It is always important to have some idea of the size and influence of random errors. The assertion that errors are independent of one another is your most immediate hazard. If they are not, as is often the case, you may be able to readjust the numerical value of n to be an estimate of the number of independent errors. We did something like this in time series analysis when we took n to be not the number of points on the time series but the number of intervals of length .The second big hazard in trying to state a confidence interval is the common assumption that, because of the central-limit theorem and for lack of better information, the errors follow a gaussian probability function. If in fact the data errors include blunders which arise from human errors or blunders from transient electronic equipment difficulties, then the gaussian assumption can be very wrong and can lead you into serious errors in geophysical interpretation.
Some useful help is found in the field of nonparametric statistics.
To begin with, it is helpful to rephrase the original question into one involving the median rather than the mean. The median m 1 is defined as that value which is expected to be less than half of the population and greater than the other half. The median is insensitive to a data point, which, by some blunder, is near infinity. In fact, median and mean are equal when the probability function is symmetrical. Now define a third random variable s as Since each y i is zero or one, then s must be an integer between zero and n .