Assessing the limitations of effective number of samples for finding the uncertainty of the mean of correlated data

The efficacy of recent and classical theories on the uncertainty of the mean of correlated data have been investigated. A variety of very large data sets make it possible to show that, under circumstances that are often too expensive to achieve, the integral time scale can be used to determine the effective number of independent samples, and therefore the uncertainty of the mean. To do so, the data set must be sufficiently large that it may be divided into many records, each of which is many integral time scales long. In this circumstance, all lags of the autocorrelation should be integrated to determine the integral scale. Some secondary findings include that the classical definition of the integral time scale goes identically to zero if a single record of any length is used and demonstration that measuring the integral scale requires ensemble averaging. Estimation of the integral time scale for a single record requires that the integration of the autocorrelation be truncated. This works well for signals where anti-correlation is not present. Additionally, for anti-correlated samples, the effective number of samples exceeds the number of acquired samples.