The Standard Normal Distribution
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Key Questions

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 kilobags 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 sugarbags will weigh between:
#musigma# 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
#mu2sigma# 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. 
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