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Re: Significant Digits for the Mean
Posted:
Jun 14, 1996 6:26 AM


In article <1996Jun11.134058@mint.hnrc.tufts.edu>, jerry@mint.hnrc.tufts.edu (Jerry Dallal) writes: >In article <4pjolu$40e7@b.stat.purdue.edu>, hrubin@b.stat.purdue.edu (Herman Rubin) writes: >> In the case of integer data, the number of "significant" digits in >> the mean depends heavily on the sample size; with an extremely large >> data set, the number grows, but not that fast. One way to see that >> this must be so is in reporting an estimate of a probability from >> Bernoulli trials. The dictum in 1. above would limit this to one >> decimal place, which is clearly absurd. > > Consider an urn containing 10 balls, an unknown proportion > of which are red. Suppose a ball is drawn from the urn > repeatedly, with replacement. > > Let Xi = 1, ith ball is red > 0, otherwise > (a Bernoulli trial!) > > Now, as the sample size increases, the sample mean can be > expressed with real meaning to more significant digits. Yet, > the larger the sample size, the more sense it makes to > estimate the population mean by reporting the sample mean to > only 1 significant digit! That is, as the sample size > increases, the sample mean expressed to only 1 significant > digit is superior to the "full precision" sample mean as an > estimate of the proportion of red balls in the urn by most > commonly used measures of accuracy.
Let the urn contain 2 balls, one red and one blue. Consider the stocastic process that draws with replacement the red ball with probability p and the blue with probability 1p. Let Xi be as defined. The mean is then p, cleary not necessarily an integer.
The described process is of course the binomial with 1 trial. With n trials the mean is np.
Consequently, there is no general rule about the number of decimal places for the mean.
Jarle



