Variance random errors, definition

Some of the concepts used in defining confidence limits are extended to the estimation of uncertainty. The uncertainty of an analytical result is a range within which the true value of the analyte concentration is expected to lie, with a given degree of confidence, often 95%. This definition shows that an uncertainty estimate should include the contributions from all the identifiable sources in the measurement process, i.e. including systematic errors as well as the random errors that are described by confidence limits. In principle, uncertainty estimates can be obtained by a painstaking evaluation of each of the steps in an analysis and a summation, in accord with the principle of the additivity of variances (see above) of all the estimated error contributions any systematic errors identified... 

Only in situations where numerical uncertainty is small compared to modeling uncertainty we can successfully validate a calculation. After minimizing numerical errors there will still be other uncertainties in calculations due to for example variations in inlet conditions or due to inherent uncertainty in tabulated material properties, etc. These can be best handled by repeating the calculations with appropriate variations in the uncertain input quantities, thus resulting in say nc calculations with seemingly n, random outcomes the mean and variance of which are donated by Xc and. S 2, Similarly there would be ne repeated experiments of the same phenomenon with a, random outcomes with the corresponding mean and variance, Xe and S2e, respectively. The estimated modeling error is by definition the difference between the experimental mean and calculation mean, i.e. 

A more rigorous definition of uncertainty (Type A) relies on the statistical notion of confidence intervals and the Central Limit Theorem. The confidence interval is based on the calculation of the standard error of the mean, Sx, which is derived from a random sample of the population. The entire population has a mean /x and a variance a. A sample with a random distribution has a sample mean and a sample standard deviation of x and s, respectively. The Central Limit Theorem holds that the standard error of the mean equals the sample standard deviation divided by the square root of the number of samples ... 

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