Lyons — The Distribution of Mean Annual Rainfall. 357 



value " or "most probable value" of tlie quantity. To determine this value 

 the observed numbers may be arranged in order of magnitude, and the 

 middle term or " median " selected as the mean ; or the observed values may 

 be divided into groups, all in eacli group having the same value. The value 

 corresponding to the most numerous group may be taken as the mean (in 

 this case sometimes called the " mode "). Lastly, the arithmetical or geo- 

 metrical mean of all the observations may be taken, or a still more elaborate 

 process of reduction adopted. In such eases as the determination of an exact 

 physical magnitude the above methods will give results almost identical for 

 tlie true value of the quantity, provided the observations are sufficiently 

 numerous and trustworthy. This agreement is to be expected, and follows 

 from the general " Theory of Errors." 



In dealing with such quantities as annual rainfall, however, the problem 

 is somewhat different. The quantity by its nature has no exact value, but 

 varies within limits more or less wide. The expressions "true value" and 

 " most probable value " are meaningless, and the results obtained by different 

 methods of reduction differ widely. It becomes a question of doubt and 

 difficulty in all such cases to decide on the most satisfactory method of 

 treating the numbers, and the mean becomes largely a matter of definition 

 and convention. The ^'mean annual rainfaU^' is generally regarded as the 

 arithmetical average of a sufficiently large number of consecutive years. 

 Suggestions have been frequently made that the geometrical mean would be 

 more satisfactory. Any theoretical advantage of one " mean " over another 

 is, however, rendered very doubtful when we consider the great and irregular 

 variation in annual rainfall and the very considerable sources of errors in all 

 rainfall measurement. 



The arithmetical average of the annual rainfall for a period of years will 

 differ for different periods, the differences being great when the length 

 of the period is short. As a general rule, as the length of the period 

 increases, the mean assumes a more constant value ; and the limiting value 

 to which it approximates for a very long record is regarded as the true 

 mean. It has been found by Sir Alexander Binnie' in a very exhaustive 

 analysis of several records, extending from 50 to 97 years, that the mean of 

 about 35 years was practically constant and probably differed from the true 

 mean by not more than 2 per cent. Means computed for periods of 40 years 

 have shown greater differences than those for 35 years ; and it appears that 

 a very much longer period than 35 years should be taken to give a more 

 satisfactory result. This interesting conclusion was also reached by Professor 



' Minutes, Proc. Inst. Civil Engineers, vol. cix., p. 92. 



