The standard error of the magnitude-frequencybvalue

abstract Estimated b values in log N = a − bM are widely used in seismicity comparisons and risk analysis, but uncertainties have been little explored. In this paper, the usual F probability density distribution for b is given and compared with an asymptotic form for temporally varying b. Convenient tables for the standard error of b are given that allow statistical tests to accompany investigations of both temporal and spatial variations of b. With large samples and slow temporal changes in b, the standard error of b is σ (b ^) = 2. 30 b 2 σ (M _), where σ 2 (M _) = ∑ i − 1 n (M i − M _) 2 / n (n − 1). In an example from central California, stable estimates of b require a space-time window containing about 100 earthquakes. From 1952 to 1978, the average b and 90 per cent confidence limits are 0. 95 (+0. 94, −0. 30). Some fluctuations of b are statistically significant but some are not. Within 90 per cent confidence limits, b changes from a low of 0. 60 (+0. 11, −0. 09) in 1955 to a high of 1. 39 (+0. 25, −0. 21) in 1967 and drops to 0. 72 (+0. 13, −0. 10) in 1975. In this example, no correlation between large earthquakes (M 5) and b variations occurred.