Statistical Investigations 401
holds, and surely he is right, that the middlemost estimate, the median, is the correct one. Every other estimate has a majority of the voters against it as either too low or too high. The correct estimate cannot be the average for, as Galton puts it, the average "gives a voting power to `cranks' in proportion to their crankiness." The average allows crankiness to swamp reasonable judgment. For such reasons Galton laid considerable stress on the median, and on various contrivances for rapidly determining it.
I have already referred (pp. 336, 385) to the use Galton made of two bows or two weights to test the strength of a group, and how he determined his median from the resulting percentages. This point is more fully dealt with in a paper on "The Median Estimate" read at the Dover meeting of the British Association in 18991. In this paper Galton applies the two weights test to determine the proper damages by a jury or a suitable grant by a committee. Two sums A and B, B being greater than A, are fixed on and then three shows or counts of hands are taken, (i) for a sum less than A, (ii) for a sum between A and B, and (iii) for a sum greater than B. The individuals have thus not to determine actual amounts, but only inequalities. Galton now assumes the "normal" distributionn of judgments and proceeds to determine the median in the manner of our footnote, p. 3852. To expedite the determination he published a table of percentiles giving the ordinates in terms of the quartile. This table is also reproduced in a paper of the following year and originally appeared in his book Natural Inheritance of 1889. It can still be used although it only gives three significant figures (two decimals), when the quartile is preferred. It has, however, been superseded for most purposes by the table of five significant figures (four decimals) provided by Dr W. F. Sheppard at the suggestion of Galton, who wrote a prefatory note to the table'. This table gives the deviate in terms of the standard deviation and proceeds by permilles not percentiles. The prefatory note is a remarkable one considering that Galton was then aged 85; he there broke a last lance for the use of the ogive curve and the median, which he had introduced 40 years earlier. He took his present biographer's data for the intelligence of Cambridge graduates and represented it on a percentile scale and not on the biographer's "normal" scale ; and he made a very good defence of his method.
1 British Association Report, 1899, pp. 638-40.
2 If band a be the fractions of the total assessors who vote "above B" and "below A" respectively, then the ordinates of the probability curve corresponding to b and a, in terms of the standard deviation as unit, can be found from a table of permilles (see Tables for Statisticians
and Biometricians, Table I). If these be a and /3 the median will be
a+/3 a+ f3
Here we suppose a and 6 both less than 50 per cent. of the total number of assessors. This is the better way of determining m; a slight modification is needed, if m be greater (or less) than both A and B. The values of a and b should correspond to more than 5 per cent. of the assessors for reasonable accuracy.
3 Biometrika, Vol. v, pp. 400-6. "Grades and Deviates (including a Table of Normal Deviates corresponding to each millesimal grade in the length of an array, and a figure)."