the sample variance. Some researchers (particularly in the field of digital image processing) have applied this quantity more broadly as an index for detecting bimodality, with a small value indicating a more bimodal distribution. Statistical tests A number of tests are available to determine if a data set is distributed in a bimodal (or multimodal) fashion. Graphical methods In the study of sediments, particle size is frequently bimodal. Empirically, it has been found useful to plot the frequency against the log( size ) of the particles. This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the logarithm is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the Krumbein (or phi) scale. An alternative method is to plot the log of the particle size against the cumulative frequency. This graph will usually consist two reasonably straight lines with a connecting line corresponding to the antimode. Statistics Approximate values for several statistics can be derived from the graphic plots. where Mean is the mean, StdDev is the standard deviation, Skew is the skewness, Kurt is the kurtosis and φx is the value of the variate φ at the xth percentage of the distribution. Unimodal vs. bimodal distribution Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions. This method required the solution of a ninth order polynomial. In a subsequent paper Pearson reported that for any distribution skewness2 + 1 < kurtosis. Later Pearson showed that where b2 is the kurtosis and b1 is the square of the skewness. Equality holds only for the two point Bernoulli distribution or the sum of two different Dirac delta functions. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1. Baker proposed a transformation to convert a bimodal to a unimodal distribution. Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences. Larkin later introduced a test based on the F test; Benett created one based on Fisher's G test. Tokeshi has proposed a fourth test. A test based on a likelihood ratio has been proposed by Holzmann and Vollmer. A method based on