Question

What does the Empirical Rule (also called the 68-95-99.7 Rule) tell you about the approximate probabilities for normally distributed values?

Answer 1

It has everything to do with standard deviation `sigma`, in other words, how much your values are spread around the mean.

Say you have a machine that fills kilo-bags of sugar. The machine does not put exactly 1000 g in every bag. The standard deviation may be in the order of 10 g.
Then you can say: mean = `mu=1000` and `sigma=10` (gram)

The empirical rule (easily verified by your GC) now says:

50% will be underweight and 50% will be overweight, by varying amounts, of course.

68% or all you sugar-bags will weigh between:
`mu-sigma` and `mu+sigma` or between 990 and 1100 gram
(so 16% will weigh more and 16% will weigh less, as the normal distribution is completely symmetrical).

95% will be between `mu-2sigma` and `mu+2sigma`
So 2,5% will be under 980 gram and 2.5% over 1020 gram.

In practice
In the case presented, you may not want to be that much under weight (a small overweight is not a problem). So most manufacturers set their machines to slightly overweight. Let's calculate this:
`mu=1010, sigma=10`
Now the 68% is all more than 1000 gram (`1010+-10`)
And only 2.5% is more than 10 gram underweight.

Challenge
Now find out what happens - and what you would have to do - if the standard deviation of your filling machine were greater or smaller.