Uncertainty Propagation Calculator (value ± u)

Use it when a result is obtained directly from one or two measured quantities and you need to report not only the calculated value, but also the uncertainty it inherits. For example, it can estimate the uncertainty of an area calculated as length × width, a density obtained as mass ÷ volume, or the difference between two temperatures. It is designed for one operation at a time and for measurements that can be treated as independent.

Enter each measurement as value ± uncertainty. You can express uncertainty in the same units as the value (Absolute mode) or as a percentage (Relative mode); switching modes re-expresses the number without changing the quantity. The law is first-order propagation: each contribution is the partial derivative of the operation times that measurement's uncertainty. For addition and subtraction this gives u_C = √(u_A² + u_B²), the same expression in both cases; for the product and the quotient the equivalent form combines the relative uncertainties; and for a power the base's relative uncertainty is multiplied by |n|, with n exact. RSS combines in quadrature and is the usual choice when measurements are independent; Conservative adds the contributions and gives a more cautious bound. The reportable value rounds the uncertainty to 2 significant figures and the value to the same decimal place; the breakdown also shows the full number at 4 or 10 significant figures.

The tool rejects negative uncertainties, a relative uncertainty on a zero value, a zero divisor, powers with no real result —a negative base with a non-integer exponent, or a zero base with an exponent ≤ 0— and a base centered at zero with non-zero uncertainty, where first-order propagation cannot estimate anything reliable. It also rejects numbers outside the representable range, including 1e-400, which in floating point would silently become an exact zero. When the A ± u range reaches the zero of a power, or when the divisor's uncertainty reaches zero, the result is shown with a warning: the number exists, but the linear approximation no longer describes it. The tool works with one operation at a time and two quantities: it does not chain operations, allow correlation between measurements, or apply coverage factors.

• Difference · 21.4 ± 0.2 °C minus 18.9 ± 0.2 °C2.50 ± 0.28 °C — the relative uncertainty jumps to 11 %
• Area · 12.5 ± 0.1 cm × 8.0 ± 0.1 cm100.0 ± 1.5 cm² with RSS · 100.0 ± 2.0 cm² conservative
• Volume · (3.00 ± 0.02 cm)³27.00 ± 0.54 cm³ — 2 % = 3 × 0.67 %