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# Average Definition

An average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Oftentimes average refers to the arithmetic mean which is simply the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and any of these methods might be considered as an average value for a set of values. Depending on the dataset and what you are trying to analyze about it, different measures of central tendency will be utilized.

## Measures of Central Tendency

Comparison of common averages of values:

 Type Description Example Result Arithmetic Mean The sum of values of a data set divided by number of values: (a1 + a2 + . . . + an) ÷ n (1 + 2 + 2 + 3 + 4 + 7 + 9) ÷ 7 4 Median The middle value separating the greater and lesser halves of a data set. 1, 2, 2, 3, 4, 7, 9 3 Mode The most frequent value in a data set. 1, 2, 2, 2, 3, 3, 4, 7, 9 2 Mid-Range The arithmetic mean of the highest and lowest values of a set. (1 + 7) ÷ 2 4

The arithmetic mean, median, mode, and mid-range are the most often used estimates of central tendency in descriptive statistics. These measures of central tendency are better defined below with more examples of how to solve for them:

• Arithmetic Mean

• The most common type of average is the arithmetic mean. If n numbers are given, each number denoted by ai (where i = 1, 2, ... , n), the arithmetic mean is the sum of the as divided by n or (a1 + a2 + . . . + an) ÷ n. The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = (2 + 8) ÷ 2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2, 8, and 11, the arithmetic mean is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11) ÷ 3 = 7.

• Median

• The median is the middle number of the group when they are ranked in order. If there are an even number of numbers, the arithmetic mean of the middle two is taken. Thus, to find the median, order the list according to its elements magnitude and then repeatedly remove the pair consisting of the highest and lowest values until either one or two values are left. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of these two. This method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13. Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this remaining list, the median is their arithmetic mean, (3 + 7) ÷ 2 = 5.

• Mode

• The mode is the most frequently occurring number in a list. For instance, the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. It may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed upon definition of mode. Some authors say they are all modes, and some say there is no mode.

• Mid-Range

• The mid-range is the arithmetic mean of the highest and lowest values of a set. For instance if the greatest value of a given set is 10 and the lowest 2, then the midrange is (2 + 10) ÷ 2 = 6.

### Sources

“Average.” Wikipedia, Wikimedia Foundation, 30 June 2020, en.wikipedia.org/wiki/Average.

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