Relative Uncertainty⚠ unverified

Understanding the Parameters

• Measured value $x$ — the magnitude of the quantity. It sits in the denominator, so the same absolute uncertainty becomes a smaller relative uncertainty on a larger value. This is why relative uncertainty blows up for measurements near zero (dividing a fixed $u$ by a vanishing $x$), and why measuring small differences of large numbers is treacherous.

• Absolute uncertainty $u$ — the standard uncertainty in the units of $x$ (millimetres, volts, grams). It and $x$ must carry the same units so the ratio is dimensionless (the registry labels both dimensionless, but they inherit the measurand's units).

• The output $u_r$ — a dimensionless fraction, usually quoted as a percentage ($u_r \times 100\%$) or in parts per million for precise work. A voltmeter good to $0.1\%$, a mass standard to $1\,\text{ppm}$ — relative uncertainty is the common currency for stating and comparing instrument accuracy across scales and units.

• Why relative matters. Many error sources are inherently proportional (gain errors, calibration scale factors), so they are naturally relative; and when quantities are multiplied or divided, it is their relative uncertainties that combine (see the derivation). Relative uncertainty is therefore the right variable for propagating uncertainty through products and ratios.

Derivation (Approaching a Proof)

The definition is a ratio, but its importance comes from how uncertainties propagate through products and quotients. Consider a result formed by multiplication and division, $y = x_1^a x_2^b \cdots$. Taking the natural log, $\ln y = a\ln x_1 + b\ln x_2 + \cdots$, and differentiating, $\frac{dy}{y} = a\frac{dx_1}{x_1} + b\frac{dx_2}{x_2} + \cdots$.

Each term $dx_i/x_i$ is a relative change. Applying the law of propagation of uncertainty (adding independent contributions in quadrature) gives the combined relative uncertainty

$$\left(\frac{u_y}{y}\right)^2 = a^2\left(\frac{u_1}{x_1}\right)^2 + b^2\left(\frac{u_2}{x_2}\right)^2 + \cdots,$$

i.e. relative uncertainties (weighted by the exponents) add in root-sum-square. The single-variable definition $u_r = u/x \qquad\blacksquare$ is the building block: it is exactly the quantity that combines cleanly through multiplicative relationships, which is why converting to relative form is the standard first step in propagating uncertainty through products.

Dimensional check. $u$ and $x$ share units, so $u_r = u/x$ is dimensionless. $\checkmark$

History and Development

• A universal accuracy metric. Expressing precision as a percentage or ppm of reading is as old as quantitative measurement; it lets a $1\,\text{ppm}$ frequency standard and a $0.5\%$ pressure gauge be compared on a common footing regardless of units or magnitude.

• Propagation through products. The result that relative uncertainties add in quadrature for products and quotients (while absolute uncertainties add for sums and differences) is a cornerstone of the GUM law of propagation of uncertainty and of every error-analysis course.

• Instrument specifications. Manufacturers routinely split accuracy specs into a relative "% of reading" term and an absolute "% of range" (offset) term, precisely because some error sources scale with the value and others do not — relative uncertainty isolates the former.

Related Concepts: Expanded Uncertainty, Coverage Factor, Gage Rr Percent, Process Capability Cpk, Normal Distribution PDF

Notes: Registry calculator relative-uncertainty (unverified). $u_r = u/x$ — correct as shipped. $u$ and $x$ share the measurand's units (registry labels both dimensionless); result usually quoted as % or ppm. Relative uncertainties add in quadrature through products/quotients; absolute ones through sums/differences. Returns $+\infty$ at $x=0$.