1. It is widely felt that any method of rejecting observations with large deviations from the mean is open to some suspicion. Suppose that by some criterion, such as Peirce’s and Chauvenet’s, we decide to reject observations with deviations greater than 4 σ, where σ is the standard error, computed from the standard deviation by the usual rule; then we reject an observation deviating by 4·5 σ, and thereby alter the mean by about 4·5 σ/
n
, where
n
is the number of observations, and at the same time we reduce the computed standard error. This may lead to the rejection of another observation deviating from the original mean by less than 4 σ, and if the process is repeated the mean may be shifted so much as to lead to doubt as to whether it is really sufficiently representative of the observations. In many cases, where we suspect that some abnormal cause has affected a fraction of the observations, there is a legitimate doubt as to whether it has affected a particular observation. Suppose that we have 50 observations. Then there is an even chance, according to the normal law, of a deviation exceeding 2·33 σ. But a deviation of 3 σ or more is not impossible, and if we make a mistake in rejecting it the mean of the remainder is not the most probable value. On the other hand, an observation deviating by only 2 σ may be affected by an abnormal cause of error, and then we should err in retaining it, even though no existing rule will instruct us to reject such an observation. It seems clear that the probability that a given observation has been affected by an abnormal cause of error is a continuous function of the deviation; it is never certain or impossible that it has been so affected, and a process that completely rejects certain observations, while retaining with full weight others with comparable deviations, possibly in the opposite direction, is unsatisfactory in principle.
Understanding the Difference Between Standard Deviation and Standard Error of the Mean, and Knowing When to Use Which