Percentile-based grain size distribution analysis tools (GSDtools) – estimating confidence limits and hypothesis tests for comparing two samples

• Published in: Earth surface dynamics Vol. 7; no. 3; pp. 789 - 806
• Main Authors: Eaton, Brett C, Moore, R. Dan, MacKenzie, Lucy G
• Format: Journal Article
• Language: English
• Published: Gottingen Copernicus GmbH 02.09.2019; Copernicus Publications
• Subjects: Approximation; Binomial distribution; Confidence intervals; Confidence limits; Distribution; Equality; Estimates; Estimation; Frequency distribution; Grain size; Grain size distribution; Gravel; Gravel beds; Mathematical analysis; Measurement; Particle size; Pebbles; Properties; River sediments; Rivers; Sampling; Sampling methods; Shape; Size distribution; Spreadsheet software; Spreadsheets; Statistical analysis; Statistics; Surface layers; Uncertainty; Workers; Germany
• ISSN: 2196-632X; 2196-6311; 2196-632X
• DOI: 10.5194/esurf-7-789-2019

Most studies of gravel bed rivers present at least one bed surface grain size distribution, but there is almost never any information provided about the uncertainty in the percentile estimates. We present a simple method for estimating the grain size confidence intervals about sample percentiles derived from standard Wolman or pebble count samples of bed surface texture. The width of a grain size confidence interval depends on the confidence level selected by the user (e.g., 95 %), the number of stones sampled to generate the cumulative frequency distribution, and the shape of the frequency distribution itself. For a 95 % confidence level, the computed confidence interval would include the true grain size parameter in 95 out of 100 trials, on average. The method presented here uses binomial theory to calculate a percentile confidence interval for each percentile of interest, then maps that confidence interval onto the cumulative frequency distribution of the sample in order to calculate the more useful grain size confidence interval. The validity of this approach is confirmed by comparing the predictions using binomial theory with estimates of the grain size confidence interval based on repeated sampling from a known population. We also developed a two-sample test of the equality of a given grain size percentile (e.g., D50), which can be used to compare different sites, sampling methods, or operators. The test can be applied with either individual or binned grain size data. These analyses are implemented in the freely available GSDtools package, written in the R language. A solution using the normal approximation to the binomial distribution is implemented in a spreadsheet that accompanies this paper. Applying our approach to various samples of grain size distributions in the field, we find that the standard sample size of 100 observations is typically associated with uncertainty estimates ranging from about ±15 % to ±30 %, which may be unacceptably large for many applications. In comparison, a sample of 500 stones produces uncertainty estimates ranging from about ±9 % to ±18 %. In order to help workers develop appropriate sampling approaches that produce the desired level of precision, we present simple equations that approximate the proportional uncertainty associated with the 50th and 84th percentiles of the distribution as a function of sample size and sorting coefficient; the true uncertainty in any sample depends on the shape of the sample distribution and can only be accurately estimated once the sample has been collected.