Hourly Temperatures and Annual Rani Hakby P. Bailey* "Frequency distributions of the various climatic elements, for *- varying periods of time, have proliferated in the last two decades, as electronic data processing was adopted by national weather services . For example, the National Weather Records Center at Asheville , North Carolina, can furnish climatic summaries for hundreds of stations, giving temperature, precipitation, humidity, and wind conditions in frequency tables singly or in combination. Interpretation of such material has lagged, perhaps because the very depth of the tabular detail is discouraging. Certainly time is required before climatologists can learn to interpret the significance of this new method of presenting climatic information. Frequency distributions of some elements, notably temperature, have been studied slightly since the 19th Century.1 Blumenstock2 attempted a frequency analysis of the difficult concept of drought. Court's analysis3 of the frequency distributions of hourly temperatures in January and July throughout the United States is in some respects the starting point for the present investigation. In particular, under study here is the distribution of all the 8,760 (or 8,784) hourly temperatures of a year, and especially the * Indispensable to the prosecution of this investigation have been the intramural research grants of the University of California, with which Dr. Bailey has been affiliated for 30 years; since 1964 he has been Professor of Geography at the University of California, Riverside, CaI., 92502. Mrs. Jane Eunice and Mr. Roy Elledge, as well as other assistants, toiled many hours with desk calculators and microfilms. Many revisions and improvements were suggested by the editor, Dr. Arnold Court, almost to the point of collaboration. 1 Hugo Meyer, Anleitung zur Bearbeitung Meteorologischen Beobachtungen für die Klimatohgie (Berlin: Julius Springer, 1891). 2 David I. Blumenstock, "Drouth Lengths and Frequencies in the United States," dissertation ( University of California, Berkeley, 1943 ) . a Arnold Court, "Temperature Frequencies in the United States," Journal of Meteorology , Vol. 8 (1951), pp. 367-380. 25 26 ASSOCIATION OF PACIFIC COAST GEOGRAPHERS relation of the standard deviation (or its square, the variance) of this distribution to the mean annual range of temperature. Both are measures of dispersion around the annual mean, but have markedly different properties. The conventional mean annual range is simply the difference between the means of the warmest and coldest months. Thus, it reflects the influence of but one-sixth the time span that affects the frequency distribution formed from hourly temperatures throughout the year. More importantly, the standard deviation of all the hourly temperatures is more sensitive to extreme observations than is the mean annual range. Similar sensitivity is provided by the extreme range, dependent upon only two occurrences, or the mean extreme range, again based on a tiny fraction of the sample of hourly temperatures. The standard deviation of all hourly observations, in fact, is a characteristic of a "total" or "complete" temperature population, and is often referred to as such hereafter. Linear Relation However, these differences should not overshadow the considerable area of agreement between the mean annual range and the standard deviation of a complete temperature population. Both are first-order measures of dispersion around the annual mean. If hourly temperatures at all places followed the same distribution, a linear relation would exist between the two measures. For 47 Air Weather Service stations (Figure 1), the correlation between the mean annual range (Am) and the standard deviation of ?§ ^ Figure 1. Locations of 47 U.S.A.F. Air Weather Service stations used in study. VOLUME 30 YEARBOOK 1968 27 Table 1. Notation for Analysis of Temperature A*, Ad, A» . B ................ Shyf Shmj Skd Sdjfj S&m ...... T ... ..Mean annual range of temperature, computed as the difference between the means of the warmest and coldest hours, days, and months, respectively. ..Mean daily range. .Standard deviation of hourly temperatures around the annual mean, the monthly mean, and daily mean, respectively; s2 is variance. .Standard deviation of daily temperatures around annual and monthly mean, respectively. ..Standard deviation of monthly temperatures around annual mean. .Mean annual temperature. ..Mean monthly temperature. hourly temperatures around the annual mean (shV) during ten years (1951-1960) is +0.984. (Notation is given in Table 1). A least squares linear regression, or straight line of best fit...