A Proof of the Variance Formula for the Total Crossing Time of a Continuous Random Sound Signal with Respect to a Fixed Level

For a quality controlled application of the percentile sound evaluation index LX , its accuracy, limited by the stochastic level fluctuations, should be quantified. This is achievable by determining first the variance of the partition of the signal amplitude with respect to a fixed level, the expectation value of LX . Then the percentile’s spread itself is accessible through the cumulative level distribution. As is still to be done, this paper presents a comprehensible proof of the partition variance formula, which is of fundamental importance for the evaluation of the LX confidence limits. Given a number of crossings within a given measurement time interval the probability that a definite total of the single continuous overshoots, i. e. “crossing up” time intervals occurs, depends as on the stochastic system’s crossings up as on its crossings down, and further on the probability density function (p. d. f.) of the crossing number. Above an easily practicable minimum crossing number the p. d. f. of the total crossing time, and so of the partition itself, can be presented explicitly and straightforward in microstatistic terms applying the Central Limit Theorem of Statistics. Then step by step integration of the variance definition equation leads to the already known and applied final result.