ENTROPY, OBSERVATIONS AND SOME STANDARD DEVIATIONS

The current manuscript discusses the relationship between the standard deviation σ of samples of size n random numbers and the standard deviation σMTE of the synthetic sample of these observations from origin samples, which have the least true errors. In addition, the relationship between the standard deviation σ of samples of size n random numbers and the standard deviation σMTEA of the combined synthetic sample of the observations with the least true errors and those averages near the true value of the quantity than observations is also analyzed. The research is based on simulations of random numbers from Normal and Uniform distributions with preliminary known parameters. Functional relationships between standard deviations σMTE and σMTEA, the sample standard deviation σ, the entropy of the standard normal distribution N(0, 1), and the sample size n, where n={1, 2, 3, …, 100}, were found. It is also shown that the inequality σMTEA < σMTE < 𝜎𝑋̅, where 𝜎𝑋̅ is the standard deviation of the mean 
of n >1 observations, is valid. Moreover, the ratios 𝜎𝑋̅ / σMTE and 𝜎𝑋̅ / σMTEA get bigger when n→100. For example, if n=3 the 𝜎𝑋̅ / σMTEA > 1.75, if n=10 𝜎𝑋̅ / σMTEA > 2.6, and if n=100 𝜎𝑋̅ / σMTEA > 6.4 in the case of Normal distribution. In the case of Uniform distribution, if n=3 the 𝜎𝑋̅ / σMTEA > 1.45, if n=10 𝜎𝑋̅ / σMTEA > 2.15, and if n=100 𝜎𝑋̅ / σMTEA > 4.6. In addition, increasing the sample size n leads to the convergence of σMTE to σMTEA. The above conclusions are valid for both Normal and Uniform distributed data, which implies that they will be valid for any arbitrary continuous distribution data with finite variance. Considering the simulation results, modern data analysis should be based on the observations with the least true errors rather than only the arithmetic mean.