tendency and statistical dispersion may be defined for both the population and a sample drawn from that population. Central tendency The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean. When data is concentrated, the median and mode may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful. Dispersion The most common measures of circular spread are: The . For the sample the circular variance is defined as: and for the population Both will have values between 0 and 1. The with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution. Note that for small , we have . The with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance. Distribution of the mean Given a set of N measurements the mean value of z is defined as: which may be expressed as where or, alternatively as: where The distribution of the mean angle () for a circular pdf P(θ) will be given by: where is over any interval of length and the integral is subject to the constraint that and are constant, or, alternatively, that and are constant. The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed. The central limit theorem may be applied to the distribution of the sample means. (main article: Central limit theorem for directional statistics). It can be shown that the distribution of approaches a bivariate normal distribution in the limit of large sample size. Goodness of fit and significance testing For cyclic data – (e.g., is it uniformly distributed) : Rayleigh test for a unimodal cluster Kuiper's