Volume 1

Collected papers of R. A. Fisher / edited by J.H. Bennett.

  • Ronald Fisher
Date:
1971-1974
Catalogue details

Licence: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)

Credit: Collected papers of R. A. Fisher / edited by J.H. Bennett. Source: Wellcome Collection.

    576/616 (page 568)
    104 Proceedings of the Royal Society of Edinburgh. [Sess. upon (i) the magnitude of the instrumental errors, (ii) the rapidity with which the true value of F varies as v is decreased. The more terms are used in the formula, and the lower the order of differences ignored, the smoother will be the resultant series of estimates of F, and the more thoroughly will instrumental errors be obliterated. On the other hand, if instrumental errors are small, and the structure of the soil in respect of velocity distribution is highly complex, with rapidly recurring maxima and minima, then high order differences should be retained and the sequence of values used should be curtailed. To obtain information as to instrumental errors, it is evidently necessary to study the records of parallel soil samples, sedimenting preferably through different heights. The ideal method of reduction may then be regarded as the one which brino's out the maximum detail common to the several series. o with the minimum of discrepancies between them. In order to study the errors themselves, however, we shall not, in the present case, carry the smoothing' so far. To illustrate the principles underlying this method we may consider the possible formulae for the first differential coefficient. If only three points were taken we should have no choice but to take R,Vi— H-\) as the differentiated coefficient at the central point, for this is the slope of the tangent to the parabola passing through the three points. With five points we may take either u, { 2 (!/■;-y-i ) + (//i - //-,)} or V_. { - (//2 - V->) + //] -if -,) }, the first representing the slope of the best fitting straight line (or parabola), and the second that of the best fitting cubic polynomial, or of the quartic which passes through the five points. The difference between these two formulae is Til - (//o+ 2(yj-//_i)} = ~ ’ ir{ (.Vo - 3.V, + :b/ 0 - //_, ) + (//, - 3 //(J + 3//_ [ - )} . in the latter form it is seen to be proportional to the mean of the two third differences derivable from the live given points. If such third differences are not negligible, then either (i) a smooth curve of a higher degree than a parabola is required to represent the data, or (ii) the data are affected by random experimental errors, and so do not give a smooth curve. In the first case the second formula is required, for the first will suffer from a svstematic error in the neglect of the third differences. In the second case both formula} are free from systematic error, but the first formula is prefer able, since it is very much the less affected by random experimental errors.