**** How about picking a true value then adding random errors to create hypothetical 'measurements'

What you are proposing is scientific fraud!

The whole POINT of random averages and the law of large numbers and central limits theorem is that YOU DO NOT KNOW the "true" mean and in most cases CAN NOT KNOW the "true" mean. The whole point of your random measurements and statistical calculations is to ESTIMATE the value of the "true" mean within a range of confidence!

There are perhaps 100 billion stars in our galaxy: what is the average mass of a star in our galaxy? There is no way for sure we can know! We measure a random selection of stars in our galaxy, estimate their average mas and declare, within a certain range of confidence, that an average mass of a star in our galaxy is 5.36 solar masses. Then somebody finds a bunch of stars that are way bigger! So we re-estimate, and so on. There is no way we can ever get the "true" average.

Here you go, we produce 10,000 cases of jelly beans per day. Each case has a gross of boxes of jelly beans. On average, how many jelly beans are in each box. We want to assure our customers that there are at least 10 jelly beans in a box and not more than 13 jelly beans in a box at least 90% of the time. Remember that we are producing 10,000 new cases of jelly beans every day! We won't ever know what the "true" mean is! We must just estimate it within a range of confidence.

Here's another one: In a certain auto rating territory, within a state, we have 3375 insureds covered for collision insurance. Last year there were 1182 accidents invoking collision coverage for a total paid loss of $2,805,645.94. What will the average loss per policy be NEXT year? You can only ESTIMATE the answer within a range of confidence! (Trust me, people using fancy statistics lose on this one every year!)