211 
1916-17.] Arithmetical Mean and “Middle” Value. 
magnitude, and the positive and negative departures from the arithmetical 
mean are equally probable. None of these properties are shown by the 
actual countings given in Table I, and it follows that the individual values 
are not grouped about only one value. It is noteworthy that the figures 
in a column can approximately be represented by two superposed theo- 
retical error curves whose zeros do not coincide. This might have been 
o 
expected in view of the fact that the temperature on a certain day is not 
only due to the heat radiated by the sun, but also to the direction of the 
wind, etc. Under these circumstances the best representative value of 
such or similar data is the middle value t' 0 , for there are even chances 
of an observation lying above or below this value. 
Let t' 0 = t 0 + m. The zero point from which r is counted must be 
changed by m so as to give equal sums on both sides of the zero, m is 
interpolated from the figures in Table I, and its values are given in the 
bottom line of the table. According to these values of m the middle 
value lies in winter up to 0°‘5 F. above the arithmetical mean, and in 
summer up to 0 o, 7 F. below the arithmetical mean. 
As to the grouping of the individual temperatures about the middle 
value, let N(t) designate the number of days of same date in the forty-eight 
years on which the maximum temperature lies within the limit t 0 -H — \ 
and tf'o + r + i, t ' o being the middle value of the maximum temperature on 
that date. Hence N(r) = M(T + m), and N(r) can be obtained from Table I 
by interpolation. The result is compiled in Table II. In the winter 
months the temperatures are scattered about 8° F. further to the negative 
side of the middle value than towards the positive side, and from March 
to October the extremes of the high temperatures lie much further from 
the middle value than the extreme low temperatures do, the scattering 
in September extending 10° F. further to the positive side than to the 
negative side. Hence the curves representing N(r) with abscissae r are 
steep for positive t in winter and for negative t in summer. 
When all the months are taken together the N(r) curve becomes 
symmetrical. And this might have been expected. For the heat of the 
sun is the one cause which operates the whole year in the same direction, 
while the other causes, wind direction, etc., act in a haphazard way during 
a year, and produce accidental departures. The average scattering of the 
temperatures in a year about their respective middle values well agrees 
with that calculated from the Theory of Errors. In applying this theory 
we might adapt its terminology to our case, and call the “ probable error,” 
r = 0'4769/A, the “ probable scattering.” The frequency numbers, N(t), are 
given by the function [ Continued on page 214. 
