Averaging in the presence of “sliding” errors

Abstract In many cases the precision with which an experiment can measure a physical quantity depends on the value of that quantity. Not having access to the true value, experimental groups are forced to assign their errors based on their own measured value. Procedures which attempt to derive an improved estimate of the true value by a suitable average of such measurements usually weight each experiment's measurement according to the reported variance. However, one is in a position to derive improved error estimates for each experiment from the average itself, provided an approximate idea of the functional dependence of the error on the central value is known. Failing to do so can lead to substantial biases. Techniques which avoid these biases without loss of precision are proposed and their performance is analyzed with examples. These techniques are quite general and can bring about an improvement even when the behavior of the errors is not well understood. Perhaps the most important application of the technique is in fitting curves to histograms.