Theory — Error propagation and uncertainty rules
Error propagation is the process of estimating how measurement uncertainty in the input quantities affects the uncertainty in a
calculated result. If a measured value \(x\) is reported as \(x \pm \Delta x\), then \(\Delta x\) describes the uncertainty interval
around the central value. When several measured quantities are combined in a formula, the resulting uncertainty depends on the way
the variables enter the expression. This is why addition, subtraction, multiplication, division, and powers each use slightly
different propagation rules.
In the most common introductory treatment, uncertainties are assumed to be independent and small enough that a first-order
approximation is accurate. Under those conditions, the standard rule is the quadrature rule, which combines independent
uncertainty contributions by taking the square root of the sum of squares. Some classrooms also use a simpler linear-sum rule as a
quick conservative estimate, especially for rough checks. The calculator supports both so you can compare the standard result with
a rough upper-bound style estimate.
Addition and subtraction
When two quantities are added or subtracted, the propagated uncertainty is naturally handled as an absolute uncertainty.
For independent measurements, the standard rule is
Standard quadrature rule for sums and differences.
\[
\begin{aligned}
z &= x \pm y \\
\Delta z &= \sqrt{(\Delta x)^2 + (\Delta y)^2}
\end{aligned}
\]
A rougher linear estimate is often written as
Linear estimate for sums and differences.
\[
\begin{aligned}
\Delta z &\approx \Delta x + \Delta y
\end{aligned}
\]
The quadrature version is the standard propagation result for independent random uncertainties. The linear version is more
conservative because it never allows cancellation between independent contributions.
Multiplication and division
For products and quotients, it is more natural to propagate relative uncertainty. The reason is that changing one factor
by a small percentage changes the product or quotient by a related small percentage. For independent uncertainties, the standard rule is
Standard quadrature rule for products and quotients.
\[
\begin{aligned}
z &= x y \quad \text{or} \quad z = \frac{x}{y} \\
\frac{\Delta z}{|z|} &= \sqrt{\left(\frac{\Delta x}{x}\right)^2 + \left(\frac{\Delta y}{y}\right)^2}
\end{aligned}
\]
A common rough estimate is
Linear relative estimate.
\[
\begin{aligned}
\frac{\Delta z}{|z|} &\approx \left|\frac{\Delta x}{x}\right| + \left|\frac{\Delta y}{y}\right|
\end{aligned}
\]
Once the relative uncertainty is known, the absolute uncertainty follows from
Convert relative to absolute uncertainty.
\[
\begin{aligned}
\Delta z &= |z| \cdot \frac{\Delta z}{|z|}
\end{aligned}
\]
Powers
For a power law \(z = x^n\) with a constant exponent \(n\), the uncertainty can be obtained from the derivative. The first-order
propagation rule is
Power rule from the derivative.
\[
\begin{aligned}
z &= x^n \\
\Delta z &\approx \left| \frac{dz}{dx} \right| \Delta x \\
&= \left| n x^{n-1} \right| \Delta x
\end{aligned}
\]
If \(x \neq 0\), this can also be written in relative form:
Relative uncertainty for powers.
\[
\begin{aligned}
\frac{\Delta z}{|z|} &= |n| \frac{\Delta x}{|x|}
\end{aligned}
\]
Worked multiplication example
Take the sample values \(x = 12.3 \pm 0.2\) and \(y = 4.5 \pm 0.1\), with \(z = xy\). First compute the central value:
Central value.
\[
\begin{aligned}
z &= 12.3 \cdot 4.5 \\
&= 55.35
\end{aligned}
\]
Next compute the relative uncertainties of the factors:
Relative input uncertainties.
\[
\begin{aligned}
\frac{\Delta x}{x} &= \frac{0.2}{12.3} \approx 0.0163 \\
\frac{\Delta y}{y} &= \frac{0.1}{4.5} \approx 0.0222
\end{aligned}
\]
Using the standard quadrature rule:
Quadrature propagation.
\[
\begin{aligned}
\frac{\Delta z}{|z|}
&= \sqrt{(0.0163)^2 + (0.0222)^2} \\
&\approx 0.0276
\end{aligned}
\]
Then the absolute uncertainty is
Absolute uncertainty in the product.
\[
\begin{aligned}
\Delta z &= 55.35 \cdot 0.0276 \\
&\approx 1.53
\end{aligned}
\]
So the propagated result is approximately
Final reported product.
\[
\begin{aligned}
z &\approx 55.35 \pm 1.53
\end{aligned}
\]
A linear-sum estimate would give a larger uncertainty because it adds the two relative contributions directly. That is why it is
useful as a quick conservative check but not usually the preferred standard result for independent random errors.
Reporting the result
In experimental reporting, the uncertainty is usually rounded first, often to one or two significant figures, and then the central
value is rounded to the same decimal place. This keeps the reported result readable and consistent with the uncertainty. The calculator
therefore shows both the raw propagated values and a recommended reported form.
Error propagation does not replace careful experimental design, but it is essential for understanding how measurement quality affects
a final calculation. It tells you whether the main limitation comes from one measurement, from several equally important contributions,
or from the mathematical form of the formula itself.