# Measurement Uncertainty¶

Independent measurement results will vary. The measurand will not always fall within the stated measurement uncertainty

No measurement is perfect. The quantity intended to be measured (the measurand) and a measurement result inevitably differ because of measurement error.

While the measurement error is never known, a range of possible values can be anticipated, which allows a statement to be made about the uncertainty of the measurement result as an estimate of the measurand.

Uncertainty statements are inherently random, because measurement errors change. So, if a particularly large error occurred in a measurement the uncertainty statement may not cover the measurand.

To allow for this randomness, some assumptions are made about the statistical distribution of errors. It is then possible to state how many uncertainty statements, on average, will include the measurand. This success-rate is called coverage probability, or level of confidence. A coverage probability of 95% means that uncertainty intervals should contain the measurand about 95 times out of 100.

## Real quantities¶

When reporting the measurement of a real-valued quantity, the uncertainty statement represents an interval that is usually centered on the best estimate of the measurand.

The interval half-width is called the expanded uncertainty and is the product of

• a coverage factor (for a specific coverage probability) and
• a standard uncertainty

In conventional notation, the standard uncertainty of a measurement result $$x$$ is written $$u(x)$$ and the letter $$k$$ is used for the coverage factor [1]. So the uncertainty interval is

$[x - k \cdot u(x), x+ k \cdot u(x)]$

or simply

$[x - U, x+ U] \; ,$

where $$U$$ is the expanded uncertainty.

For example, an RF power measurement is reported with an expanded uncertainty $$U$$ and a coverage probability of 95%:

$p = 12.1 \; \mu \mathrm{W} \; , \; U =0.5 \; \mu \mathrm{W} \;,$

which means that the true power level is likely to be within the interval $$p \in [11.6,12.6]\;\mu \mathrm{W}$$.

## Complex quantities¶

The measurement of a complex quantity is really a simultaneous measurement of the real and imaginary components. Each measurement will have an uncertainty and, in addition, the results may be correlated.

Information about the uncertainties and the correlation between results can be summarized in a covariance matrix

$\begin{split}\mathbf{V} = \begin{bmatrix} v_{11} & v_{12}\\ v_{21} & v_{22} \end{bmatrix} \;.\end{split}$

The diagonal matrix elements are squared standard uncertainties, $$v_{11} = u_\mathrm{re}^2$$ and $$v_{22} = u_\mathrm{im}^2$$, and the off-diagonal element is the covariance $$v_{12} = v_{21} = u_\mathrm{re}\, r\, u_\mathrm{im}$$, where $$r$$ is the correlation coefficient between the real and imaginary component estimates.

The covariance matrix is used to calculate uncertainty, but it is more natural to think in terms of standard uncertainties $$u_\mathrm{re}$$ and $$u_\mathrm{im}$$. Fortunately, the covariance matrix can be decomposed into the product of a diagonal uncertainty matrix and a simple correlation matrix

$\begin{split}\mathbf{V} = \begin{bmatrix} u_\mathrm{re} & 0\\ 0 & u_\mathrm{im} \end{bmatrix} \begin{bmatrix} 1 & r\\ r& 1 \end{bmatrix} \begin{bmatrix} u_\mathrm{re} & 0\\ 0 & u_\mathrm{im} \end{bmatrix} \; .\end{split}$

The uncertainty associated with a measured complex quantity is region

In general, an uncertainty statement for a complex quantity represents a region of the complex plane. Different types regions are possible, although the conventional choice is an ellipse. The shape of the ellipse depends on the form of the covariance matrix.

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Footnotes

 [1] The coverage factor depends on the coverage probability and the number of degrees of freedom. The notion of degrees of freedom is borrowed from classical statistics, where it is related to size of the sample used to estimate the variance. The interpretation of degrees of freedom is less strict in measurement problems than in classical statistics. When degrees of freedom are infinite, the standard deviation of the distribution of measurement error is assumed known. However, the coverage factor increases as the degrees of freedom become smaller, increasing the expanded uncertainty.