Propagate uncertainty for any expression

  • Converts +- into ±
  • Propagates uncertainty as the standard error, δf(x,y,...)=(fxδx)2+(fyδy)2+...\delta f(x,y,...)=\sqrt{\left(\frac{\partial f}{\partial x} \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \delta y \right)^2 + ...}
  • Assumes measurements are normal and independent
  • Supports variables, sqrt(), exp(), e, pi, log(), and trigonometric functions (sin, csc, asin, ...)

Error analysis

Error analysis is how we account for uncertainty in our calculations. Error analysis works by treating measurements as probability distributions. This allows us to calculate the final quantity's probability distribution, and thus know the range of possible values.

Propagating uncertainty

We propagate uncertainty by calculating the final quantity's probability distribution. To calculate the uncertainty of an expression directly, we can use the general form of Summation in Quadrature, δf(x,y,...)=(fxδx)2+(fyδy)2+...\delta f(x,y,...)=\sqrt{\left(\frac{\partial f}{\partial x} \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \delta y \right)^2 + ...}. This calculator derives and evaluates this expression for you. To learn more about why uncertainty exists and how to propagate it through equations, check out the guide!

This calculator treats all measurements as normal distributions that are independent from each other. This is common practice and often works well.

Sources of uncertainty

Uncertainty in our measurements with real numbers is inevitable. The most common source of uncertainty is our measurement tools. Even with a theoretically perfect tool, we will see variation in our measurements because the objects themselves vary. For example, temperature and humidity will alter the length of wood and steel.

When measuring with high enough precision, defining what exactly to measure becomes problematic. For example, where exactly along the object is the length? What if the object has layers of dust or oxidation? Fortunately, we don't need to eliminate uncertainty. We only need to ensure the uncertainty is low enough for our use-case.


How do I type "±"?
Type "+-" and it will be converted.
Why doesn't my decimal "," work?
Use a period decimal "." instead of a comma ","
How do I enter a percent uncertainty, like "5±5%"?
This isn't supported. If you need this, please leave some feedback!
How do I enter a log base 10?
Type "log(x,10)" where x is your value.
How do I use degrees in my trigonometric function?
Trigonometric functions use radians by default. Type "sin(x deg)" where x is your value.