A. Dinsmore 9/2006

Propagating Uncertainties

All experiments have systematic uncertainties and random uncertainties (sometimes known as ‘error’). 

Random uncertainties arise from unknown or uncontrolled randomness in measurements (e.g. in defining the edge of an object when measuring its size), or from randomness in the physical process itself (e.g. in counting oscillations or nuclear decays).  A key point is that random uncertainties affect each data point separately, so that some of them might be overestimated and others underestimated.  Therefore, acquiring more data leads to a more precise answer (small uncertainty).  This principle explains why we often take many data points (>2) and fit to them.

Generally, these can (and should) be estimated quantitatively.  Moreover, they should be ‘propagated’ to find the total uncertainty in the final result of the experiment.  The rules for this are summarized below.

Systematic uncertainties arise from unknown or uncontrolled aspects of a measurement.  These usually affect all data points in the same way (e.g. if a calibration is incorrect, then  all data points might be 25% too large).  Systematic error can, of course, arise from mistakes (typically not an acceptable source) or a lack of detailed knowledge of the experimental system.  Often, the data can be quantitatively corrected to account for systematic errors (e.g. by accounting for the thermal expansion of a device if the room gets hot).  Many ILab experiments will require that systematic errors be identified and that the data be corrected.

Some rules for propagating random uncertainties:

Suppose a group of experimenters measures three quantities: A ± DA, B ± DB, and C ± DC.

DA is an absolute uncertainty; it has the same unit as A.

DA/A is a fractional uncertainty and has no units.

When adding or subtracting measured quantities, add the absolute uncertainties in quadrature:

Y = A + B - C

DY = sqrt{ (DA)² + (DB)² + (DC)² } (note that the units are OK)

When multiplying or dividing measured quantities, add the relative uncertainties in quadrature:

Y = (A × B)/C

DY/Y = sqrt{ (DA/A)² + (DB/B)² + (DC/C)² } (note that the units are OK)

When raising a measured quantity to a power, multiply the relative uncertainties:

Y = Aⁿ

DY/Y = n(DA/A)

When  doing several of these things, break the arithmetic into individual steps like the above.

Y =  A + Bⁿ/C can be broken into three steps: (1) J = Bⁿ, (2) K = J/C, and (3) Y = A + K.  The rules above can provide DJ, DK and DY.