The Circular Normal Distribution: Theory and Tables

THE recognition of periodicities, their amplitude and length, is one of the main tasks of the application of statistics to long-range data and especially to economic phenomena. Although numerous theorems have been developed, the question of how to recognize a cycle remains open, since the complicated mathematical methods used in the periodogram and correlogram analysis may generate both real and factitious cycles and may conceal real ones. Therefore, the study of cycles is a fertile field for nonscientific procedures. Since even a constant value in a certain interval can easily be reproduced by a Fourier series, cycles with superimposed epicycles and third and fourth cycles have been "found" in nearly every domain that may be represented by numbers. Different observers have found different cycles in the same series-not to mention the great cycles of history leading to forecasts, good advices and policies advocated by their prophets. This abuse has discredited the whole theory of cycles. In contrast to these procedures, we examine here a specialized distribution function. Applications to recurrent phenomena of known period will be given in a second article. Since the distribution function has only two disposable parameters, and since we consider using it only in connection with such phenomena as seasonal or diurnal variations-where the length of period is definitely established from nonstatistical considerations-we feel that the chance of generating false cycles is small indeed. The distribution function under discussion was tabulated because it possesses important theoretical properties of the linear normal distribution and because it has practical applications to such meteoric phenomena as the direction of the wind, which is an angular or "circular" variable. Wind directions are represented clockwise on the map from north to east, south, west, and back to north in the same way as an angle varies from zero to 2ir. Each direction or angle has a