Estimating confidence intervals for gravel bed surface grain size distributions

Abstract. 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 of the percentile estimates. We present a simple method for estimating the confidence intervals about the grain size percentiles derived from standard Wolman or pebble count samples of bed surface texture. Our approach uses binomial probability theory to generate confidence intervals for all grain sizes in the distribution. We find that the standard sample size of 100 observations is associated with errors ranging from about ±15 % to ±30 %, which may be unacceptably large for many applications. In comparison, a sample of 500 stones produces an uncertainty 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 median size and the 84th percentile of the distribution as a function of the sample size and the standard deviation of the distribution, assuming that the underlying distribution is log-normal. However, the true uncertainty of any sample can only be accurately estimated once the sample has been collected, so these simple equations complement – but do not replace – the basic uncertainty analysis using binomial probability theory.