equals , i.e., true diversity as calculated with . The original Simpson's index hence equals the corresponding basic sum. The interpretation of λ as the probability that two entities taken at random from the dataset of interest represent the same type assumes that the first entity is replaced to the dataset before taking the second entity. If the dataset is very large, sampling without replacement gives approximately the same result, but in small datasets, the difference can be substantial. If the dataset is small, and sampling without replacement is assumed, the probability of obtaining the same type with both random draws is: where is the number of entities belonging to the th type and is the total number of entities in the dataset. This form of the Simpson index is also known as the Hunter–Gaston index in microbiology. Since the mean proportional abundance of the types increases with decreasing number of types and increasing abundance of the most abundant type, λ obtains small values in datasets of high diversity and large values in datasets of low diversity. This is counterintuitive behavior for a diversity index, so often, such transformations of λ that increase with increasing diversity have been used instead. The most popular of such indices have been the inverse Simpson index (1/λ) and the Gini–Simpson index (1 − λ). Both of these have also been called the Simpson index in the ecological literature, so care is needed to avoid accidentally comparing the different indices as if they were the same. Inverse Simpson index The inverse Simpson index equals: This simply equals true diversity of order 2, i.e. the effective number of types that is obtained when the weighted arithmetic mean is used to quantify average proportional abundance of types in the dataset of interest. The index is also used as a measure of the effective number of parties. Gini–Simpson index The Gini-Simpson Index is also called Gini impurity, or Gini's diversity index' in the field of Machine Learning. The original Simpson index λ equals the probability that two entities taken at random from the dataset of interest (with replacement) represent the same type. Its transformation 1 − λ, therefore, equals the probability that the two entities represent different types. This measure is also known in ecology as the probability of interspecific encounter (PIE'') and the Gini–Simpson index. It can be expressed as a transformation of the true diversity of