New approximating polynomials for the treatment of measurement results

Conclusions

Methods are examined and recommended in this article for approximating physical relationships by polynomials which can be found with a considerably smaller number of calculations that those required for finding polynomials which provide the best approximation with respect to the minimum of the deviations squared. However, in such a case it becomes necessary to accept either a certain loss of information (a reduction in precision) or a certain rise in the number of experiments.

It is shown in this article that polynomials of the second and third degree can be taken through their mean points and corrected, without any appreciable loss of precision, by a constant or by a linear additional polynomial.

In the first half of the article it is shown that a polynomial of any k-th degree taken through the mean points should be corrected by a polynomial of degree k-2; moreover, even if the latter is obtained by the least squares method, the number of calculations is reduced considerably.

In conclusion, we should like to point out a method which can be used in cases when only limited precision is required. An approximating curve is then plotted according to any reasonable and simple rule (it can even be drawn “by eye” and defined from a table), and the deviations of the empirical points from it are approximated by one of the above-mentioned polynomials. It is also often possible to approximate with a linear polynomial which can easily be found by the least squares method. A comparison of residual variances can serve, as usual, as a criterion for selecting a suitable method.