Daily mean temperature formulas

The sum of the twenty-four hourly temperature observations divided by twenty-four is accepted as the true daily mean. However, most of the individual observers in the nineteenth and twentieth century climate record could not make observations every hour of every day. Instead, climate networks prescribed convenient observation times and devised formulas to acceptably approximate the true daily mean. Over twenty different formulas were used at one time or another over the past two hundred years. For this paper, hourly data from thirty geographically disparate stations in the United States were used to produce the true daily mean for each day and month of the year 2008. Geographically distributed stations that had hourly temperature data were selected. Twenty-one surrogate daily mean formulas were applied to the hourly data from each of those stations. Surrogate daily means, rounded to two decimal places were calculated in an S-Plus script using each formula. Daily, monthly, and annual means were calculated from each of them. The magnitude of temperature bias induced by each surrogate formula was determined. The accuracy and consistency of the formulas relative to the true mean were analyzed to identify variations by locations and climatic regions. Maps of the magnitude of the annual bias induced by each of the twenty-one surrogate formulas were prepared to examine the spatial variation of the results. The biases that were discovered can be used to adjust the temperature means using the same contemporaneous formula that the observational network prescribed for the observer.