Arithmetic Mean

Reference work entry

The arithmetic mean is a measure of central tendency. It allows us to characterize the center of the frequency distribution of a quantitative variable by considering all of the observations with the same weight afforded to each (in contrast to the weighted arithmetic mean).

It is calculated by summing the observations and then dividing by the number of observations.


The arithmetic mean is one of the oldest methods used to combine observations in order to give a unique approximate value. It appears to have been first used by Babylonian astronomers in the third century BC. The arithmetic mean was used by the astronomers to determine the positions of the sun, the moon and the planets. According to Plackett (1958), the concept of the arithmetic mean originated from the Greek astronomer Hipparchus.

In 1755 Thomas Simpson officially proposed the use of the arithmetic mean in a letter to the President of the Royal Society.


Let \( { x_1, x_2, \ldots, x_n } \)

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  1. 1.
    Plackett, R.L.: Studies in the history of probability and statistics. VII. The principle of the arithmetic mean. Biometrika 45, 130–135 (1958)Google Scholar
  2. 2.
    Simpson, T.: A letter to the Right Honorable George Earl of Macclesfield, President of the Royal Society, on the advantage of taking the mean of a number of observations in practical astronomy. Philos. Trans. Roy. Soc. Lond. 49, 82–93 (1755)CrossRefGoogle Scholar
  3. 3.
    Simpson, T.: An attempt to show the advantage arising by taking the mean of a number of observations in practical astronomy. In: Miscellaneous Tracts on Some Curious and Very Interesting Subjects in Mechanics, Physical-Astronomy, and Speculative Mathematics. Nourse, London (1757). pp. 64–75Google Scholar

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