Efficiencies of Percentile Measures for Describing the Mean Size and Sorting of Sedimentary Particles

Abstract

Present methods for describing the size distribution of sediments include the use of graphic measures, functions of the percentiles obtained from the cumulative size frequency distribution curve. In estimating the mean size and sorting of sediments, most graphic measures used are linear combinations of the percentiles. Considering a specified percentile as a random variable, the distribution of the values observed for this percentile tends to a normal distribution as the sample size approaches infinity. A similar relation exists for any graphic measure which is a linear combination of percentiles. On the assumption that the logarithm of the grain diameter is normally distributed, the statistical efficiency of a graphic measure as described above can be calculated. The statistical efficiency of a graphic measure is the ratio of the variance of the distribution of the corresponding efficient estimate and the variance of the limiting distribution of the graphic measure. Regarding the efficiency of a graphic measure as an index of precision, graphic measures with greater efficiencies are preferable to those with lower efficiencies. Efficiencies of graphic measures currently in use to estimate mean size are: $$1/2(\phi_{16} + \phi_{84})$$, 74 per cent; $$1/2(\phi_{25} + \phi_{75})$$, 81 per cent; $$1/3(\phi_{16} + \phi_{50} + \phi_{84})$$, 88 per cent; to estimate sorting are: $$1/1.35(\phi_{75} - \phi_{25})$$, 37 per cent;$$ l/2(\phi_{84} - \phi_{16})$$, 54 per cent; $$ l/3.3(\phi_{95} - \phi_{5})$$, 64 per cent;$$ l/4(\phi_{84} - \phi_{16}) + l/6.6(\phi_{96} - \phi_{5})$$, 79 per cent. Suggested graphic measures to estimate mean size are: $$ 1/3(\phi_{20} + \phi_{50} + \phi_{80})$$, 88 per cent; $$1/5(\phi_{10} + \phi_{30} + \phi_{50} + \phi_{70} + \phi_{90})$$, 93 per Cent; $$1/10(\phi_{5} + \phi_{15} + \phi_{25} + \phi_{35} + \phi_{45} + \phi_{55} + \phi_{65} + \phi_{75} + \phi_{85} + \phi_{95})$$, 97 per cent; to estimate sorting are: $$1/5.4(\phi_{85} + \phi_{95} - \phi_{5} - \phi_{15})$$, 79 per cent; $$1/9.1(\phi_{70} + \phi_{80} + \phi_{90} + \phi_{97} -\phi_{3} - \phi_{10} - \phi_{20} - \phi_{30}))$$, 87 per cent.