Statistical measures of the central tendency for H+ activity and pH

Abstract

Despite the numerous papers on the statistical analyses of pH, there is no explicit opinion on the use of arithmetic mean as a measure of the central tendency for pH and H+ activity. The problem arises because the transformation of the arithmetic mean for one does not give the arithmetic mean for the other. The paper presents 1) the theoretical considerations on the distribution of pH and H+ activity and relation between them, properties of these distributions, the choice of distributions which should be consistent with the distribution of pH and the distribution of H+ activity and measures of central tendency for features of such distributions and 2) examples of calculations of measures of central tendency for pH and H+ activity based on the literature data on soil and lake water pH. These data analyses included distributions of pH and H+ activities, properties of distribution, descriptive statistics for pH and for the H+ activity and comparison of arithmetic mean with the geometric mean. From the results, it could be concluded that a uniform approach to the choice of measure for the central tendency of pH and H+ activity requires the determination of the type of measure (mean) for one of them and then consistent transformation of this measure. The choice of measure of the central tendency for the variable should be preceded by determination of its distribution. Normal probability distribution of pH and thus lognormal distribution of H+ activity indicate that the arithmetic mean, and its corresponding geometric mean should be used as proper measures of the central tendency for pH and for H+ activity. Besides, the position statistic that is a median can be used for each of those variables, irrespective of their probability distributions.