Uncertainty Calculator

Whether you’re trying to learn how to calculate uncertainty in physics labs or managing quality control for an engineering project, this uncertainty calculator has been put together to make short work of determining your data’s margin of error. In the following page, we will cover:

• What uncertainty is and why it is important;
• How to use our uncertainty calculator; and
• How to calculate uncertainty step-by-step.

We intend to demystify the process, explain the propagation-of-uncertainty formula, and provide straightforward examples so that anyone can learn to perform uncertainty calculations with ease.

There is always a certain amount of doubt about how exact a number is when you record a measurement, be it the voltage on a circuit, the weight of something, or the length of a table. This doubt is what we call measurement uncertainty.

It is not a mistake or an error; rather, it is a frank assessment of the precision at hand. Uncertainty gives us a range where the true value is likely to lie. Take a table measured at 1.20 m ± 0.01 m, for instance. The figure shows that the actual length is between 1.19 m and 1.21 m and indicates our level of confidence in the result.

To understand how to calculate uncertainty, the first thing to do is examine your source of data. In an experimental setting, you will often deal with a raw error margin before moving on to calculating percentage uncertainty to see how large that error is relative to the total measurement.

But once you are combining several measurements, you have to contend with propagated uncertainty. Manually working through the math to track those errors can become a chore, which is why an online propagation of uncertainty calculator (or an uncertainty propagation calculator) is such a handy tool

We have built our uncertainty calculator to follow standard rules of uncertainty propagation, combining the uncertainties of individual measurements into the total for your final result. You will see that the math varies by operation, though you don’t need to worry about the details since this calculator handles it for you.

Addition and subtraction

For $Z = X + Y$ or $Z = X − Y$, the absolute uncertainties are added in quadrature:

Multiplication and division

Here, you must use relative (fractional) uncertainties to achieve the correct propagating uncertainty for $Z = X \times Y$ or $\frac{X}{Y}$.

Power functions

When $Z = X^{Y}$, the propagation of uncertainty formula becomes:

Our tool is meant to be simple and spare you from any tedious calculus. If you want to know how to do uncertainty calculations using our system, follow these three steps:

1. Pick your function, be it power, division, addition, and so on.
2. Enter input values for X and Y and their associated uncertainties (ΔX and ΔY).
3. Finally, the output for your calculated Z and the combined ΔZ will appear at once.

To give you an idea of what the uncertainty calculator is doing under the hood, let’s run through a manual calculation to see uncertainty principles in at work. Say you are in a lab setting and want to see how to calculate uncertainty in physics by adding two lengths:

First, find the core value:

Then apply the addition rule for the final uncertainty:

The answer is:

To compare the size of your measurement’s error directly against a known target or true value, check out our percent error calculator (or relative error calculator).