Mode

Definition

The mode is generally defined as the most frequent observation or element in the distribution. Unlike the mean and the median, there can be more than one mode. A sample or a population with one mode is unimodal. One with two modes is bimodal, one with three modes is trimodal, and so forth. In general, if there is more than one mode, one can say that a sample or a distribution is multimodal.

History

The mode is an unusual statistic among those defined in the field of statistics and probability. It is a counting term, and the concept of counting elements in categories dates back prior to human civilization. Recognizing the maximum of a category, be it the maximum number of predators, the maximum number of food sources, and so forth, is evolutionarily advantageous.

The mathematician Karl Pearson is often cited as the first person to use the concept of the mode in a statistical context. Pearson, however, also used a number of other descriptors for the concept, including the “maximum of theory” and the “ordinate of maximum frequency.”

Calculation

In order to calculate the mode of a distribution, it is helpful first to group the data into like categories and to determine the frequency of each observation. For small samples, it is often easy to find the mode by looking at the results. For example, if one were to roll a die 12 times and get the following results,

it is fairly easy to see that the mode is 3.

However, if one were to roll the die 40 times and list the results, the mode is less obvious:

In Table 1, the data are grouped by frequency, making it obvious that the mode is 4.

Often, a statistician will be faced with a table of frequencies in which the individual data have been grouped into ranges of categories. Table 2 gives an example of such a table. We can see that the category of incomes between $40,000 and $60,000 has the largest number of members. We can call this the modal class. Yadolah Dodge has outlined a method for calculating a more precise estimate for the mode in such circumstances:

where L1 = lower value of the modal category; d1 = difference between the number in the modal class and the class below; d2 = difference between the number in the modal class and the class above; and c = length of the interval within the modal class. (This interval length should be common for all intervals.)

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