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Combined Standard Uncertainty
and Propagation of Uncertainty

- Terms
- Uncertainty
- PDFs
  - Gaussian
  - Uniform
  - Triangular
- Procedure
- Combined
- Example
- References

For the uncertainty to be truly meaningful, it must address the entire measuring process, which may have uncertainties associated with factors such as equipment calibration, operator skill, sample variation, and environmental factors. When a measurement has more than one identifiable source of measurement uncertainty, then the combined standard uncertainty (uc) must be calculated.

Calculating the combined standard uncertainty is a two step process. The first step is to determine the uncertainties measured directly and the second step is combine the uncertainties using summation in quadrature, which is also known as root sum of the squares. For example, is a measurement of a measurand x, has three sources of uncertainty for which the three standard uncertainties u1(x) , u2(x) and u3(x) have been determined, then the combined standard uncertainty uc(x) for the measurement is given by:

Uncertainty contributions from both Type A and Type B evaluations may be combined as long as they are expressed in similar terms before they are combined. Thus, all the uncertainties must be expressed as one standard uncertainty and in the same units.

Propagation of Uncertainty

Many physical quantities are not determined from a single direct measurement but instead are calculated by combining two or more separate measurements. Therefore, it is important to understand how measurement uncertainty propagates when mathematical operations are performed on measured quantities, so that a final combined uncertainty can be calculated. Consider the determination of the velocity of a sound wave as it travels through a medium. The velocity (V) is calculated by dividing the measured distance (d) traveled by the measured time (t) that it took the sound to travel the distance. This calculation of velocity is easy enough but the measured quantities (d and t) each have a measurement uncertainty that must be combined to arrive at an uncertainty for the velocity calculation.

The propagation of uncertainty is treated differently depending on the mathematical operation(s) performed. The simplest case is where the result is the sum of a series of measured values (either added together or subtracted).  The combined standard uncertainty is found by squaring the uncertainties, adding them all together, and then taking the square root of the total. For more complicated cases, such as multiplication and division where mixed units are often involved, it is necessary to work in terms of relative uncertainties. The formula for calculating the combined standard uncertainty for basic mathematical operations are shown in the table below.

Mathematical Operation Performed Equation Form   Formula for Calculation of Combined Standard   Uncertainty
Addition or Subtraction

  Z± u(z) = (X ± u(x)) + (Y ± u(y)) or
  Z± u(z) = (X ± u(x)) - (Y ± u(y))

Multiplication or Division   Z ± u(z) = (X ± u(x)) x (Y ± u(y)) or
  Z ± u(z) = (X ± u(x)) / (Y ± u(y))
See Note 1
Power (Squared)   Xn± u(xn)                 X2 ± u(x2)

See Note 1

(Square Root)
See Note 1
Mixed (Addition, Division, Square, and Square Root)
See Note 1

Note 1: The result of this calculation is the relative combined uncertainty. The absolute combined uncertainty can be calculated by multiplying uc by the best approximation of the measurand.


The equations in the table above or only valid if the contributing uncertainties are not correlated. Factors leading to measurement error are often independent, but sometimes they are correlated of inter-related. For example, a temperature shift could have a similar effect on several uncertainty contributors. If two or more sources of uncertainty are believed to be correlated, consult the references for additional information on dealing with the correlation.

The Uncertainty Budget

An uncertainty budget is simply a way of organizing and summarizing the uncertainty analysis in tabular form. An uncertainty budget lists all the contributing components of uncertainty and these components are used to calculate the combined standard uncertainty for the measurement. The table can consist of as few as two columns, one for listing the source of uncertainty and the second for recording the standard uncertainty. However, more involved tables such as the one shown below can be helpful.

Source of Uncertainty   Value ±   Shape of pdf   Divisor Standard Uncertainty
Calibration uncertainty (@k=2)   1.50 mm   Normal
(to reduce k=2 to 1)
  0.75 mm
Resolution (size of divisions)   0.5 mm   Uniform
  0.29 mm
Standard uncertainty of mean
(10 repeated readings)
  0.38 mm   Normal
  0.39 mm
Combined standard uncertainty     Assumed normal     0.90 mm

Expanded uncertainty (k=2)

    Assumed normal       1.80 mm

Last updated on August 17, 2011