William Sealy Gosset

“ Any experiment may be regarded as forming an individual of a “population” of experiments which might be performed under the same conditions. A series of experiments is a sample drawn from this population.
Now any series of experiments is only of value in so far as it enables us to form a judgment as to the statistical constants of the population to which the experiments belong. ”

“ All the four individuals of the sample will occasionally come from the same group, giving a zero value for the standard deviation. Now this leads to an infinite value of and is clearly due to too wide a grouping, for although two men may have the same height when measured by inches, yet the finer the measurements the more seldom will they be identical, till finally the chance that four men will have exactly the same height is infinitely small. ”

“ Accordingly we find that the mean value of the standard deviation is quite close to that calculated, while in each case the variability is sensibly greater. The fit of the curve is not good, both for this reason and because the frequency is not evenly distributed owing to effects (2) and (3) of grouping. On the other hand the fit of the curve giving the frequency of is very good and as that is the only practical point the comparison may be considered satisfactory.
The following are the figures for height:—
Standard deviation of standard deviations:—
Comparison of Fit. ”

“ If we had smaller grouping the zero values of the standard deviation might be expected to increase, and a similar consideration will show that the smaller values of the standard deviation would also be likely to increase, such as .436, when 3 fall in one group and 1 in an adjacent group, or .50 when 2 fall in two adjacent groups. ”

“ In point of fact a large scale experiment confirmed the result, the small corn yielding about 15 per cent. more than the large.
I will conclude with an example which comes beyond the range of the tables, there being eleven experiments.
To test whether it is of advantage to kiln-dry barley seed before sowing, seven varieties of barley were sown (both kiln-dried and not kiln-dried) in 1899 and four in 1900; the results are given in the table.
It will be noticed that the kiln-dried seed gave on an average the larger yield of corn and straw, but that the quality was almost always inferior. ”

“ In order to see whether such odds are sufficient for a practical man to draw a definite conclusion, I take another set of experiments in which Dr Voelcker compares the effects of different artificial manures used with potatoes on the large scale. ”

“ The average gain by the use of sulphate of potash was 15.25 cwt. and the S.D. 9 cwt., whence, if we want the odds that the conclusion given below is right, corresponding, when , to or odds of 32:1; this is midway between the odds in the former example. Dr Voelcker says ‘It may now fairly be concluded that for the potato crop on light land 1 cwt. per acre of sulphate of potash is a better dressing than kainit.’ As an example of how the tables should be used with caution, I take the following pot culture experiments to test whether it made any difference whether large or small seeds were sown. ”

“ In this case I propose to use the approximation given by the normal curve with standard deviation and therefore use Sheppard’s tables, looking up the difference divided by . The probability in the case of yield of corn per acre is given by looking up in Sheppard’s tables. This gives , or the odds are about 14:1 that kiln-dried corn gives the higher yield.
Similarly , corresponding to , [9] so that the odds are very great that kiln-dried seed gives barley of a worse quality than seed which has not been kiln-dried. ”

“ The height and left middle finger correlation table was chosen because the distribution of both was approximately normal and the correlation was fairly high. Both frequency curves, however, deviate slightly from normality, the constants being for height , , and for left middle finger lengths , , and in consequence there is a tendency for a certain number of larger standard deviations to occur than if the distributions were normal. ”

“ Now Wallis’s expression for derived from the infinite product value of sin is
.
If we assume a quantity which we may add to the in order to make the expression approximate more rapidly to the truth, it is easy to show that etc. ”