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QUESTION

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.