natural distributions are geometric for exampleconsider the populations of the largest cities in the united states if you plot these populations on number linemost of them will be clustered toward the low sidewith few outliers on the high side this is an example of zipf distribution (see section viewed the other waythe home town for given person is far more likely to be particular large city than particular small town collected data are likely to be skewed in some way field samples might be rounded tosaythe nearest ( all numbers end in or if the input is collection of common english wordsthe beginning letter will be poorly distributed note that in each of these exampleseither highor low-order bits of the key are poorly distributed when designing hash functionswe are generally faced with one of two situations we know nothing about the distribution of the incoming keys in this casewe wish to select hash function that evenly distributes the key range across the hash tablewhile avoiding obvious opportunities for clustering such as hash functions that are sensitive to the highor low-order bits of the key value we know something about the distribution of the incoming keys in this casewe should use distribution-dependent hash function that avoids assigning clusters of related key values to the same hash table slot for exampleif hashing english wordswe should not hash on the value of the first character because this is likely to be unevenly distributed below are several examples of hash functions that illustrate these points example consider the following hash function used to hash integers to table of sixteen slotsint (int xreturn( )the value returned by this hash function depends solely on the least significant four bits of the key because these bits are likely to be poorly distributed (as an examplea high percentage of the keys might be even numberswhich means that the low order bit is zero)the result will also be poorly distributed this example shows that the size of the table can have big effect on the performance of hash system because this value is typically used as the modulus