295 Research Article Received: 8 February 2010, Revised: 27 July 2010, Accepted: 3 August 2010, Published online in Wiley Online Library: 17 March 2011 (wileyonlinelibrary.com) DOI: 10.1002/cem.1350 Assessing the coefﬁcient of variations of chemical data using bootstrap method Saeid Amiri a* and Silvelyn Zwanzig a The coefﬁcient of variation is frequently used in the comparison and precision of results with different scales. This work examines the comparison of the coefﬁcient of variation without any assumptions about the underlying dis- tribution. A family of tests based on the bootstrap method is proposed, and its properties are illustrated using Monte Carlo simulations. The proposed method is applied to chemical experiments with iid and non-iid observations. Copyright © 2011 John Wiley & Sons, Ltd. Keywords: bootstrap method; coefﬁcient of variation; Monte Carlo simulation 1. INTRODUCTION The coefﬁcient of variation is well recognized as pertinent to the interpretation of variability of quantitative experiments. It is the ratio of the population standard deviation to the popu- lation mean,  = / where > 0, that can be estimated from the sample. The dimensionless property of the dispersion is ex- pressed sometimes as a percentage. Therefore, it is often used to compare the variability when different measurement scales are presented. This is a very useful characteristics in many situ- ations, especially to check the variability between populations. It is readily interpretable as opposed to other commonly used measures, such as standard deviation, that are more informative. For instance, consider  1 = 2 and  2 = 3; the direct comparison of them can be problematic, since the standard deviation does not include . This descriptive measures, ˆ  , of relative dispersion, which is regarded as stability and uncertainty, can be found vir- tually in most introductory statistical books. This measure is also called relative standard deviation (see Reference [1]). Its application can be found in different disciplines (see, for example, References [2] and [3]). It attracts special attention to assessing variability of quantitative experiments. The chemical experiments produce continuous-type values. The comparison of them using  has special appeal compared to  because experi- ments generally increase or decrease proportionally as the mean increases or decreases; hence, division by the mean removes it as a factor in the variability. Therefore,  is a standardization of  that allows comparison of variability estimate regardless of magnitude of analyst concentration, at least throughout most of the working range of the quantitative experiment. Hence, it can be considered as a suitable tool for the chemometrican (see Reference [4]). There are some researches that used  in the quantity assay as param- eter of variability (see for example References [5–7]). Although in the context of quantitative experiments, the application of  may be preferred to  as a measure of precision, the lack of appro- priate inference makes it less a favorite, which is discussed in the following. Unfortunately inference concerning the population  is often considered by making assumptions on the shape and the dis- tribution parameter [2], [3]. Because of these difﬁculties the re- ported inferences in literature are generally based on the para- metric model, particularly for the normal population. However, the exact distribution of the sample ˆ  is quite difﬁcult to obtain even for a normal distribution, even more difﬁcult for the skewed distribution [3]. Therefore, there are many researches in the liter- ature to overcome this problem. An approximation of the distribution of sample ˆ  has been considered in Mckay [8] that has  2 distribution if < 1/3. Note that most of the given tests are held under this condition. The literature of the coefﬁcient of variation can be found in Nairy and Rao [2], Pang et al. [3], and Mahmoudvand and Hassani [9] and the references therein. Here we propose a test to examine the equality of coefﬁcient of variation of two populations. This test is based on the resampling approach. The bootstrap method has provided a new branch in statistics in the form of nonparametric approach to inference. This work focuses on the nonparametric bootstrap, since if the chosen distribution of parametric bootstrap is correct then it works well, but it suffers when the underlying assumption is violated. There exist some research papers that tried to discuss the difference be- tween parametric and nonparametric bootstrap in other regards (see, for example, Reference [10]). Here, we look at the following objects: (1) Present a test for  that does not depend on the Mckay’s limitation, < 1/3, since many chemical data violate this, see Section 4. (2) The proposed test is a nonparametric test. Thus, it does not need the underlying assumptions of the parametric method, e.g. normality of observations. * Correspondence to: Saeid Amiri, Department of Mathematics, Uppsala Univer- sity, P.O. Box 480, 751 06 Uppsala, Sweden. E-mail: saeid@math.uu.se a Saeid Amiri, Silvelyn Zwanzig Department of Mathematics, Uppsala University, P.O.Box 480, 751 06 Uppsala, Sweden J. Chemometrics 2011; 25: 295–300 Copyright © 2011 John Wiley & Sons, Ltd.