arithmetic mean

(noun)

The measure of central tendency of a set of values computed by dividing the sum of the values by the number of values; commonly called the "average."

Related Terms

Examples of arithmetic mean in the following topics:

  • Averages

    • The arithmetic mean, or average, of a set of numbers indicates the "middle" or "typical" value of a data set.
    • The arithmetic mean, or "average" is a measure of the "middle" or "typical" value of a data set.
    • While it is often referred to simply as "mean" or "average," the term "arithmetic mean" is preferred in some contexts because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.
    • For example, per capita income is the arithmetic mean income of a nation's population.
    • The arithmetic mean $A$ is defined via the expression:

  • Sums and Series

    • The "$i=m$" under the summation symbol means that the index $i$ starts out equal to $m$.
    • If you add up all the terms of an arithmetic sequence (a sequence in which every entry is the previous entry plus a constant), you have an arithmetic series.
    • You understand that this trick will work for any arithmetic series.
    • If we apply this trick to the generic arithmetic series, we get a formula that can be used to sum up any arithmetic series.
    • Every arithmetic series can be written as follows:

  • Arithmetic Sequences

    • An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
    • An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
    • For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.
    • The behavior of the arithmetic sequence depends on the common difference $d$.
    • Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences

  • Summing Terms in an Arithmetic Sequence

    • An arithmetic sequence which is finite has a specific formula for its sum.
    • An arithmetic progression or arithmetic sequence is an ordered list of numbers such that the difference between the consecutive terms is constant.
    • The sum of the members of a finite arithmetic sequence is called an arithmetic series.
    • An infinite arithmetic series is exactly what it sounds like: an infinite series whose terms are in an arithmetic sequence.
    • The general form for an infinite arithmetic series is:

  • Introduction to Sequences

    • An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term.
    • An explicit definition of an arithmetic sequence is one in which the $n$th term is defined without making reference to the previous term.
    • This is more useful, because it means you can find (for instance) the 20th term without finding all of the other terms in between.
    • To find the explicit definition of an arithmetic sequence, you begin writing out the terms.

  • Recursive Definitions

    • When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way:
    • In this equation, one can directly calculate the nth-term of the arithmetic sequence without knowing the previous terms.
    • This means that using a recursive formula when using a computer to manipulate a sequence might mean that the calculation will be finished quickly.

  • Applications and Problem-Solving

    • Arithmetic series can simplify otherwise complex addition problems by decreasing the number of terms to be added.
    • Using equations for arithmetic sequence summation can greatly facilitate the speed of problem solving.
    • This trick applies to all arithmetic series.
    • As long as you go up by the same amount as you go down, the sum will stay the same—and this is just what happens for arithmetic series.

  • The Order of Operations

    • The order of operations is an approach to evaluating expressions that involve multiple arithmetic operations.
    • The order of operations is a way of evaluating expressions that involve more than one arithmetic operation.
    • These rules means that within a mathematical expression, the operation ranking highest on the list should be performed first.
    • This means that multiplication and division operations (and similarly addition and subtraction operations) can be performed in the order in which they appear in the expression.

  • Rules for Solving Inequalities

    • Arithmetic operations can be used to solve inequalities for all possible values of a variable.
    • To solve an inequality means to transform it such that a variable is on one side of the symbol and a number or expression on the other side.
    • This means that we must also change the direction of the symbol:

  • Addition, Subtraction, and Multiplication

    • Indeed, if i is treated as a number so that di means d time i, the above multiplication rule is identical to the usual rule for multiplying the sum of two terms.
    • Discover the similarities between arithmetic operations on complex numbers and binomials