Average average, the notion whereby several quantities are reduced to a single representative magnitude, hath long served as a compass in the arithmetic of diverse phenomena. In its simplest and most venerable form it is the arithmetick mean, that which is obtained by the addition of the given numbers and the division of the sum by their count. This operation, though elementary, renders a measure of central tendency that is both convenient and, in many circumstances, sufficient for the purpose of comparison. The arithmetick mean may be expressed, in the language of the time, as the quotient of the total and the number of terms, and it is this quotient which furnishes what is commonly called the average. In the art of reckoning the mean is not the sole means of summarising a collection. When the quantities in question are multiplicative rather than additive, the geometric mean assumes a more apt character. By multiplying the numbers together and then extracting the root whose degree equals the number of factors, one obtains a value that reflects the proportional growth or diminution inherent in the set. Thus, for a sequence of ratios or rates, the geometric mean supplies a more faithful representation than the arithmetick mean, which would otherwise be distorted by the asymmetry of multiplication. A third, less frequently employed, but nevertheless noteworthy, is the harmonic mean, derived by taking the reciprocal of each term, forming their arithmetick mean, and then inverting the result. The harmonic mean proves useful when the quantities are rates of work or speed, wherein the reciprocal relation mirrors the underlying physical law. The employment of the average in the doctrine of chances, as first systematised by the likes of Huygens, de Moivre, and later refined by the present author, rests upon the principle that the mean of a large number of observations converges upon a value that may be regarded as the most probable. When a die is cast repeatedly, each face being equally likely, the sum of the observed outcomes divided by the number of throws tends, as the number of throws increases, towards the arithmetick mean of the possible faces, namely three and a half. This convergence, though never exact in a finite series, is made ever nearer by the increase of trials, and thus the average serves as an estimator of the underlying chance. The reliability of such estimation may be examined through the notion of error or deviation from the true mean. In the eighteenth‑century discourse, the term error is employed to denote the difference between an observed value and the expected value, the latter being the average of the possible outcomes. When many observations are taken, the errors tend to distribute themselves in a manner that is symmetric about zero, a property which underlies the method of least squares later expounded by Legendre and Gauss. Though the present treatise does not dwell upon the algebraic intricacies of that method, it is sufficient to observe that the average, by virtue of its minimising property for the sum of squared deviations, occupies a privileged station in the theory of errors. In the application to the inference of unknown proportions, the average assumes a role akin to that of a prior belief. Suppose a collection of urns, each containing balls of two colours, is such that the proportion of red balls in each urn is unknown but is believed, a priori, to be uniformly distributed between zero and one. By drawing a number of balls from a particular urn and recording the proportion of reds thus obtained, the average of those observations furnishes a posterior estimate of the true proportion. In the language of the doctrine of chances, this estimate may be regarded as the most probable value given the observed data and the assumed uniform prior. The reasoning follows the same pattern as that employed in the computation of posterior probabilities for events such as the occurrence of a particular number of successes in a series of Bernoulli trials. The use of the average in the analysis of population data, though more recent, can be traced to the same principles. When the ages of a multitude of individuals are recorded, the arithmetick mean of those ages provides a succinct summary, yet it is but one facet of the distribution. The median and mode may be introduced to complement the mean, but the average remains the primary indicator of central tendency. In agricultural statistics, for instance, the average yield per acre, obtained by dividing the total harvest by the number of cultivated acres, offers a measure by which the productivity of differing methods may be compared. Such comparisons, when extended over several seasons, reveal the tendency of the average to stabilise, thereby granting a basis for prudent judgment in the management of estates. The method of indirect reasoning , whereby one infers the value of an unknown quantity from the known averages of related quantities, is a further illustration of the versatility of the average. Consider a merchant who knows the average price of a bundle of wheat over a year and the average price of a bundle of barley, and who wishes to determine the average price of a mixed shipment containing known proportions of the two grains. By the linearity of the arithmetick mean, the average price of the mixture is the weighted sum of the individual averages, the weights being the respective proportions. This principle, often termed the law of total expectation in modern parlance, was already implicit in the works of early mathematicians who treated the problem of allocating portions of a whole. When the quantities under consideration are not directly comparable, the average may be employed after an appropriate normalisation . In the study of probabilities arising from games of chance, it is customary to express the outcomes in units of expectation , that is, the product of the payoff and its probability. The average of such expectations over a series of trials yields the mean gain or mean loss per trial, a figure of paramount importance to the prudent gambler. The same technique may be applied to the assessment of insurance premiums, wherein the average loss per insured individual, computed from observed claims, guides the determination of a fair rate. The stability of the average under the addition of further observations is a property of great practical import. If a set of numbers possesses an average \( \mu \) and a further number \( x \) is appended, the new average becomes \( \frac{n\mu + x}{n+1} \), where \( n \) is the original count. This recursive formula permits the continual updating of the average without the necessity of recomputing the entire sum, a convenience that has been embraced by those who keep accounts or conduct experiments over extended periods. The formula also reveals that the influence of a single new observation diminishes as the number of prior observations grows, thereby confirming the intuition that the average becomes more resistant to fluctuations as the sample enlarges. The concept of weighted average extends the foregoing idea by assigning to each term a coefficient reflecting its relative importance. In the calculation of a composite price index, for instance, each commodity may be allotted a weight proportional to its share in the total consumption. The weighted average then furnishes a single figure that summarises the overall movement of prices, a practice that has been adopted by the merchants of London and the clerks of the Exchequer alike. The mathematical justification of the weighted average lies in the same linearity that undergirds the unweighted case, the only alteration being the introduction of the weights into the summation. The average, when considered in the context of probability distributions, may be identified with the expected value of a random variable. The expected value, defined as the sum of each possible value multiplied by its probability, coincides with the arithmetick mean of a large number of independent observations drawn from the distribution. This identification provides a bridge between the theoretical realm of chance and the empirical realm of measurement. In the case of a binomial experiment, such as the repeated throwing of a die to obtain a particular face, the expected number of successes after \( n \) trials is \( n p \), where \( p \) denotes the probability of success on a single trial. The average number of successes observed in practice will, by the law of large numbers, approach this expected value as \( n \) increases. The law of large numbers, first articulated in a rudimentary form by Jacob Bernoulli and later refined by the author of this essay, asserts that the average of a sequence of independent, identically distributed random variables converges in probability to their common expectation. Though the formal proof requires a delicate handling of limits and inequalities, the essential idea is that the fluctuations of the average about the expectation become arbitrarily small as the number of observations grows without bound. This principle vindicates the use of the average as a reliable estimator in the practice of gambling, insurance, and scientific observation, for it assures that the observed mean will not wander far from the true mean when a sufficient quantity of data is gathered. In the realm of inverse probability , the average assumes a role of inferential significance. When the probability of an event is unknown, yet the frequencies of its occurrence are observed, the observed average frequency serves as the posterior probability, provided a uniform prior is adopted. This reasoning, which underlies the celebrated theorem concerning the probability of a cause given an observed effect, demonstrates that the average is not merely a descriptive statistic but also a conduit for logical deduction. The theorem, wherein the probability of a cause is expressed as the ratio of the average occurrence of the effect under that cause to the overall average occurrence of the effect, exemplifies the intimate connection between the average and the doctrine of chances. The limitations of the average must likewise be acknowledged. In distributions that are markedly skewed, the arithmetick mean may be drawn away from the bulk of the data, rendering it a poor representative. In such cases the median, the value that separates the lower half of observations from the upper, may convey a more faithful picture of central tendency. Nevertheless, the average retains a privileged status in the theory of probability, for it is the quantity most directly linked to the linearity of expectation and the additive nature of independent events. To summarise, the average, in its various guises—arithmetick, geometric, harmonic, weighted, and expected—constitutes a fundamental instrument in the calculation of chances, the estimation of unknown quantities, and the summarisation of empirical data. Its properties of linearity, convergence, and recursive update render it indispensable to the mathematician, the merchant, and the physician alike. Though its simplicity may belie its depth, the average remains the cornerstone upon which much of the modern calculus of probability and statistics is erected, and its study continues to yield insight into the regularities that underlie the apparently erratic motions of chance. [role=marginalia, type=heretic, author="a.weil", status="adjunct", year="2026", length="46", targets="entry:average", scope="local"] Yet the average, by erasing singularity, becomes an instrument of oppression; it reduces each soul to a datum, privileging the middle while consigning the extreme to invisibility. True knowledge must attend to the outlier, for the divine is found in the exception, not in the mean. [role=marginalia, type=objection, author="a.dennett", status="adjunct", year="2026", length="38", targets="entry:average", scope="local"] The entry’s equation of “average” with the arithmetic mean obscures the fact that, for skewed or heavy‑tailed data, the mean can be a poor representative; the median or a trimmed mean often yields a more robust central tendency. [role=marginalia, type=objection, author="a.dennett", status="adjunct", year="2026", length="43", targets="entry:average", scope="local"] The average as “most likely outcome” is a misstep—it’s often the least likely single value. Averages are statistical summaries, not probabilistic predictions. Confusing them with modal or expected outcomes risks reifying abstraction, betraying the very anti-essentialism that probability theory was meant to uphold. [role=marginalia, type=clarification, author="a.freud", status="adjunct", year="2026", length="45", targets="entry:average", scope="local"] The average masks the unconscious drift of individual destinies—each death, each fortune, bears the trace of repressed conflict. Statistical regularity is not nature’s law, but the psyche’s projection: order imposed upon chaos to evade the terror of randomness. The average is the superego of numbers. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:average", scope="local"] I remain unconvinced that the average fully captures the complexities of human cognition and bounded rationality. While averages provide a useful estimation, they often gloss over the individual variability and the cognitive limitations that influence our perception and interpretation of data. From where I stand, more attention must be paid to the ways in which our mental processes shape and limit our understanding of statistical constructs. See Also See "Measurement" See "Number"