Chasing rainbows： The elusive normal and lognormal distribution in geochemistry

For decades geochemists have debated the nature of the distribution of geochemical dements. The general observation is that major elements tend to approach normal distribution, whereas trace elements tend towards lognormal distribution. In order to establish geochemical baselines and anomalies, the distr/bution of geochemical （typically univariate） data is often compared to normal or lognormal distribution. Geochemical data--particularly when the data set is large-usually fails to fit either distribution. Departures from these distributions are commonly attributed to outliers or mixtures of different geochemical populations. This approach fails for two reasons. First, formal tests for normality are a function of power and effect size. For large data sets, even small departures from normality can result in statistical significance, but not necessarily in practical significance. Second, and more importantly, distr/bution tests on ＂raw＂ or log-transformed data fail to recognize the nature of compositional data. Such data are by definition constrained to positive values whose components （variables） are closed to sum to 100%. Components are not independent, i.e., any variation in any component must be compensated by changes in other components. Log-ratio transformations, developed by John Aitchison to deal with these constraints, have existed for over 25 years, but geochemists continue largely to ignore or dismiss the problem of closure. Log-ratio techniques rid compositional data of their component dependence and the constraint of closure by allowing data to assume any value in Euclidean space. We examined the issue of geochemical data distribution with a large set of stream sediments. The logratio transformation not only reconciles the apparent differences between major and trace element distributions, but also more readily enables the recognition of potentially contaminating populations. Moreover, observations below detection limits, which usually cause pronounced skew in ＂raw＂ or log-transformed data, are readily accommodated with a substitution method, typically decreasing skewness.