objections to example given; ambiguous and false 

"The probability of taking data more than two standard deviations from the mean is roughly .05. Six measurements were taken, so the probability that one should be so far from the mean is .05*6 = .3"

I have several objections to this statement: the first sentence above is true in the case of data from a normal distribution for which the true mean and true variance are known. In general it is not true. Furthermore, in the example given, the observed (estimated) values of these parameters are used. One can't just replace true values of the parameters with the estimated values and expect whatever probabilistic statement one's trying to make to remain true (and not necessarily even "roughly" true as stated, especially for a sample of size 6).

Furthermore, even IF the data are from a normal distribution and IF one does happen to know the parameters' true values (or good estimates thereof), the probabilistic statement made (the second sentence above) is ambiguous and false: the probability that "one should be so far from the mean" (in the exact sense) is in fact zero, since the normal distribution has no point masses. If the event whose probability is meant to be calculated is "exactly one should be so far or further from the mean", then that value is actually , not 0.3 as claimed. Or, the probability that "at least one should be so far or further from the mean is actually , not 0.3 as claimed. 

I would fix the article myself but first I'd like to hear what others think about all this.  30 June 2005 22:46 (UTC)