Measurement and Analysis of Biodiversity observations are scattered over most of the classes Therefore, we wind up with small sample sizes for the rare observations where we need to characterize variation the most In the most extreme case, imagine a community in which every species is represented by just one individual Then, for any sample size below the true number of species we shall always observe fewer than the true number of species Of course, in this case we could observe that the number of species was a linear function of the number of individuals This might suggest to us a way to estimate the number of undetected species if we knew how many individuals were present in the study area This example illustrates the strategy followed by richness estimators: Model the regularity in the behavior of the number of species detected as a function of sample size Then, use this model to predict the number of species when the sample size becomes large The regularity could involve a clear relationship between the average sample size and the average number of species observed; this is the basis of extrapolation-based estimators On the other hand, the regularity could mean that we can replace a complicated sampling process with a more tractable model; estimators based on sampling theory follow this approach Article Overview An exhaustive review of all methods used for estimations of species richness is beyond the scope of this article Therefore, this article reviews the most widely used methods for the estimation of species richness Definition of Symbols defines the symbols used in this article Theoretical Properties of Richness Estimators lists some theoretical properties of richness estimators Practical Sampling Considerations discusses practical considerations of using community sampling to estimate species richness The next two sections present the estimators: richness estimators based on fine-scale, theoretical models of sampling are detailed in Estimators Based on Sampling Theory, whereas those based on coarse-scale, global modeling of species accumulation (extrapolation techniques) are discussed in Estimators Based on Extrapolation Which Estimation Method to Use? addresses the practical problem of evaluating and selecting estimators for use on new data sets Definition of Symbols We adopt the convention that random variables are written as uppercase symbols, whereas lowercase is reserved for deterministic variables or fixed constants The definitions we shall need are given in Table Table s n xi ri Sobs Sest N C t Xi Xi,j Yi Yi,j Ri Fi Pi qi I(o A A) E [X] VAR[X] COV[X, Y] CORR[X, Y] g(Z) 179 Symbols Used Number of species in a community Number of individuals in a community Number of individuals of species i in a community Number of species with i individuals in a community Number of species in a census Estimate of the number of species in a community using given method Number of individuals in a census The sample coverage of the census Number of samples in a census Number of individuals of species i in a census Number of individuals of species i in sample j Number of samples containing species i in a census The presence (1) or absence (0) of species i in sample j Number of species containing exactly i individuals in a census Number of species occurring in exactly i samples in a census Probability of detecting species i in a sample Probability that an individual is of species i if o A A and otherwise The expectation of the random variable X The variance of the random variable X The covariance of the random variables X and Y The correlation of the random variables X and Y Coefficient of variation of the random variable Z Sampling assumptions: Many estimators make assumptions about the sampling process (e.g., invariance of capture probabilities across all samples in a census) Parametric/nonparametric: The estimator Sest ¼ c(X) is nonparametric if the function c does not depend on a given distribution of X Bias: Bias measures average deviation from the true value We define bias E[c(X)]–s An estimator of s is unbiased in the strict statistical sense if E[c(X)]–s ¼ for every s Variance: An estimator is a random variable Therefore, we can use variance as a measure of the uncertainty in an estimate Sufficiency: Let w ¼ c(x) be an estimator of s The statistic W ¼ c(X) is sufficient for s if P{XAA9W, s) ¼ P{XAA9W) Suppose Sest ¼ c(X) is sufficient for s Then there are no other estimators (other than functions of c) that we could compute using X that could increase or decrease our confidence about our estimate of s Practical Sampling Considerations Theoretical Properties of Richness Estimators Salient features of richness estimators include the following: Type of data: Estimators differ in whether they use incidence or abundance data The incidence of species i is the P number of samples in which i occurs Formally, Yi ¼ tj¼1 Yi;j The abundance of species i is the number of individuals of i P contained in the census Formally, Xi ¼ tj¼1 Xi;j Limiting the Influence of Sampling Biases We lack the resources for complete censusing of communities Thus, sampling is the window through which we view the ecological world The species we observe when we sample are determined both by underlying ecological processes and by biases associated with sampling The simplest of these is bias due to sample size If we plot the number of species observed