arithmetic mean

Algebra

(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

Statistics

(noun)

the measure of central tendency of a set of values computed by dividing the sum of the values by their number; commonly called the mean or the average

Related Terms

Examples of arithmetic mean in the following topics:

  • Mean: The Average

    • The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.
    • When we think of means, or averages, we are typically thinking of the arithmetic mean.
    • The arithmetic mean is defined via the expression:
    • Comparison of the arithmetic, geometric and harmonic means of a pair of numbers.
    • Define the average and distinguish between arithmetic, geometric, and harmonic means.

  • Log Transformations

    • The comparison of the means of log-transformed data is actually a comparison of geometric means.
    • This occurs because, as shown below, the anti-log of the arithmetic mean of log-transformed values is the geometric mean.Table 1 shows the logs (base 10) of the numbers 1, 10, and 100.
    • The arithmetic mean of the three logs is
    • Therefore, if the arithmetic means of two sets of log-transformed data are equal then the geometric means are equal.

  • 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:

  • The Sample Average

    • The sample average (also called the sample mean) is often referred to as the arithmetic mean of a sample, or simply, $\bar{x}$ (pronounced "x bar").
    • For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.
    • The arithmetic mean is the "standard" average, often simply called the "mean".
    • For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:
    • The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode).

  • Exercises

    • If the arithmetic mean of log10 transformed data were 3, what would be the geometric mean?

  • Measures of Central Tendency

    • The arithmetic mean is the most common measure of central tendency.
    • $\text{"}\mu\text{"}$ is used for the mean of a population.The symbol "M" is used for the mean of a sample.
    • The mean number of touchdown passes thrown is 20.4516 as shown below.
    • Although the arithmetic mean is not the only "mean" (there is also a geometric mean), it is by far the most commonly used.
    • Therefore, if the term "mean" is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean.

  • Average Value of a Function

    • The average of a list of numbers is a single number intended to typify the numbers in the list, which is called the arithmetic mean.
    • If $n$ numbers are given, each number denoted by $a_i$, where $i = 1, \cdots , n$, the arithmetic mean is the sum of all $a_i$ values divided by $n$:
    • The first mean value theorem for integration states that if $G : [a, b] \to R$ is a continuous function and $\varphi$ is an integrable function that does not change sign on the interval $(a, b)$, then there exists a number $x$ in $(a, b)$ such that:
    • The value $G(x)$ is the mean value of $G(t)$ on $[a, b] $ as we saw previously.

  • Which Average: Mean, Mode, or Median?

    • If elements in a sample data set increase arithmetically, when placed in some order, then the median and arithmetic mean are equal.
    • The mean is 2.5, as is the median.
    • However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as $\{1,2,4,8,16\}$, the median and arithmetic mean can differ significantly.
    • In this case, the arithmetic mean is 6.2 and the median is 4.
    • 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 (values that are very much larger or smaller than most of the values).

  • 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