Statistical tools systematic/random errors

The inspection of the fit residuals, that is, the (normalized) differences between the experimental and fitted data point, is a reliable tool to check for deviations from the fitted model. Residuals should be statistically noncorrelated and randomly distributed around zero. For example, if a bi-exponential decay is fitted to a single exponential function, the residuals will show systematic errors. Therefore, correlations in the residuals may indicate that another fit model should be used. 

In Chapter 3 it was discussed how the presence of a random error can be handled by statistical tools. The precautions which must be taken by the experimenter not to violate the assumption of independencies of the experimental error is randomization, which allows certain time-dependent systematic errors to be broken down and turned into random errors. There are, however, sources of error which can be suspected to produce systematic deviations which cannot be counteracted by randomization. In such cases, forseeable sources of systematic variation can be brought under control by dividing the whole set of experiments into smaller blocks which can be run under more homogeneous conditions. By a proper arrangement of these blocks, the systematic variation can be isolated through comparison of the between-block variation. Some examples where splitting the series of experiments into blocks is appropriate are ... 

Statistical methods provide an approach that yields quantitative estimates of the random uncertainties in the raw data measurements themselves and also in the conclusions drawn from them. Statistical methods do not detect systematic errors (e.g. bias) present in an assay nor do they give a clear-cut answer to the question as to whether or not a particular experimental result is acceptable. An acceptability criterion must be chosen a priori based on the underlying assumption that the data follow a Gaussian (normal) distribution. A common acceptability criterion is the 95 % confidence level, corresponding to a p-value of 0.05. Because work is with small data sets in trace quantitative analyses, as opposed to the infinitely large data sets required for idealized statistical theory, use is mode of tools and tests based on the (t) distribution (Sudent s t distribution) developed specifically for the statistical analysis of small data sets. 

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