The measurement uncertainty of the measuring instrument is a new concept which takes into account during the calibration not only the errors or deviations found but also its resolution of indication as well as the uncertainty of the measurement standard used in the calibration itself.


By measurement uncertainty we mean:

  • according to ISO-IMV (Internat. Metrology Vocabulary): “non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used”;
  • according to ISO-GUM (Guide to Uncertainty of the Measurement): “result of the estimation that determines the amplitude of the field within which the true value of a measurand must lie, generally with a given probability, that is, with a determined level of confidence”.

From the above definitions we can deduce two fundamental concepts of measurement uncertainty:

  1. Uncertainty is the result of an estimate, which is evaluated according to the following two types:
  • Category A: when the evaluation is done by statistical methods, that is through a series of repeated observations, or measurements.
  • Category B: when the evaluation is done using methods other than statistical, that is, data that can be found in manuals, catalogs, specifications, etc.

2. The uncertainty of the estimate must be given with a certain probability, which is normally provided in the three following expressions (see also Table 1):

  • Standard uncertainty (u): at the probability or confidence level of 68% (exactly 68.27%).
  • Combined uncertainty (uc): the standard uncertainty of a measurement when the result of the estimate is obtained by means of the values of different quantities and corresponds to the summing in quadrature of the standard uncertainties of the various quantities relating to the measurement process.
  • Expanded uncertainty (U): uncertainty at the 95% probability or confidence level (exactly 95.45%), or 2 standard deviations, assuming a normal or Gaussian probability distribution.

Standard uncertainty u(x) (a)

The uncertainty of the result of a measurement expressed as a standard deviation u(x) º s(x)

Type A evaluation (of uncertainty)

Method of evaluation of uncertainty by the statistical analysis of series of observations

Type B evaluation (of uncertainty)

Method of evaluation of uncertainty by means other than the statistical analysis of series of observations

Combined standard uncertainty uc(x)

Standard uncertainty of the result of a measurement when that result is obtained from the values of a number of other quantities, equal to the positive square root of a sum of terms, the terms being the variances or covariances of these other quantities weighted according to how the measurement result varies with changes in these quantities

Coverage factor k

The numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded uncertainty (normally is 2 for probability @ 95% and 3 for probability @ 99%)

Expanded uncertainty U(y) = k . uc(y) (b)

Quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand (normally is obtained by the combined standard uncertainty multiplied by with a coverage factor k = 2, namely with the coverage probability of 95%)

(a)   The standard uncertainty u (y), ie the mean square deviation s (x), if not detected experimentally by a  normal or Gaussian distribution, can be calculated using the following relationships:

u(x) = a/Ö3, for rectangular distributions, with an amplitude of variation ± a, eg Indication errors

u(x) = a/Ö6, for triangular distributions, with an amplitude of variation ± a, eg Interpolation errors

(b)   The expanded measurement uncertainty U (y) unless otherwise specified, is to be understood as provided or calculated from the uncertainty composed with a coverage factor 2, ie with a 95% probability level.


Table 1- Main terms & definitions related to measurement uncertainty according to ISO-GUM


Author: Eng. Alessandro Brunelli

Book: Manuale di Strumentazione