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

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.