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Random numbers calculus.
We, all of us, are custom to arithmetic of exact numbers. Two by two is four. That's it. End of story. But what we are counting? Perfect ideas of something that could be defined as a piece. Two apples plus three bananas equal 5 fruits. Correct? Why not.
Problem. Thousand nuts weight 1.34 pounds. What weight of 500 nuts? The answer is most possible 0.67. And definitely between 0 and 1.34. Zero? How it could be? For example if we have very different nuts and more then half of them are so small that sum of their weight is less then sensitivity of our scale. Impossible? Ridiculous? Maybe. But logical.
Every experimental measurements number made with some accuracy. If we using such experimental values for calculating by some other formulas a derivative quantities we have to expect that accuracy of them must be defined too. But rules for it much more complex then primitive arithmetic.
The problem. We measured weight of big number of nuts and find that average of one is 2.04±0.24 where 0.24 is standard deviation (variance) of measurement. What is the weight of two nuts from the same pocket?
To find a solution we could use LeoStatistic in such way. Inserting three parameters as values 10000 records by formulas:
Then in tab "View" will set distribution style and get the picture:
Result that we have is:
2.041 (±0.2381) + 2.043(±0.2404)=4.085(±0.338)
What is unexpectedly interesting is that reactive variance of sum smaller then variance sum of variances... Analytical solution most definitely could be found of course but LeoStatistic gives it in minutes and visible correct.
By the way what is most amazing is the result of division if variance for both variables are quite noticeable. Repeat the same operation as previous time but wit the formulas:
Here is a distributions:
The final formula:
2 (±0.6951) / 2.01(±0.7006)=1.18(±3.14)
is really counterintuitive. Not only variation dividend is much large then variations of quotient and divisor, not only a shape of probability of distribution of dividend is is not normal and obviously asymmetric; what is the shocking - the mean of dividend in random numbers arithmetic is not equal to obtained from classical operations.
Random numbers arithmetic is far from just calculation of variations of result. The result itself are affected also.
So summarizing how to use LeoStatistic to find results of operations with random numbers one has to follow these steps:
Much more reliable and faster then any other procedure. Enjoy.
Screenshots of the LeoStatistic software:
click on picture to enlarge
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