Good day! Im currently dealing with the formula in determining the expected number of real roots of random algebraic polynomial,,, and I want to solve for the formula of the standard deviation. I have collected a number of real roots from 100,000 different random polynomials with degree of 25. I have the following results:
$$Number of real roots==>Number of samples$$ $$1 ==> 2479$$$$3==>24 808$$$$5==>46350$$$$7==>23110$$$$9==>3168$$$$11==>83$$$$13==>2$$
Hence, the expected number of real roots of random algebraic polynomial is asymptotically close to $$\sqrt{n} = \sqrt{25} = 5$$ wherein the actual result based on the given above is $$4.99874$$ Surprisingly, as the number of samples increases, the distribution become more normal.
But the results stated above are arbitrary, but again with sufficiently large number of samples, the result will be close to $$\sqrt{n}.$$
At this point, I want to solve for the formula of standard deviation but since the points are arbitrary, it gives me a hard time to solve for the standard deviation... because points changes everytym a new trial will be perform. Is there a feasible way how to deal with standard deviation with unknown points?