The Effect of Likely Biases in Estimating the Variance of Long Time Averages of Climatological Data

Bias in estimating the variance of independent identically distributed (IID) random variables with an unknown mean is well known and readily handled in the time domain by the n-1 factor (e.g., Mood and Graybill 1963, 183). A similar factor can be derived for autocorrelated data (Trenberth 1984) to adjust for bias. Variance can also be determined from the frequency spectrum of a time series, and there are interesting analogous adjustments that can be made to compensate for bias (Trenberth 1984, his appendix). These adjustments depend on the spectral densities near zero frequency, which cannot be estimated directly from the data because they are at timescales longer than the length of the time series. Interestingly, when computing the variance of time averages, these near-zero frequency spectral densities, which must be based on assumptions, become more important as the averaging time becomes longer (Jones 1975). In this paper we consider the effect of various choices or assumptions about the unresolved low-frequency spectral densities on estimates of the variance of averages.