POPULATION ECOLOGY Quantifying change in distributions: a new departure index that detects, measures and describes change in distributions from population structures, size-classes and other ordered data K. M. Menning Æ J. J. Battles Æ T. L. Benning Received: 3 March 2006 / Accepted: 26 June 2007 / Published online: 11 August 2007 Ó Springer-Verlag 2007 Abstract Many statistics are available to compare dis- tributions. Some are limited to nominal data while others, such as skew, Kullback–Leibler, Kolmogorov–Smirnov and the Gini coefficient, are useful for providing infor- mation about ordered distributions. While many of these tests are useful for determining properties of data in his- tograms, there has not been a test until now that allows for the detection of differences between distributions, describes the difference and is sensitive to the location of the departures. Such a test could be critical for comparing pre-and post-event distributions, such as a change in the distribution of biomass due to fire, for example, or for comparing data from different locations, such as soil size distributions, and even for evaluating economic disparity or examining differences in age demographics. We present a new statistic, a departure index, which allows a test dis- tribution to be compared with any reference distribution. The resulting index contains information about the loca- tion, magnitude and direction of departure from the reference distribution to the test distribution. The departure index in turn provides a standardized response range that allows for a comparison of results from different analyses. A case study of actual fire data demonstrates the sensitivity and range of the test. Keywords Change detection  Quantitative analysis  Size class distributions  Structure  Statistics Introduction In science in general and ecology in particular, frequency distributions are commonly used to summarize the under- lying structure of populations or resources (Begon et al. 1996). Data – whether they be size, age, performance rat- ings or other measures – are organized into defined categories or bins, and the numbers of organisms (or objects) per bin are examined in a histogram. With nominal data, the order of categories does not matter: for example, the order of species does not matter when calculating diversity indices such as the Shannon-Wiener Diversity Index (Zar 1999). In contrast, order does matter with ordi- nal, interval or ratio data (i.e. ordered data). With tree size classes, for example, order is inherent in the categories being considered. Often, the challenge in an ecological analysis is to quantify the changes in ordered distributions that result from a natural or experimental process. The ideal comparison would not only detect a difference between distributions but also describe the nature of the difference. A number of statistical measures test differences between distributions (Wiegand et al. 2000; Yang et al. 2004). Some analyses are limited to nominal data, such as the Shannon-Wiener. Some, such as the chi-square good- ness-of-fit test (Payette et al. 2000; Gaymer et al. 2001), the Communicated by Marc Mangel. Electronic supplementary material The online version of this article (doi:10.1007/s00442-007-0810-3) contains supplementary material, which is available to authorized users. K. M. Menning (&)  J. J. Battles Division of Ecosystem Sciences, Department of Environmental Science, Policy and Management, University of California, Berkeley, 137 Mulford Hall #3114, Berkeley, CA 94720, USA e-mail: kmenning@nature.berkeley.edu T. L. Benning Department of Environmental Science, Harney Science Center, University of San Francisco, 2130 Fulton St, San Francisco, CA 94117, USA 123 Oecologia (2007) 154:75–84 DOI 10.1007/s00442-007-0810-3