Log Transformation and the Effect on Estimation, Implication, and Interpretation of Mean and Measurement Uncertainty in Microbial Enumeration.

Estimation of measurement uncertainty (MU) has been extensively addressed in documents from standard authorities. In microbiology, bacterial counts are log transformed to get a more normal distribution. Unfortunately, the difference between using original and log-transformed data appears to not have been investigated even in publications focusing on MU estimation. Statistical formulae inferencing and estimation of MU using real bacterial enumeration datasets. Both mean and SD calculated from original data carry the same scale and unit as the original data. However, the mean of log-transformed data becomes a geometric mean in log, and the SD becomes the logarithm of a ratio. Furthermore, calculation of RSD obtained by dividing the SD by the mean is meaningless and misleading for log-transformed data. The ratio, the antilog of the SD of log-transformed data, copes with multiplicative and divisive relationships to geometric mean (without log), instead of the arithmetic mean. The ratio can be converted to an analog ratio, which is similar or almost identical to the RSD of the untransformed data, especially when the within-subject variation is small. When MU is estimated from multiple samples with different measurands, the calculated RSD of original data is independent of the mean and can be pooled; however, for log-transformed data, the SD can be combined to estimate the common uncertainty. Calculation and use of RSD of log-transformed data are meaningless and misleading. Procedures outlining the estimation and interpretation of MU from log-transformed data require re-evaluation.