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Measures of Central Tendency

Main Concept

In statistics, there are three main measures of central tendency, which describe the way quantitative data tends to cluster around a particular, the "central value." These measures are:



(Arithmetic) Mean: The mean is the average value of the data set. It is calculated by taking the sum of all of the numbers in the set and dividing by the size of the set. Given a data set of a1, a2, ... , an, the formula for the arithmetic mean is A = 1n i=1nai .

        Example 1: The mean of the sample set {4, 2, 3, 1, 5, 5, 7} is A = 4+2+3+1+5+5+77=2773.86.



Median: The median is the middle value of the data set, or the numerical value which separates the higher half from the lower half of the data. Given a finite set of data, the median can be found by arranging the values from lowest to highest, then choosing the middle value. If a set contains an even number of values, the median is then defined as the mean of the two middle values.

        Example 1: The median of the sample set {4, 2, 3, 1, 5, 5, 7} is 4 because when arranged in increasing order, we see that 4 is the middle value {1, 2, 3, 4, 5, 5, 7}.

        Example 2: The median of the sample set {9, 10, 13, 6} is 9.5 because when arranged in increasing order, we see that 9 and 10 are the two middle values, {6, 9, 10, 13}, and so their mean is 9+102=9.5 .



Mode: The mode is the value which occurs the most frequently in the data set. Unlike the mean and median, the mode of a data set may not be unique.  If no numbers occur more than once, there is said to be "no mode".  However, if multiple values occur with the same frequency, then the set is considered to be "multi-modal".

        Example 1: The mode of the sample set {4, 2, 3, 1, 5, 5, 7} is 5 because only the number 5 occurs twice in the data set.

        Example 2: The sample set {1, 2, 3, 4} has no mode because all the values occur only once.

        Example 3: The modes of the sample set {7, 8, 7, 7, 10, 2, 2, 15, 1, 2} are 2 and 7 because both of these values occur three times in the data set.


In the past, it was common to refer to all measures of central tendency by the ambiguous term: "averages", but this label has fallen into disuse. Now, the term "average" is often used to describe the mean of a data set.




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