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In a binomial experiment, if we observe $x=0$ positive individual among $n$ individuals, then the proportion of positive individuals is significantly lower than $3/n$ with a type 1 error less than and very close to $5\%$. This fact, sometimes called the "rule of three", is a consequence of the inequalities $$\exp\left(-\frac{np}{1-p}\right) \leq \Pr(X=0) \leq \exp(-np).$$

Do you know other such basic easy rules for statistics? I find them very interesting and useful. This principle is not really a "rule of thumb" because it has a reliable theoretical foundation, but I don't see another tag for this question (I hope it is not off-topic)

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"Normally, more than two-thirds of the people are average" (meaning within one standard deviation of the mean)? – Dilip Sarwate Jun 8 '12 at 11:42
One very simple one that comes to mind is how the variance of a sample proportion of successes out of $n$ Bernoulli trials is no more than $1/4n$ (which is achieved when the success probability is $1/2$) – Macro Jun 8 '12 at 12:11
Many "rules of thumb" are based on theoretically rigorous analyses or approximations. – whuber♦ Jun 8 '12 at 13:02

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Check out Gerald van Belle's book "Statistical Rules of Thumb" a very nice little paperback text loaded with examples of rules of thumb and explanations including the "Rule of three" that you mention above.

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(+1) That seems like a nice book. Not only does he give rule-of-thumbs, but he also motivates them and discuss their validity. – MånsT Jun 8 '12 at 12:13
Nice! I will ask to my boss to buy it :) – Stéphane Laurent Jun 8 '12 at 12:13
Ahah ! I've just send an email to my boss and actually we already have this book :) – Stéphane Laurent Jun 8 '12 at 12:20
Yes, it's a great book. It's fun just to read it. – jbowman Jun 8 '12 at 13:28
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