XI.—CORRELATION : MISCELLANEOUS THEOREMS. 225 
of the children in different schools might be obtained on the 
basis of a standard school population of given age and sex 
composition, or indeed of given composition as regards hair and 
eye-colour as well. 
20. In §§ 14-17 we have dealt only with the theory of 
the weighted arithmetic mean, but it should be noted that 
any form of average can be weighted. Thus a weighted median 
can be formed by finding the value of the variable such that 
the sum of the weights of lesser values is equal to the sum 
of the weights of greater values. A weighted mode could be 
formed by finding the value of the variable for which the sum 
of the weights was greatest, allowing for the smoothing of 
casual fluctuations. Similarly, a weighted geometric mean could 
be calculated by weighting the logarithms of every value of the 
variable before taking the arithmetic mean, i.e. 
7 
log GC, = (WV. log X) 
tA) 
REFERENCES. 
Effect of Grouping Observations. 
(1) SHEPPARD, W. F., ““On the Calculation of the Average Square, Cube, etc., 
of a large number of Magnitudes,” Jour. Roy. Stat. Soc, vol. lx., 1897, 
. 698. 
(2) Suen, W. F., “On the Calculation of the most probable Values of 
Frequency Constants for Data arranged according to Equidistant 
Divisions of a Scale,” Proc, Lond. Math. Soc., vol. xxix. p. 353. (The 
result given in eqn. (4) for the correction of the standard-deviation is 
Sheppard’s result.) i: 
(8) SueppARD, W. F., “The Calculation of Moments of a Frequency-distribu- 
tion,” Biometrika, v., 1907, p. 450. 
(4) PEARSON, KarL, and others [editorial], “On an Elementary Proof of 
Sheppard’s Formule for correcting Raw Moments, and on other allied 
points,” Biometrika, vol. iii., 1904, p. 308. 
(5) PEARSON, KARL, ‘ On the Influence of ¢ Broad Categories’ on Correlation,” 
Biometrika, vol. ix., 1913, pp. 116-139. 
Effect of Errors of Observation on the Correlation-coefficient, 
(6) SprARMAN, C., “The Proof and Measurement of Association between Two 
Things,” Amer. Jour. of Psychology, vol. xv., 1904, p., 88. 
(Formula (8).) 
(7) SrearMAN, C., “‘ Demonstration of Formule for True Measurement of 
Correlation,” Amer. Jour. of Psychology, vol. Xviii., 1907, p. 161. 
(Proof of formula (8), but on different lines to that given in the text, 
which was communicated to Spearman in 1908, and published by 
Brown and by Spearman in (8) and (10).) 
(8) SrEARMAN, C., “Correlation calculated from Faulty Data,” British Jour. 
of Psychology, vol. iii., 1910, p. 271. 
15