The Standard Normal Distribution

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 $\mu - 2 \sigma$ and $\mu + 2 \sigma$
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 ($1010 \pm 10$)
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.