XV.—BINOMIAL DISTRIBUTION AND NORMAL CURVE. 315 
Jour. Roy. Stat. Soc., vol. Ixxiii., 1910, p. 26. (A binomial distribu- 
tion with negative index, and the related curve, i.e. a special case of 
one of Pearson's curves, ref. 13.) 
The Resolution of a Distribution compounded of two Normal 
Curves into its Components. 
(18) PEARsoN, KARL,“ Contributions to the Mathematical Theory of Evolu- 
tion (on the Dissection of Asymmetrical Frequency Curves),” Phil. 
Trans. Roy. Soc., Series A, vol. clxxxv., 1894, PZ 
(19) Epceworrn, F. Y., “On the Representation of Statistics by Mathema- 
tical Formule,” part ii., Jour. Roy. Stat. Sec., vol. Ixii., 1899, p. 125. 
(20) PearsoN, KARL, “On some Applications of the Theory of Chance to 
Racial Differentiation,” Phil. Jay. 6th Series, vol. i., 1901, p. 110. 
(21) HELGUERO, FERNANDO DE, *‘ Per la risoluzione delle curve dimorfiche,” 
Biometrika, vol. iv., 1905, p. 230. Also memoir under the same title 
in the Transactions of the Reale Accademia dei Lincei, Rome, vol. vi., 
1906. (The first is a short note, the second the full memoir, ) 
See also the memoir by Charlier, cited in (2), section vi. of that 
memoir dealing with the problem of dissection. 
Testing the Fit of an Observed to a Theoretical or 
another Observed Distribution. 
(22) PEARSON, KARL, “On the Criterion that a given System of Deviations 
from the Probable, in the Case of a Correlated System of Variables, is 
such that it can be reasonably supposed to have arisen from random 
sampling,” Phil. Mag., 5th Series, vol. 1., 1900, p- 157. 
(23) Pearson, KARL, “On the Probability that Two Independent Distribu- 
tions of Frequency are really Samples from the same Population,” 
Biometrika, vol. viil., 1911, p- 250 ; also Biometrika, vol. x., 1914, 
pp. 85-143, 
EXERCISES. 
1. Calculate the theoretical distributions for the three experimental cases 
(1), (2), and (8) cited in § 7 of Chapter XIII. 
2. Show that if np be a whole number, the mean of the binomial coincides 
with the greatest term. 
3. Show that if two symmetrical binomial distributions of degree n (and 
of the same number of observations) are so superposed that the rth term of 
the one coincides with the (r+1)th term of the other, the distribution 
formed by adding superposed terms is a symmetrical binomial of degree n+ 1. 
[Note : it follows that if two normal distributions of the same area and 
standard-deviation are superposed so that the difference between the means is 
small compared with the standard-deviation, the compound curve is very 
nearly normal. ] 
4. Culculate the ordinates of the binomial 1024 (05+ 05)", and compare 
them with those of the normal curve. 
5. Draw a diagram showing the distribution of statures of Cambridge 
students (Chap. VI., Table VII), and a normal curve of the same area, 
mean, and standard-deviation superposed thereon. 
6. Compare the values of the semi-interquartile range for the stature 
distributions of male adults in the United Kingdom and Cambridge students, 
(1) as found directly, (2) as calculated from the standard-deviation, on the 
assumption that the distribution is normal.