Arithmetic mean
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection.
[1] The collection is often a set of results from an experiment, an observational study, or a survey.
The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic.
In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent.
While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others).
For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle".
In that case, robust statistics, such as the median, may provide a better description of central tendency.
In simpler terms, the formula for the arithmetic mean is:
The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid.
The statistician Churchill Eisenhart, senior researcher fellow at the U. S. National Bureau of Standards, traced the history of the arithmetic mean in detail.
In the modern age it started to be used as a way of combining various observations that should be identical, but were not such as estimates of the direction of magnetic north.
In 1668, a person known as “DB” was quoted in the Transactions of the Royal Society describing “taking the mean” of five values:[3] In this Table, he [Capt.
Sturmy] notes the greatest difference to be 14 minutes; and so taking the mean for the true Variation, he concludes it then and there to be just 1. deg.
27. min.The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency.
These include: The arithmetic mean may be contrasted with the median.
, the median and arithmetic average can differ significantly.
In contrast, a weighted mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as
If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero.
In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the mean of the probability distribution.
The most widely encountered probability distribution is called the normal distribution; it has the property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms[7]), are equal.
Particular care is needed when using cyclic data, such as phases or angles.
Taking the arithmetic mean of 1° and 359° yields a result of 180°.
This is incorrect for two reasons: In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range.
A solution to this problem is to use the optimization formulation (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).
The arithmetic mean is often denoted by a bar (vinculum or macron), as in
[4] Some software (text processors, web browsers) may not display the "x̄" symbol correctly.
PR is the diameter of a circle centered on O; its radius AO is the arithmetic mean of a and b . Using the geometric mean theorem , triangle PGR's altitude GQ is the geometric mean . For any ratio a : b , AO ≥ GQ.
