Hayashi, Y., 1986: Statistical interpretations of ensemble-time mean predictability. Journal of the Meteorological Society of Japan, 64 (2), 167-181.

Abstract: Statistical interpretation of ensemble-time mean forecasts by the use of a dynamical model with unchanging external conditions are discussed. For this purpose, three kinds of variances are defined and their interrelations are clarified. It is proposed to define the predictability limit of the ensemble-time mean forecasts as the period when their error variance surpasses that of the climate-time mean forecasts. It is shown that, for a large ensemble of forecasts, this limit is close to Shukla's (1981) limit of individual time mean forecasts. The latter limit is defined as the period when the variance of the time mean forecasts with slightly different initial perturbations approaches that of the time mean forecasts from widely different basic initial conditions.
The statistical significance of ensemble-time mean predictability is also discussed and the interpretation of the analysis of variance is clarified. It is emphasized that a null hypothesis of unpredictability should not be readily accepted unless the confidence intervals are sufficiently small. It is shown by the use of confidence intervals that the number of Shukla's predictability experiments with a general circulation model is too small to statistically support his conclusion that the 31-60 day means are not dynamically predictable.