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

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(φ₁₆ + φ₈₄), 74 per cent; 1/2(φ₂₅ + φ₇₅), 81 per cent; 1/3(φ₁₆ + φ₅₀ + φ₈₄), 88 per cent; to estimate sorting are: 1/1.35(φ₇₅ - φ₂₅), 37 per cent;l/2(φ₈₄ - φ₁₆), 54 per cent; l/3.3(φ₉₅ - φ₅), 64 per cent;l/4(φ₈₄ - φ₁₆) + l/6.6(φ₉₆ - φ₅), 79 per cent. Suggested graphic measures to estimate mean size are: 1/3(φ₂₀ + φ₅₀ + φ₈₀), 88 per cent; 1/5(φ₁₀ + φ₃₀ + φ₅₀ + φ₇₀ + φ₉₀), 93 per Cent; 1/10(φ₅ + φ₁₅ + φ₂₅ + φ₃₅ + φ₄₅ + φ₅₅ + φ₆₅ + φ₇₅ + φ₈₅ + φ₉₅), 97 per cent; to estimate sorting are: 1/5.4(φ₈₅ + φ₉₅ - φ₅ - φ₁₅), 79 per cent; 1/9.1(φ₇₀ + φ₈₀ + φ₉₀ + φ₉₇ -φ₃ - φ₁₀ - φ₂₀ - φ₃₀)), 87 per cent.