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·
93 on Readers’
ability
distribution
known.
is
exactly
Unfortunately, very often, in dealing with short statistical series, the distribution function is not known beforehand. The present inconsiderate habit to base “confidence limits” straightaway on the normal distribution function may lead to deceptions and isa danger for the firm establishment of statistical methods in analytical chemistry. Alas! This is not the only trouble with "confidence.” At present there is a growing tendency to recommend “standard confidence limits” by convention {e.g., 95% confidence, corresponding to the range X ± 2 etc. ? ? ?). I fear I must be very outspoken to make clear that this leads into a dead road. The idea to agree generally upon such numerical values as confidence, significance, and risk is absurd. These terms originate from practical ethics, not from science or mathematics; they indicate the boundary between life and theory. It is remarkable that statistics enables us to describe numerically some aspects of the complex of confidence or that of risk; however, “how much confidence” is required, which risk ca,n be taken, cannot be settled by convention, but is determined solely by the facts and tasks of life. For instance, 95% confidence or only 5% risk to fail may be reasonable for the routine work of an analyst if no serious decisions are at stake. But what about an analysis in a case of murder? Or would anyone be satisfied by a confidence limit of 95%, 99%, or even 99.9% in testing the safety of an airplane or in the evaluation of a radar measurement decisive for war
or
peace?
The wrong
approach to agree upon confidence in advance is caused by the emotional warmth of this word, by the tendency to escape the desert of abstraction. Yet all the statistical information which is necessary to make decisions in practical life is contained in the distribution function or in compact form in its parameters, or their estimates. There is no need to force a special interpretation upon a man who uses this material for his decisions. Give him the mental tools to understand the meaning, but leave
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ANALYTICAL CHEMISTRY, VOL. 42, NO. 4, APRIL 1970
to him the responsibility how to use it; he may calculate confidence limits for his special case. Now let us come back to the question of distribution functions in
analytical chemistry: The “noror Gaussian distribution plays an important role which will be considered in the next chapter. But I dare say that a normal distribution mal”
is not the normal case in chemical
analysis. We have already mentioned the binomial and the Poisson distribution. Frequently, distributions with several peaks are found if the sample material is mixed from different batches. Inhomogeneity in parts of the material may lead to a distribution with a narrow peak and broad wings. Preselection according to technical specifications may cause highly asymmetric distributions one wing may be cut off. Obviously, in some of these examples the frequency distribution is determined not only by the accidental errors of the measurements but also by the statistical fluctuations of sample composition. Let us suppose that we have made a large number of measurements on the same strictly homogeneous material and thereby have found the distribution function giving a survey of the relative frequency of ocof all possible accidental currence errors. Certainly this function characterizes the used analytical procedure as such. Now we may have found a nice normal distribution around the average with a low standard deviation indicating a high precision. We report the average as analytical result, give confidence limits (which we should not do!) based on that standard deviation, and according to prescribed specifications, lean back and feel ;
happy.......
Everything
seems
fine—and yet
statements may be wrong! The distribution function (of any type), gained by a finite series of repeated measurements, gives the and inaverage x of the measures formation about their scatter around this average, but it does not say how near this particular average may be to the “true value,” which is supposed to be equal to the mean X of the whole population if no systematic errors, causing bias, are Tschebyscheff’s inpresent. —our
