Calculate uncertainty for any expression

  • Converts +- into ±
  • Calculates 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 sqrt(), exp(), e, pi, log(), trigonometric functions (sin, csc, asin, ...), and variables
  • Converts measurements (1±0.1) into variables (m1=1,δm1=0.1,...m_1=1, \delta m_1=0.1, ...)

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.

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

Unfortuantely, uncertainty in our measurements with real numbers is inevitable. The most common source of uncertainty are our measurement tools. Even with a theoretically perfect tool, the objects we measure often show variation when you measure closely enough. 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.

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 Quarature, 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!