From pg8121020 of Chapter 8 “Sufficiency, Ancillarity, And All That” of Probability Theory: The Logic of Science by E.T. Jaynes:
The classical example showing the error of this kind of reasoning is the fable about the height of the Emperor of China. Supposing that each person in China surely knows the height of the Emperor to an accuracy of at least ±1 meter, if there are N=1,000,000,000 inhabitants, then it seems that we could determine his height to an accuracy at least as good as
(849)
merely by asking each person’s opinion and averaging the results.
The absurdity of the conclusion tells us rather forcefully that the rule is not always valid, even when the separate data values are causally independent; it requires them to be logically independent. In this case, we know that the vast majority of the inhabitants of China have never seen the Emperor; yet they have been discussing the Emperor among themselves and some kind of mental image of him has evolved as folklore. Then knowledge of the answer given by one does tell us something about the answer likely to be given by another, so they are not logically independent. Indeed, folklore has almost surely generated a systematic error, which survives the averaging; thus the above estimate would tell us something about the folklore, but almost nothing about the Emperor.
We could put it roughly as follows:
error in estimate = (850)
where S is the common systematic error in each datum, R is the RMS ‘random’ error in the individual data values. Uninformed opinions, even though they may agree well among themselves, are nearly worthless as evidence. Therefore sound scientific inference demands that, when this is a possibility, we use a form of probability theory (i.e. a probabilistic model) which is sophisticated enough to detect this situation and make allowances for it.
As a start on this, equation (850) gives us a crude but useful rule of thumb; it shows that, unless we know that the systematic error is less than about of the random error, we cannot be sure that the average of a million data values is any more accurate or reliable than the average of ten^{1}. As Henri Poincare put it: “The physicist is persuaded that one good measurement is worth many bad ones.” This has been well recognized by experimental physicists for generations; but warnings about it are conspicuously missing in the “soft” sciences whose practitioners are educated from those textbooks.
Or pg10191020 Chapter 10 “Physics of ‘Random Experiments’”:
…Nevertheless, the existence of such a strong connection is clearly only an ideal limiting case unlikely to be realized in any real application. For this reason, the law of large numbers and limit theorems of probability theory can be grossly misleading to a scientist or engineer who naively supposes them to be experimental facts, and tries to interpret them literally in his problems. Here are two simple examples:
 Suppose there is some random experiment in which you assign a probability p for some particular outcome A. It is important to estimate accurately the fraction f of times A will be true in the next million trials. If you try to use the laws of large numbers, it will tell you various things about f; for example, that it is quite likely to differ from p by less than a tenth of one percent, and enormously unlikely to differ from p by more than one percent. But now, imagine that in the first hundred trials, the observed frequency of A turned out to be entirely different from p. Would this lead you to suspect that something was wrong, and revise your probability assignment for the 101’st trial? If it would, then your state of knowledge is different from that required for the validity of the law of large numbers. You are not sure of the independence of different trials, and/or you are not sure of the correctness of the numerical value of p. Your prediction of f for a million trials is probably no more reliable than for a hundred.
 The common sense of a good experimental scientist tells him the same thing without any probability theory. Suppose someone is measuring the velocity of light. After making allowances for the known systematic errors, he could calculate a probability distribution for the various other errors, based on the noise level in his electronics, vibration amplitudes, etc. At this point, a naive application of the law of large numbers might lead him to think that he can add three significant figures to his measurement merely by repeating it a million times and averaging the results. But, of course, what he would actually do is to repeat some unknown systematic error a million times. It is idle to repeat a physical measurement an enormous number of times in the hope that “good statistics” will average out your errors, because we cannot know the full systematic error. This is the old “Emperor of China” fallacy…
Indeed, unless we know that all sources of systematic error  recognized or unrecognized  contribute less than about onethird the total error, we cannot be sure that the average of a million measurements is any more reliable than the average of ten. Our time is much better spent in designing a new experiment which will give a lower probable error per trial. As Poincare put it, “The physicist is persuaded that one good measurement is worth many bad ones.”^{2} In other words, the common sense of a scientist tells him that the probabilities he assigns to various errors do not have a strong connection with frequencies, and that methods of inference which presuppose such a connection could be disastrously misleading in his problems.
I excerpted & typed up these quotes for use in my DNB FAQ appendix on systematic problems; the applicability of Jaynes’s observations to things like publication bias is obvious. See also http://lesswrong.com/lw/g13/against_nhst/

If I am understanding this right, Jaynes’s point here is that the random error shrinks towards zero as N increases, but this error is added onto the “common systematic error” S, so the total error approaches S no matter how many observations you make and this can force the total error up as well as down (variability, in this case, actually being helpful for once). So for example, ; with N=100, it’s 0.43; with N=1,000,000 it’s 0.334; and with N=1,000,000 it equals 0.333365 etc, and never going below the original systematic error of . This leads to the unfortunate consequence that the likely error of N=10 is 0.017<x<0.64956 while for N=1,000,000 it is the similar range 0.017<x<0.33433  so it is possible that the estimate could be exactly as good (or bad) for the tiny sample as compared with the enormous sample, since neither can do better than 0.017!↩

Possibly this is what Lord Rutherford meant when he said, “If your experiment needs statistics you ought to have done a better experiment”.↩
Maybe it's just that it's late, but what he's saying in this quote isn't making sense to me.
To demonstrate the validity of the italicized comment, he should give an example where the data values are causally independent, but not logically independent. But the example he gives, of shared conversations and folklore, strike me as not causally independent at all, so while it supports the basic point about systematic error, it doesn't support this particular comment.
What's more, that's not even the problem. Let's say you ask the Chinese people to estimate the height of the king of Samoa. Here they've never talked of him, and know nothing about him other than that he's human. You still can't magic the information out of repeated estimation.
I was confused about that too. I think he must mean conditionally causally independent, but then he has to say conditional on what.
Maybe he means that each interview of a citizen is causally independent, since interviewing one of them won't causally affect the answer of another.
You could analyze the interview as adding a perturbation to people's "pre" responses, and per jsalvatier's comment, say that those perturbations are conditionally independent, as conditioned by the pre responses.
But it's the independence of the response that matters, and it's not independent of the stories and folklore.
Maybe my confusion can be clarified by showing a case where you have logical dependence but causal independence. I'm not seeing it. Jaynes uses inferential reasoning using backward in time urn draws as his go to example for causal independence but logical dependence. But that still seems a case where there is shared causal dependence on the number and kinds of balls in the urn originally.
It surprises and disappoints me that this is one of those well upvoted, but poorly commented threads.
Technical threads are like that.
I'm trying to follow the argument in footnote 1, but I got stuck on the above sentence  where did these values (especially the 0.017) come from?
IIRC, they're the 'plus' and the 'minus' range of the equation Jaynes gave using ±.
Aren't there lots of cases where the systematic error is small relative to the random error? Medicine seems like an example, simply because the random error is often quite large.
I actually would have said the opposite: medicine is one of the areas most likely to conduct wellpowered studies designed for aggregation into metaanalyses (having accomplished the shift away from pvalue fetishism into focusing on effect sizes and confidence intervals), and so have the least problem with random error and hence the most with systematic error.
Certainly you can't compare medicine with fields like psychology; in the former, you're going to get plenty of results from studies with hundreds or thousands of participants, and in the latter you're lucky if you ever get one with n>100.
Interesting. I did not know that medicine currently mostly does a good job of having well powered studies.
It would actually be pretty interesting, to go through many fields and determine how important systematic vs random error is.
Here's some guesses based on nothing more than my intuition:
What would you need to do this better? A sample of studies from prestigious journals for each field with N, size of random error, lower bound on effect size considered interesting.
Does engineering use these sorts of statistics? I read so few engineering papers I'm not really sure what statistics I would expect them to look like or how systematic vs random error would play out there.
That would be an interesting approach; typically power studies just look at estimating beta, not beta and bounds on effect size.
If you read any narrow/weak/specific/whatever AI papers, then I'd say you do read engineering papers  that's how I mostly think of my field, computational linguistics, anyway.
The "experiments" I'm doing at the moment are attempts to engineer a better statistical parser of English. We have some human annotated data, and we divide it up into a training section, a development section, and an evaluation section. I write my system and use the training portion for learning, and evaluate my ideas on the development section. When I'm ready to publish, I produce a final score on the evaluation section.
In this case, my experimental error is the extent to which the accuracy figures I produce do not correlate with the accuracy that someone really using my system will see.
Both systematic and random error abounds in these "experiments". I'd say a really common source of systematic error comes from the linguistic annotation we're trying to replicate. We evaluate on data annotated by the same people according to the same standards as we trained on, and the scientific standards of the linguistics behind that are poor. If some aspects of the annotation are suboptimal for applications of the system, that won't be reflected in my results.