Arithmetic mean

The arithmetic mean (also called the mean or average) is the sum of a collection of numbers divided by the count of numbers in that collection, and it is the most widely used measure of central tendency in statistics and everyday quantitative reasoning. Formally, for a data set of n values x₁, x₂, …, xₙ, the arithmetic mean is the quantity x̄ = (x₁ + x₂ + ⋯ + xₙ)/n. It represents the "center of mass" or balancing point of the data, and it serves as the foundation for numerous statistical concepts, including variance, standard deviation, expected value, and the method of least squares. In ordinary language, when people speak of an "average," they almost always mean the arithmetic mean, distinguishing it from other kinds of means such as the geometric and harmonic means.

Definition and notation

For a finite data set {x₁, x₂, …, xₙ} of real numbers, the arithmetic mean is defined as:

x̄ = (x₁ + x₂ + ⋯ + xₙ) / n = (1/n) Σᵢ xᵢ

The symbol x̄ (an "x" with a bar over it, read as "x-bar") is the conventional notation for a sample mean. When the mean refers to an entire population rather than a sample drawn from it, it is commonly denoted by the Greek letter μ (mu) and defined as μ = (1/N) Σᵢ xᵢ over all N population members.

Two important generalizations exist:

• Weighted arithmetic mean. When observations carry different degrees of importance, each value xᵢ is assigned a weight wᵢ ≥ 0, and the weighted mean is x̄ = (Σ wᵢxᵢ) / (Σ wᵢ). Grade-point averages, index numbers, and survey estimates are typically weighted means. The ordinary mean is the special case in which all weights are equal.
• Expected value. For a random variable X with probability distribution, the arithmetic mean is interpreted as the expectation E[X]. For a discrete variable, E[X] = Σ xᵢ pᵢ; for a continuous variable with probability density f, E[X] = ∫ x f(x) dx. In classical probability theory, the expectation of a random variable is often called its "mean."

Historical development

The theory of means originated in ancient Greek mathematics. The Pythagorean school of the sixth century BCE, motivated largely by investigations into music, proportion, and number theory, distinguished the arithmetic, geometric, and harmonic means, and examined the relationships among them. In an arithmetic mean b of two quantities a and c, the differences are equal (a − b = b − c); the Pythagoreans recognized that string lengths related by such means produce consonant musical intervals. Plato discussed the arithmetic and harmonic means in the Timaeus, and Euclid's Elements developed the theory of proportion that underlies these notions. In the second century CE, Nicomachus of Gerasa's Introduction to Arithmetic gave a systematic treatment of the three principal means, a formulation that dominated medieval and early Renaissance mathematics.

A separate tradition of averaging arose in commercial and legal practice. The English word "average" derives, through Anglo-Norman and Old French avarie, from the Arabic ʿawāriyya, referring to damage to, or the apportionment of losses on, maritime cargo; the term gradually shifted in eighteenth-century English usage to denote the arithmetical mean of quantities.

The systematic use of the arithmetic mean to combine repeated scientific observations developed comparatively late. Ancient and medieval astronomers generally reconciled discrepant measurements by selecting a "middle" observation rather than computing an average. Historians of statistics generally trace the regular employment of the arithmetic mean of observations to sixteenth- and seventeenth-century astronomy, becoming standard practice among observers such as the first Astronomer Royal, John Flamsteed, by the late seventeenth century. The theoretical justification came in the early nineteenth century: Adrien-Marie Legendre published the method of least squares in 1805, and Carl Friedrich Gauss developed its probabilistic foundations, showing that the arithmetic mean is the value that minimizes the sum of squared deviations and connecting it to the theory of errors. At the same time, Jacob Bernoulli's Ars Conjectandi (1713) had established the law of large numbers, linking averages of repeated trials to underlying probabilities. In the nineteenth century, Adolphe Quetelet applied the mean to social data and coined the notion of the "average man" (l'homme moyen), while later statisticians such as Francis Galton and Karl Pearson embedded the mean at the center of modern statistical methodology.

Mathematical properties

The arithmetic mean possesses a set of properties that account for both its utility and its prominence:

• Zero sum of deviations. The deviations of the observations from their mean sum to zero: Σᵢ (xᵢ − x̄) = 0. Equivalently, the total "excess" above the mean exactly balances the total "deficit" below it, which is why the mean is regarded as the balance point or center of gravity of the data.
• Characterization by equal sums. The mean is the unique number m for which the sum of the deviations equals zero.
• Least squares property. Among all real numbers m, the arithmetic mean x̄ minimizes the sum of squared deviations Σᵢ (xᵢ − m)². (The median, by contrast, minimizes the sum of absolute deviations.)
• Sum preservation. The sum of the data equals n times the mean: Σᵢ xᵢ = n x̄.
• Affine equivariance. If a constant c is added to every observation, the mean increases by c; if every observation is multiplied by a constant c, the mean is multiplied by c. That is, mean(xᵢ + c) = x̄ + c and mean(c·xᵢ) = c·x̄.
• Combining groups. If one sample of size n₁ has mean x̄₁ and a second sample of size n₂ has mean x̄₂, the mean of the pooled data is (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂), an overall mean that is itself a weighted mean of the group means.
• AM–GM–HM inequality. For non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean; equality holds only when all the numbers are identical. This inequality, of great importance in pure mathematics, is often proved by induction or by convexity arguments.

The arithmetic mean is also the first raw moment of a data set or distribution and appears in the very definition of variance, which is the mean of the squared deviations from the mean.

Relation to other measures of central tendency

The arithmetic mean is one of several competing summaries of the "center" of a distribution, alongside the median (the middle value) and the mode (the most frequent value). In symmetric, unimodal distributions such as the normal distribution, all three coincide approximately. In skewed distributions they diverge: for right-skewed data such as household incomes, a small number of very large values pulls the mean above the median, so the median is often preferred as a summary of typical income. For this reason, statistical practice pairs the mean with the standard deviation when the data are approximately symmetric, and relies on medians and quantiles for strongly skewed data.

Because a single extreme value can shift the arithmetic mean without bound, the mean has a breakdown point of zero and is classified as non-robust. Robust alternatives include the trimmed mean (discarding a fixed fraction of the highest and lowest observations), the Winsorized mean (replacing extreme values with less extreme ones), and the midrange (the mean of the minimum and maximum, used mainly for small uniform samples).

Comparison with other means

The arithmetic mean belongs to a family of generalized (power) means. For positive numbers and a real parameter p, the power mean is Mₚ = ((1/n) Σ xᵢᵖ)^(1/p). The arithmetic mean corresponds to p = 1; the harmonic mean (appropriate for averaging rates over fixed distances or quantities) to p = −1; the root mean square (used for alternating currents and error magnitudes) to p = 2; and, in the limit as p → 0, the geometric mean (appropriate for averaging ratios and growth rates). Choosing the arithmetic mean when another mean is appropriate leads to well-known errors—for example, averaging speeds over equal distances requires the harmonic mean, and averaging multiplicative growth factors requires the geometric mean.

Role in probability and statistics

In statistical inference, the sample mean x̄ is the standard estimator of the population mean μ. It is unbiased—its expectation equals μ—and, by the law of large numbers, it converges to μ as the sample size grows. The central limit theorem establishes that for large samples the sampling distribution of the mean is approximately normal, with standard deviation (the standard error) equal to σ/√n, where σ is the population standard deviation. These results justify the mean's role in confidence intervals, hypothesis tests, and virtually all classical estimation procedures, including analysis of variance and linear regression, whose least-squares estimators are effectively conditional arithmetic means.

Certain theoretical limits are worth noting: some heavy-tailed distributions, such as the Cauchy distribution, possess no finite mean, so that averaging repeated draws fails to stabilize around any value.

Applications

The arithmetic mean pervades quantitative practice. In economics, per-capita figures such as GDP per capita and mean income are arithmetic means. In education, course grades and grade-point averages are (often weighted) means. In meteorology, daily and annual mean temperatures summarize climate; in physics and engineering, mean values underpin measurements of error, energy, and signal levels. In sports, averages of performance measures are standard statistics. In astronomy and surveying, the mean of repeated readings remains the basic method of reducing observational error, exactly as in the era of Gauss and Legendre. In calculus, the average value of a function f over an interval [a, b] is defined as (1/(b − a)) ∫ₐᵇ f(x) dx, the continuous analogue of the finite arithmetic mean, and forms the substance of the mean value theorem for integrals.

Limitations and criticisms

Despite its centrality, the arithmetic mean is not universally appropriate. It presumes that the data are measured on an interval or ratio scale; applying it to ordinal categories (such as ranked satisfaction ratings) can be misleading, though the practice is common in social research. It is undefined or meaningless for periodic quantities such as clock times or compass bearings, where the answer may fall outside the data range; such data require the circular mean. It is also inapplicable when the values are not commensurable or when averaging must respect a nonlinear transformation (as in pH values, decibels, or star magnitudes, which are already logarithmic).

Substantively, the mean can obscure distributional reality: a population may show a rising mean income while typical incomes stagnate, if growth concentrates at the top. Consequently, responsible statistical reporting usually presents the mean alongside measures of spread and skewness, or alongside the median, so that the summary conveys the shape of the distribution rather than a single, potentially unrepresentative, number.

Significance

The arithmetic mean occupies a unique position at the intersection of mathematics, science, and ordinary reasoning. Its ancient pedigree in Greek proportion theory, its commercial origins in the law of averages, its adoption by astronomers as the canonical method of combining observations, and its theoretical vindication by the theory of probability have together made it the default concept of a "typical value" in modern culture. As the pivot of least squares, expectation, variance, and the central limit theorem, it is arguably the single most consequential quantity in statistics, simultaneously a simple piece of arithmetic accessible to a schoolchild and a deep structural concept in the theory of inference.