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 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.

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