of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their standard deviations to produce bimodality when the two distribution curves are combined. Bimodal distributions have the peculiar property that – unlike the unimodal distributions – the mean may be a more robust sample estimator than the median. This is clearly the case when the distribution is U shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails. Moments of mixtures Let where gi is a probability distribution and p is the mixing parameter. The moments of f(x) are where and Si and Ki are the skewness and kurtosis of the ith distribution. Mixture of two normal distributions It is not uncommon to encounter situations where an investigator believes that the data comes from a mixture of two normal distributions. Because of this, this mixture has been studied in some detail. A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two normal distributions with equal standard deviations is bimodal only if their means differ by at least twice the common standard deviation. Estimates of the parameters is simplified if the variances can be assumed to be equal (the homoscedastic case). If the means of the two normal distributions are equal, then the combined distribution is unimodal. Conditions for unimodality of the combined distribution were derived by Eisenberger. Necessary and sufficient conditions for a mixture of normal distributions to be bimodal have been identified by Ray and Lindsay. A mixture of two approximately equal mass normal distributions has a negative kurtosis since the two modes on either side of the center of mass effectively reduces the tails of the distribution. A mixture of two normal distributions with highly unequal mass has a positive kurtosis since the smaller distribution lengthens the tail of the more dominant normal distribution. Mixtures of other distributions require additional parameters to be estimated. Tests for unimodality When the components of the mixture have equal variances the mixture is unimodal if and only if or where p is the mixing parameter and and where μ1 and μ2