"mean" usually refers to the argument of the complex mean. The nth raw moment of z is: where the integral is over any interval of length 2π. In calculating the above integral, we use the fact that z = cos(nx) + i sin(nx) and the Bessel function identity: The mean of the complex exponential z is then just and the circular mean value of the angle x is then taken to be the argument μ. This is the expected or preferred direction of the angular random variables. The variance of z, or the circular variance of x is: Limiting behavior When is large, the distribution resembles a normal distribution. More specifically, for large positive real numbers , where σ2 = 1/ and the difference between the left hand side and the right hand side of the approximation converges uniformly to zero as goes to infinity. Also, when is small, the probability density function resembles a uniform distribution: where the interval for the uniform distribution is the chosen interval of length (i.e. when is in the interval and when is not in the interval). Estimation of parameters A series of N measurements drawn from a von Mises distribution may be used to estimate certain parameters of the distribution. The average of the series is defined as and its expectation value will be just the first moment: In other words, is an unbiased estimator of the first moment. If we assume that the mean lies in the interval , then Arg will be a (biased) estimator of the mean . Viewing the as a set of vectors in the complex plane, the statistic is the square of the length of the averaged vector: and its expectation value is In other words, the statistic will be an unbiased estimator of and solving the equation for will yield a (biased) estimator of . In analogy to the linear case, the solution to the equation will yield the maximum likelihood estimate of and both will be equal in the limit of large N. For approximate solution to refer to von Mises–Fisher distribution. Distribution of the mean The distribution of the sample mean for the von Mises distribution is given by: where N is the number of measurements and consists of intervals of in the variables, subject to the constraint that and are constant, where is the mean resultant: and is the mean angle: Note that