Error Propagation & Significant Figures Calculator

Propagates the uncertainty (±) of measured values through a formula and rounds the result to sensible significant figures. It also counts significant figures and gives the mean ± standard error of repeated measurements for a lab report.

How to use

1. On "Error propagation", enter a "Name", "Value" and "Uncertainty ±" for each variable. Type 2% for a relative uncertainty.
2. Write the "Formula" with those names, for example a*b/c^2. The result, the "Relative uncertainty" and each variable's "Share of the variance" appear at once.
3. On "Significant figures", enter a number to count its significant figures, or type a sum such as 12.11 + 18.0 under "Calculation" to see the rules step by step.
4. On "Repeated measurements", paste your readings to get the mean, standard deviation and standard error of the mean.

How it works

• The general formula is σf = √(Σ(∂f/∂xᵢ × σᵢ)²). The variables are assumed to be independent; correlation is not handled.
• The uncertainty is rounded to 1 or 2 digits ("Digits of the uncertainty") and the value to the same decimal place. Exact halves follow "Round half up" or "Round half to even", as you choose.
• Your inputs are saved in this browser on this device.

Questions and answers

How do you propagate error for multiplication and division?

Add the relative uncertainties in quadrature. If a is ±3% and b is ±4%, a×b is ±√(3² + 4²) = ±5%. For sums and differences, add the absolute uncertainties the same way.

What are the rules for significant figures in calculations?

Addition and subtraction keep the fewest decimal places; multiplication and division keep the fewest significant figures. 12.11 + 18.0 = 30.11 is written as 30.1.

How many significant figures does 1500 have?

Trailing zeros in a whole number are ambiguous, so 2 to 4. By default they are not counted. Writing 1.500×10³ shows clearly that there are 4.

What is the difference between standard deviation and standard error?

The standard deviation is the spread of single readings. The standard error, standard deviation ÷ √n, is the uncertainty of the mean.