210 
Proceeding's of the Royal Society of Edinburgh. [Sess, 
XIII. — The Arithmetical Mean and the “ Middle ” Value of Certain 
Meteorological Observations. By L. Becker, Ph.D., Regius 
Professor of Astronomy in the University of Glasgow. 
(MS. received April 30, 1917. Read May 7, 1917.) 
The arithmetical mean of similar meteorological observations is usually 
regarded as their representative value. The object of this paper is 
to show that in a certain case this assumption is not borne out by 
observations. 
The investigation refers to the maximum temperature in the shade 
as observed at Glasgow Observatory in the forty-eight years 1868 to 1916. 
For each of the 73 periods of 5 days the arithmetical mean of the (240) 
maximum temperatures observed in the forty-eight years was calculated, 
and the average maximum temperature, t 0 , was interpolated from these 
means for each day of the year. Let M(t) designate the number of days 
of same date on which the maximum temperature lies within the limits 
a. 
t 0 + riki degree. For each day of the year M(r) was counted, r being 
successively all the positive and negative integers. These numbers change 
from day to day in a regular way for the same value of r, and no appre- 
ciable error is committed by combining the numbers counted on all the days 
of the month and ascribing the result to the middle of the month. Table 1 
contains the M(r) per 1000 days. The figures for January are based on 
1488 ( = 48 x 31) observations, and similarly for the other months. 
We may assume that when the same cause is acting on different occa- 
sions, and is disturbed in a haphazard way, the effects will not be identical, 
but they will be grouped round an average effect, which effect is the most 
probable to occur. In the event of the effect having been measured and 
expressed in figures, the most probable value of the effect is the arith- 
metical mean of all these figures. It then follows that the individual 
values will be arranged round the arithmetical mean according to the 
Law of Errors. Hence the probability <p(r) of a departure ±t from the 
arithmetical mean is expressed as follows : 
<£(t) = llTT " e ^ T . 
Thus if there is only one constant cause to an effect, the arithmetical mean 
coincides with the “ middle ” value, i.e., that value which occupies a middle 
position when the figures expressing the effect are arranged in order of 
