312 THEORY OF STATISTICS. 
The standard-deviation is /0'5 x 0-5 x 100 = 5. Taking the 
distribution as normal, y,= 797-9. 
The mean number of heads being 50, 65 —50=30c. The 
frequency of a deviation of 3¢ is given at once by the table (p. 303) 
as 7979 x 0111 . . . . =8'86, or nearly 9 throws in 10,000. A 
throw of 65 heads will therefore be expected about 9 times. 
The frequency of throws of 65 heads or more is given by the 
area table (p. 310), but a little caution must now be used, owing 
to the discontinuity of the distribution. A throw of 65 heads is 
equivalent to a range of 64'5-65'5 on the continuous scale of the 
normal curve, the division between 64 and 65 coming at 64°5. 
64:5 — 50 = + 2:90, and a deviation of + 2:9. or more, will only 
occur, as given by the table, 187 times in 100,000 throws, or, say, 
19 times in 10,000. 
Fxample ii.—Taking the data of the stature-distribution of fig. 
49 (mean 67°46, standard-deviation 2-57 in.), what proportion of 
all the individuals will be within a range of + 1 inch of the 
mean { 
1 inch =0-3890. Simple interpolation in the table of p. 310 
gives 0'65129 of the area below this deviation, or a more extended 
table the more accurate value 0:65136. Within a range of 
+ 0°389¢ the fraction of the whole area is therefore 0:30272, or the 
statures of about 303 per thousand of the given population will lie 
within a range of +1 inch from the mean. 
Example iii.—In a case of crossing a Mendelian recessive by a 
heterozygote the expectation of recessive offspring is 50 per cent. 
(1) How often would 30 recessives or more be expected amongst 50 
offspring owing simply to fluctuations of sampling? (2) How many 
offspring would have to be obtained in order to reduce the probable 
error to 1 per cent. ? 
The standard error of the percentage of recessives for 50 
observations is 50 A/ 1/50 =7'07. Thirty recessives in fifty is 
a deviation of 5 from the mean, or, if we take thirty as representing 
29'5 or more, 4'5 from the mean; that is, 0'636.0. A positive 
deviation of this amount or more occurs about 262 times in 1000, 
so that 30 recessives or more would be expected in more than a 
quarter of the batches of 50 offspring. We have assumed 
normality for rather a small value of n, but the result is sufficiently 
accurate for practical purposes. 
As regards the second part of the question we are to have 
"6745 x 50 /1/n=1, 
n being the number of offspring. This gives n=1137 to the 
nearest unit.