Statistical Background
In principle, if a measurement has a resolution of, then if the experimenter averages independent measurements the average will have a resolution of (this is the central limit theorem of statistics). This is an important experimental technique used to reduce the impact of randomness on an experiment's outcome. This requires that the measurements be statistically independent; there are several reasons why they may not be. If independence is not satisfied, then the average may not actually be a better statistic but may merely reflect the correlations among the individual measurements and their non-independent nature.
The most common cause of non-independence is systematic errors (errors affecting all measurements equally, causing the different measurements to be highly correlated, so the average is no better than any single measurement). Experimenter bias is another potential cause of non-independence.
Read more about this topic: Experimenter's Bias
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“Pilate with his question “What is truth?” is gladly trotted out these days as an advocate of Christ, so as to arouse the suspicion that everything known and knowable is an illusion and to erect the cross upon that gruesome background of the impossibility of knowledge.”
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