Example: uncertainty of the instrument scale offset

In our scheme any quantity of influence of which we do not know the exact value is a source of systematic error. It will change the final distribution of $\mu$ and hence its uncertainty. We have already discussed the most general case in Section  . Let us make a simple application making a small variation to the example in Section  : the ``zero'' of the instrument is not known exactly, owing to calibration uncertainty. This can be parametrized assuming that its true value $Z$ is normally distributed around 0 (i.e. the calibration was properly done!) with a standard deviation $\sigma_Z$. Since, most probably, the true value of $\mu$ is independent of the true value of $Z$, the initial joint probability density function can be written as the product of the marginal ones:

$$f_\circ(\mu,z)=f_\circ(\mu)\,f_\circ(z)= k\,\frac{1}{\sqrt{2\,\pi}\,\sigma_Z} \,\exp{\left[-\frac{z^2}{2\,\sigma_Z^2}\right]}\,.$$ (5.66)

Also the likelihood changes with respect to ():

$$f(x_1\,\vert\,\mu,z) = \frac{1}{\sqrt{2\,\pi}\,\sigma_1} \,\exp{\left[-\frac{(x_1-\mu-z)^2}{2\,\sigma_1^2}\right]}\,.$$ (5.67)

Putting all the pieces together and making use of () we finally get

$$f(\mu\,\vert\,x_1, \ldots,f_\circ(z)) = \frac{ \int \frac{1}{\sqr... ...ma_Z} \exp{\left[-\frac{z^2}{2\,\sigma_Z^2}\right]} \,\rm {d}\mu\,\rm {d}z }\,.$$

Integrating^5.8we get

$$f(\mu) = f(\mu\,\vert\,x_1, \ldots,f_\circ(z)) = \frac{1}{\sqrt{2... ...a_Z^2}} \,\exp{\left[-\frac{(\mu-x_1)^2}{2\,(\sigma_1^2+\sigma_Z^2)}\right]}\,.$$ (5.68)

The result is that $f(\mu)$ is still a Gaussian, but with a larger variance. The global standard uncertainty is the quadratic combination of that due to the statistical fluctuation of the data sample and the uncertainty due to the imperfect knowledge of the systematic effect:

$$\sigma_{tot}^2 = \sigma_1^2+\sigma_Z^2\,.$$ (5.69)

This result is well known, although there are still some ``old-fashioned'' recipes which require different combinations of the contributions to be performed.

It must be noted that in this framework it makes no sense to speak of ``statistical'' and ``systematical'' uncertainties, as if they were of a different nature. They have the same probabilistic nature: $\overline{Q}_{n_1}$ is around $\mu$ with a standard deviation $\sigma_1$, and $Z$ is around 0 with standard deviation $\sigma_Z$. What distinguishes the two components is how the knowledge of the uncertainty is gained: in one case ($\sigma_1$) from repeated measurements; in the second case ($\sigma_Z$) the evaluation was done by somebody else (the constructor of the instrument), or in a previous experiment, or guessed from the knowledge of the detector, or by simulation, etc. This is the reason why the ISO Guide[3] prefers the generic names Type A and Type B for the two kinds of contribution to global uncertainty. In particular, the name ``systematic uncertainty'' should be avoided, while it is correct to speak about ``uncertainty due to a systematic effect''.