arithmetic mean

(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.

  • 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.

  • 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).

  • Shapes of Sampling Distributions

    • This fact holds true when we repeatedly take samples of a given size from a population and calculate the arithmetic mean for each sample.
    • An alternative to the sample mean is the sample median.
    • When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal; although, it may be close for large sample sizes.
    • Sample distributions, when the sampling statistic is the mean, are generally expected to display a normal distribution.

  • Additional Measures of Central Tendency

    • where the symbol Π means to multiply.
    • The arithmetic mean of the three logs is 1.
    • The anti-log of this arithmetic mean of 1 is the geometric mean.
    • Therefore the trimmed mean is a hybrid of the mean and the median.
    • This mean is 20.16.

  • The Root-Mean-Square

    • The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers.
    • The root-mean-square, also known as the quadratic mean, is a statistical measure of the magnitude of a varying quantity, or set of numbers.
    • Standard deviation being the root-mean-square of a signal's variation about the mean, rather than about 0, the DC component is removed (i.e. the RMS of the signal is the same as the standard deviation of the signal if the mean signal is zero).
    • $A$ is the arithmetic mean of scalars $a$ and $b$.
    • $G$ is the geometric mean, $H$ is the harmonic mean, $Q$ is the quadratic mean (also known as root-mean-square).

  • What Is a Sampling Distribution?

    • You would not expect your sample mean to be equal to the mean of all women in Houston.
    • Assume we repeatedly take samples of a given size from this population and calculate the arithmetic mean for each sample.
    • This statistic is then called the sample mean.
    • An alternative to the sample mean is the sample median.
    • You would not expect your sample mean to be equal to the mean of all women in Houston.