cannot be negative. The calculator returns the absolute uncertainty in the same units as your measured value. To turn it into a range, add and subtract it from the measured value.

Questions people ask

What is absolute uncertainty?

The uncertainty of a measurement expressed in its own units, telling you directly how much the true value might differ from what you recorded.

How is relative uncertainty different?

It expresses the same doubt as a percentage of the measurement, so it has no units. A one-metre uncertainty on 50 metres is a 2 percent relative uncertainty.

How do I convert between them?

Multiply the measured value by the relative uncertainty as a fraction to get the absolute uncertainty. Divide the absolute uncertainty by the measured value to go back.

How does it give a range?

Add and subtract the absolute uncertainty from the measured value. A value of 50 with an absolute uncertainty of 1 means a range of 49 to 51.

References

A quick note on where the methods here come from. Measurement uncertainty and its expression are a core topic of the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference, and NIST is the recognised authority on measurement uncertainty. OpenStax provides free, widely used science and statistics textbooks covering measurement and uncertainty.

1. NIST/SEMATECH e-Handbook of Statistical Methods (measurement uncertainty). https://www.itl.nist.gov/div898/handbook/
2. OpenStax (free science and statistics textbooks covering measurement uncertainty). https://openstax.org/subjects/math