Unit Conversion And Measurement Uncertainty

Why Convert the Uncertainty Too?

A measurement is not just a number. It also has a unit and, often, an uncertainty that describes how far the actual value could be from the reported value. If a thermometer reports \(68\pm1.8^\circ\mathrm{F}\), converting only the number \(68\) to Celsius does not give a complete converted measurement. The uncertainty must be expressed in Celsius as well.

The important distinction is between a scale factor and an offset. A scale factor changes the size of a difference between two measurements. An offset shifts all measurements by the same amount, so it changes the reported value but not the size of its uncertainty. This tutorial develops that distinction for linear unit conversions and checks it with concrete measurements.

Measurements and Linear Conversions

Suppose a measurement in one unit is reported as \(x_0\pm\Delta x\), where \(x_0\) is the reported central value and \(\Delta x\geq0\) is its absolute uncertainty in the same unit. When the uncertainty is being used as a bound, this notation means that the value lies in the interval from \(x_0-\Delta x\) to \(x_0+\Delta x\).

A linear conversion between the numerical values in two units has the form \(y=ax+b\), where \(a\) and \(b\) are constants. The corresponding converted measurement is \(y_0=ax_0+b\). The uncertainty in the converted value is \(|a|\Delta x\), not \(a\Delta x+b\). The absolute value is needed because uncertainty is nonnegative even when the conversion reverses order.

The units must be carried through each calculation. If \(a\) converts the numerical value in units \(U\) to the numerical value in units \(V\), then \(a\Delta x\) is an uncertainty in \(V\). The offset \(b\) is also expressed in the numerical scale for \(V\).

Why the Offset Does Not Affect Absolute Uncertainty

The rule follows directly from comparing the converted central value with a possible value at either edge of the original uncertainty interval. The two endpoints in the original unit are \(x_0-\Delta x\) and \(x_0+\Delta x\). Under \(y=ax+b\), their converted values are \(a(x_0-\Delta x)+b\) and \(a(x_0+\Delta x)+b\). Subtracting the converted central value \(ax_0+b\) gives deviations of \(-a\Delta x\) and \(a\Delta x\). Their magnitudes are both \(|a|\Delta x\).

The same conclusion holds if \(a<0\): the endpoints exchange order, but their distances from the converted central value are still \(|a|\Delta x\). Thus the transformed interval has the converted central value as its midpoint and \(|a|\Delta x\) as its half-width. This is an exact result for an affine conversion, not a first-order approximation.

If \(\Delta x\) represents a standard uncertainty rather than a guaranteed bound, the same scale rule applies to the standard deviation under the affine transformation \(y=ax+b\): adding a constant shifts the distribution without changing its spread, and multiplying by \(a\) scales the spread by \(|a|\). The interpretation of the uncertainty should still be stated when it matters.

Worked Examples

Absolute and Relative Uncertainty Are Not the Same

Absolute uncertainty is expressed in the same units as the measurement. For example, \(20\pm1^\circ\mathrm{C}\) has an absolute uncertainty of \(1^\circ\mathrm{C}\). Relative uncertainty compares the absolute uncertainty with the magnitude of the reported value; when the reported value is nonzero, it is \(\frac{\Delta x}{|x_0|}\). These two kinds of uncertainty do not always behave the same way under conversion.

Under a pure scale conversion \(y=ax\), with \(a\ne0\), both the measurement magnitude and absolute uncertainty are multiplied by \(|a|\). Their ratio is unchanged. Under an affine conversion \(y=ax+b\) with a nonzero offset, the absolute uncertainty still becomes \(|a|\Delta x\), but the converted central value is \(ax_0+b\). Consequently, the relative uncertainty can change. This is why a Celsius-to-Fahrenheit conversion preserves neither the numerical temperature nor, in general, its relative uncertainty.

A Common Conversion Mistake

A frequent error is to apply the entire temperature formula, including its offset, to the uncertainty. For example, for Fahrenheit-to-Celsius conversion, the incorrect calculation might be written

This is wrong for two reasons. An uncertainty is a distance, so it cannot be negative, and subtracting \(32\) makes no sense for a distance between two temperatures. The correct conversion uses only the scale factor:

The offset disappears because it is added to both the central temperature and each possible temperature. Their differences are unaffected by adding the same constant.

A Reliable Procedure

For a measurement reported with an absolute uncertainty, the following sequence keeps the conversion and its uncertainty consistent:

This procedure matters whenever converted measurements are compared, reported, or used in later calculations. A correct central value paired with an uncertainty in the wrong unit can give a misleading impression of precision. For a linear conversion, the uncertainty transformation is simple and exact, provided the conversion formula and units are handled consistently.