In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.

The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers.

Notably, for skewed distributions, the arithmetic mean may not accord with one's notion of " middle ", and robust statistics such as the median may be a better description of central tendency.

Then the arithmetic mean is defined via the equation

The arithmetic mean has several properties that make it useful, especially as a measure of central tendency.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

* For a normal distribution, the arithmetic mean is equal to both the median and the mode, other measures of central tendency.

The arithmetic mean may be misinterpreted as the median to imply that most values are higher or lower than is actually the case.

In mathematics, the arithmetic – geometric mean ( AGM ) of two positive real numbers and is defined as follows:

First compute the arithmetic mean of and and call it.

These two sequences converge to the same number, which is the arithmetic – geometric mean of and ; it is denoted by, or sometimes by.

To find the arithmetic – geometric mean of and, first calculate their arithmetic mean and geometric mean, thus:

An " elementary " proof can be given using the fact that geometric mean of positive numbers is less than arithmetic mean

# Let P ( x ) be a first-order formula in the language of Presburger arithmetic with a free variable x ( and possibly other free variables ).

Prefix notation has seen wide application in Lisp s-expressions, where the brackets are required due to the arithmetic operators having variable arity.

The long division algorithm for arithmetic is very similar to the above algorithm, in which the variable x is replaced by the specific number 10.

S2's structure closely resembles that of most imperative programming languages, and includes basic instructions such as variable assignments, arithmetic operations, conditional flow control and for loops over finite sets ( however, it distinctly lacks while loops ).

Typically, these well-known functions are defined to be elementary functions — constants, one variable x, elementary operations of arithmetic (+ − × ÷), nth roots, exponent and logarithm ( which thus also include trigonometric functions and inverse trigonometric functions ).

In this form, the bc language contains single letter variable, array and function names and most standard arithmetic operators as well as the familiar control flow constructs, (, and ) from C. Unlike C, an clause may not be followed by an.

The 360s also included instructions for variable length decimal arithmetic for commercial applications, so the practice of using word lengths that were a power of two quickly became commonplace, though some 36-bit computer systems are still sold to this day, e. g., the Unisys ClearPath Dorado series, which is the continuation of the UNIVAC 1100 / 2200 series of mainframe computers.

If an optimizing compiler or assembler is able to pre-calculate offsets to each individually referenced array variable, these can be built into the machine code instructions directly, therefore requiring no additional arithmetic operations at run time ( note that in the example given below this is not the case ).

In general, in languages offering this feature, most operators that can take a variable as one of their arguments and return a result of the same type have an augmented assignment equivalent that assigns the result back to the variable in place, including arithmetic operators, bitshift operators, and bitwise operators.

Perhaps the most obvious examples were the bit-addressable memory, the variable size arithmetic logic unit ( ALU ), and the ability to OR in data from a register into the instruction register allowing very efficient instruction parsing.

A variable of packed array type maps 1: 1 onto an integer arithmetic quantity.

`` All right, if you can't do your arithmetic during school hours you can do it after school is out '', Miss Langford said firmly, not smiling.

The abacus ( plural abaci or abacuses ), also called a counting frame, is a calculating tool used primarily in parts of Asia for performing arithmetic processes.

It is worth mentioning that the Nepōhualtzintzin amounted to the rank from 10 to the 18 in floating point, which calculated stellar as well as infinitesimal amounts with absolute precision, meant that no round off was allowed, when translated into modern computer arithmetic.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

A key problem in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.

In Aristotle this is categorized as the difference between ' arithmetic ' and ' geometric ' ( i. e. proportional ) equality.

One of his most celebrated achievements is the discovery of the first arithmetic Weil cohomology theory: the ℓ-adic étale cohomology.

The latter is more cumbersome to use, so it's only employed when necessary, for example in the analysis of arbitrary-precision arithmetic algorithms, like those used in cryptography.

In 1929, Mojżesz Presburger showed that the theory of natural numbers with addition and equality ( now called Presburger arithmetic in his honor ) is decidable and gave an algorithm that could determine if a given sentence in the language was true or false.

( Aside from its historic role as a total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessly as do variants of Ackermann's function that are specifically designed for that purpose — such as Goodstein's hyperoperation sequence.

Büttner, gave him a task: add a list of integers in arithmetic progression ; as the story is most often told, these were the numbers from 1 to 100.

An electronic calculator is a small, portable, often inexpensive electronic device used to perform both basic and complex operations of arithmetic.

Various arithmetic operations are often added at the end ; is three six-sided dice plus four to the outcome.

These often process data using fixed-point arithmetic, though some more powerful versions use floating point arithmetic.

In keeping with their social orientation, numbers are usually low in magnitude, often under ten, and any arithmetic in the game is typically trivial.

The arithmetic mean is the " standard " average, often simply called the " mean ".

Accuracy of readings obtained is also often compromised by miscounting division markings, errors in mental arithmetic, parallax observation errors, and less than perfect eyesight.

* 1-operand ( one-address machines ), so called accumulator machines, include early computers and many small microcontrollers: most instructions specify a single right operand ( that is, constant, a register, or a memory location ), with the implicit accumulator as the left operand ( and the destination if there is one ): load a, add b, store c. A related class is practical stack machines which often allow a single explicit operand in arithmetic instructions: push a, add b, pop c.

The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8

Most computers of the era operated in bit-serial mode, using single-bit arithmetic and feeding in large words, often 48 or 60 bits in size, one bit at a time.

In computer science, denormal numbers or denormalized numbers ( now often called subnormal numbers ) fill the underflow gap around zero in floating point arithmetic.

It is possible to do computations using an exact fractional representation of rational numbers and keep all significant digits, but this is often prohibitively slower than floating-point arithmetic.

Processor registers are typically divided into several groups: integer, floating-point, SIMD, control, and often special registers for address arithmetic which may have various uses and names such as address, index or base registers.

* Fixed-point arithmetic is often used to speed up arithmetic processing

The term " empty product " is most often used in the above sense when discussing arithmetic operations.

When division is explained at the elementary arithmetic level, it is often considered as splitting a set of objects into equal parts.

Amongst his most original contributions were: his " Conjecture II " ( still open ) on Galois cohomology ; his use of group actions on Trees ( with H. Bass ); the Borel-Serre compactification ; results on the number of points of curves over finite fields ; Galois representations in ℓ-adic cohomology and the proof that these representations have often a " large " image ; the concept of p-adic modular form ; and the Serre conjecture ( now a theorem ) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Loops can be unrolled ( for lower loop overhead, although this can often lead to lower speed if it overloads the CPU cache ), data types as small as possible can be used, integer arithmetic can be used instead of floating-point, and so on.

It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions.

Arbitrary precision arithmetic is often used to establish these values to a high degree of precision – typically 100 significant figures or more.