Inmathematics andstatistics, thearithmetic mean (/ˌærɪθˈmɛtɪk/ⓘarr-ith-MET-ik),arithmetic average, or just themean oraverage 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 anexperiment, anobservational study, or asurvey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such asgeometric andharmonic.
While the arithmetic mean is often used to reportcentral tendencies, it is not arobust statistic: it is greatly influenced byoutliers (values much larger or smaller than most others). Forskewed distributions, such as thedistribution 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 themedian, may provide a better description ofcentral tendency.
Further information:summation for an explanation of the summation operator
The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by the total number of observations. Symbolically, for a data set consisting of the values, the arithmetic mean is defined by the formula:[2]
In simpler terms, the formula for the arithmetic mean is:
For example, if the monthly salaries of employees are, then the arithmetic mean is:
Example
Person
Salary
A
2500
B
2700
C
2300
D
2650
E
2450
Average
2520
If the data set is astatistical population (i.e. consists of every possible observation and not just asubset of them), then the mean of that population is called thepopulation mean and denoted by theGreek letter. If the data set is astatistical sample (a subset of the population), it is called thesample mean (which for a data set is denoted as).
The arithmetic mean can be similarly defined forvectors in multiple dimensions, not onlyscalar values; this is often referred to as acentroid. More generally, because the arithmetic mean is aconvex combination (meaning its coefficients sum to), it can be defined on aconvex space, not only avector space.
StatisticianChurchill Eisenhart, senior researcher fellow at theU. 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 asestimates of the direction ofmagnetic north. In 1635, mathematicianHenry Gellibrand described as "meane" themidpoint of a lowest and highest number, not quite the arithmetic mean. In 1668, a person known as "D. B." was quoted in theTransactions 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:
If numbers have a mean, then. Since is the distance from a given number to the mean, one way to interpret this property is by saying that the numbers to the left of the mean are balanced by the numbers to the right. The mean is the only number for which theresiduals (deviations from the estimate) sum to zero. This can also be interpreted as saying that the mean istranslationally invariant in the sense that for any real number,.
If it is required to use a single number as a "typical" value for a set of known numbers, then the arithmetic mean of the numbers does this best since it minimizes the sum of squared deviations from the typical value: the sum of. The sample mean is also the best single predictor because it has the lowestroot mean squared error.[4] If the arithmetic mean of a population of numbers is desired, then the estimate of it that isunbiased is the arithmetic mean of a sample drawn from the population.
The arithmetic mean is independent of scale of the units of measurement, in the sense that So, for example, calculating a mean of liters and then converting to gallons is the same as converting to gallons first and then calculating the mean. This is also calledfirst order homogeneity.
The arithmetic mean differs from themedian, which is the value that separates the higher half and lower half of a data set. When the values in a data set form an arithmetic progression, the median and arithmetic mean are equal. For example, in the data set, both the mean and median are.
In other cases, the mean and median can differ significantly. For instance, in the data set, the arithmetic mean is, while the median is. This occurs because the mean is sensitive to extreme values and may not accurately reflect the central tendency of most data points.
This distinction has practical implications across different fields. For example, since the 1980s, themedian income in theUnited States has increased at a slower rate than the arithmetic mean income.[5]
Similarly, in climate studies, daily mean temperature distributions tend to approximate a normal distribution, whereas annual or monthly rainfall totals often display a skewed distribution, with some periods having unusually high totals while most have relatively low amounts. In such cases, the median can provide a more representative measure of central tendency.[6]
A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation.[7] For example, the arithmetic mean of and is, or equivalently. In contrast, aweighted 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. Here the weights, which necessarily sum to one, are and, the former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number ( in the above example and in a situation with numbers being averaged).
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 theprobability of a number falling into some range of possible values can be described by integrating acontinuous 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 themean of theprobability distribution. The most widely encountered probability distribution is called thenormal 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[8]), are equal. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here.
Particular care is needed when using cyclic data, such asphases orangles. Taking the arithmetic mean of 1° and 359° yields a result of 180°. This is incorrect for two reasons:
Angle measurements are only defined up to an additive constant of 360° ( or, if measuring inradians). Thus, these could easily be called 1° and -1°, or 361° and 719°, since each one of them produces a different average.
In this situation, 0° (or 360°) is geometrically a betteraverage value: there is lowerdispersion about it (the points are both 1° from it and 179° from 180°, the putative average).
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°).
