The rule:
In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within 3 standard deviations of the mean in a normal distribution.
About 68.27% of the values lie within 1 standard deviation of the mean. Similarly, about 95.45% of the values lie within 2 standard deviations of the mean. Nearly all (99.73%) of the values lie within 3 standard deviations of the mean.
So suppose that I have a set of values (measurements) which has the normal distribution property.
Let's call it S.
When they say "about 68.27% of the values" what values do they mean? Do they mean that the standard deviation of any 68.27 % of the elements of S is smaller than 1? Do they mean something more? Could someone give me a precise mathematical statement that is equivalent to this "68–95–99.7 rule".
I've posted this on math.stackexchange because I would like a mathematical answer.
Best Answer
The mathematical statement of the "within one standard deviation" rule is that
$$\Pr(\mu-\sigma < X < \mu + \sigma) =\frac{1}{\sqrt{2 \pi} \sigma} \int_{\mu - \sigma}^{\mu + \sigma} \exp \left( - \frac{(x-\mu)^2}{2 \sigma^2} \right) \; dx = \frac{1}{\sqrt{2 \pi}} \int_{-1}^1 \exp \left( - \frac{u^2}{2} \right) \; du \approx 0.682689$$
(In the integral, just make the substitution $u = (x-\mu)/\sigma)$.)
Is that what you had in mind? The other statements are similar, just replacing $\sigma$ with $2 \sigma$ or $3 \sigma$.