Recently opened experimental regularity typical for hydrogen-like atoms and ions (having only one outer shell electron) was presented in several papers, for example [1] and [2]. Chief matter of the opening consists in the fact that effective main quantum numbers of outer electron, i.e. in consideration of quantum defect (Rydberg correction), along with integers can be expressed in common fractions, having substantial constraint on denominator number. Sensational results of the study caused quite expected alert reaction and even criticism of many readers, because it affects basic foundation of modern physics – atom theory. Main objection of opponents can be reduced to the assertion that cited facts can be explained with series of random coincidences. So it was exercised supplementary mathematical treatment of data array of experimental characteristics not only of atoms and ions in ground state, but also of hydrogen-like atoms in excited states. This statistical treatment completely confirms the nonrandom character of discovered experimental law, and is subject of this paper.