Using Statistics to Reject a Projection

  Instead of using an integer multiple of the standard deviation, s, for the limits of a test one can use an arbitrary multiple known as a z-value defined by z = [a-E(a)]/s where a is a limiting value for the difference of the anomaly from the smoothed anomaly or its fit. Using z = 2.0 for the monthly global land anomaly with the three harmonic fit we find that the probability for observing an outlier is q = 5.848% so out of the 1604 months we would expect about 94 outliers. The observed number for the anomaly data of 92 is close to the expected number which is reasonable since we chose the probability function to fit the data. For z =2.0 we would expect the observed number of outliers to range from 75 to 113.

  How can we decide whether or not it is safe to reject a projection when we have collected a number of observations after the prediction? In ten years one would have 120 months of data to work with and suppose for arguments sake we observe k_obs outliers. Using the binomial distribution we can compute the probability, P_k, of observing k outliers and also the probability, 1- P_k, of not observing that number. If the level of certainty is set at 95% and the probability of not observing k_obs outliers is greater than this value we can confidently reject the projection. We would have to pass the projection if k_obs was between 4 and 10 inclusively.