Arithmetic mean

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In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪkˈmn/ air-ith-MET-ik) or arithmetic average, or simply just the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. [1] The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

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In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population.

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). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of "middle", and robust statistics, such as the median, may provide better description of central tendency.

Definition

Given a data set , the arithmetic mean (or mean or average), denoted (read bar), is the mean of the values . [2]

The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation, divided by the total number of observations. Symbolically, if we have a data set consisting of the values , then the arithmetic mean is defined by the formula:

[3]

(for an explanation of the summation operator, see summation.)

For example, consider the monthly salary of 10 employees of a firm: 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400. The arithmetic mean is

If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean , and denoted by the Greek letter . If the data set is a statistical sample (a subset of the population), then we call the statistic resulting from this calculation a sample mean (which for a data set is denoted as ).

The arithmetic mean can be similarly defined for vectors in multiple dimension, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (coefficients sum to 1), it can be defined on a convex space, not only a vector space.

Motivating properties

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

Contrast with median

The arithmetic mean may be contrasted with the median. The median is defined such that no more than half the values are larger than, and no more than half are smaller than, the median. If elements in the data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample . The average is , as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as , the median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2, while the median is 4. In general, the average value can vary significantly from most values in the sample, and can be larger or smaller than most of them.

There are applications of this phenomenon in many fields. For example, since the 1980s, the median income in the United States has increased more slowly than the arithmetic average of income. [4]

Generalizations

Weighted average

A weighted average, or weighted mean, is an average in which some data points count more heavily than others, in that they are given more weight in the calculation. [5] For example, the arithmetic mean of and is , or equivalently . In contrast, a weighted mean in which the first number receives, for example, twice as much weight as the second (perhaps because it is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as . Here the weights, which necessarily sum to the value one, are and , the former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all the weights are equal to each other (equal to in the above example, and equal to in a situation with numbers being averaged).

Continuous probability distributions

Comparison of two log-normal distributions with equal median, but different skewness, resulting in different means and modes Comparison mean median mode.svg
Comparison of two log-normal distributions with equal median, but different skewness, resulting in different means and modes

If a numerical property, and any sample of data from it, could take on any value from a continuous range, instead of, for example, just integers, then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range, even when the naive probability for a sample number taking one certain value from infinitely many is zero. The analog of a weighted average in this context, in which there are an infinite number of possibilities for the precise value of the variable in each range, is called the mean of the probability distribution. A most widely encountered probability distribution is called the normal distribution; it has the property that all measures of its central tendency, including not just the mean but also the aforementioned median and the mode (the three M's [6] ), are equal to each other. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here.

Angles

Particular care must be taken when using cyclic data, such as phases or angles. Naively taking the arithmetic mean of 1° and 359° yields a result of 180°. This is incorrect for two reasons:

In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (viz., define the mean as the central point: the point about which one has the lowest dispersion), and redefine the difference as a modular distance (i.e., the distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°).

Proof without words of the inequality of arithmetic and geometric means:
PR is a diameter of a circle centred on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem, triangle PGR's altitude GQ is the geometric mean. For any ratio a:b, AO >= GQ. AM GM inequality visual proof.svg
Proof without words of the inequality of arithmetic and geometric means:
PR is a diameter of a circle centred on O; its radius AO is the arithmetic mean of a and b. Using the geometric mean theorem,triangle PGR's altitude GQ is the geometric mean. For any ratio a:b,AO GQ.

Symbols and encoding

The arithmetic mean is often denoted by a bar, (a.k.a vinculum or macron), for example as in (read bar). [2]

Some software (text processors, web browsers) may not display the x̄ symbol properly. For example, the x̄ symbol in HTML is actually a combination of two codes - the base letter x plus a code for the line above (̄ or ¯). [7]

In some texts, such as pdfs, the x̄ symbol may be replaced by a cent (¢) symbol (Unicode &#162), when copied to text processor such as Microsoft Word.

See also

Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two positive numbers a and b QM AM GM HM inequality visual proof.svg
Geometric proof without words that max(a,b)> root mean square (RMS) or quadratic mean (QM)> arithmetic mean (AM)> geometric mean (GM)> harmonic mean (HM)>min(a,b) of two positive numbers a and b

Related Research Articles

In statistics, a central tendency is a central or typical value for a probability distribution. It may also be called a center or location of the distribution. Colloquially, measures of central tendency are often called averages. The term central tendency dates from the late 1920s.

In probability theory, the expected value of a random variable , often denoted , , or , is a generalization of the weighted average, and is intuitively the arithmetic mean of a large number of independent realizations of . The expectation operator is also commonly stylized as or . The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment. Expected value is a key concept in economics, finance, and many other subjects.

Geometric mean N-th root of the product of n numbers

In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x1, x2, ..., xn, the geometric mean is defined as

In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Sometimes it is appropriate for situations when the average rate is desired.

Median Middle quantile of a data set or probability distribution

In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result.

There are several kinds of mean in mathematics, especially in statistics.

Normal distribution Probability distribution

In probability theory, a normaldistribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is

Standard deviation Measure of the amount of variation or dispersion of a set of values

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.

Variance Statistical measure of how far values spread from their average

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

The weighted arithmetic mean is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory.

In mathematics and its applications, the root mean square is defined as the square root of the mean square . The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle.

Law of large numbers Averages of repeated trials converge to the expected value

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.

Pearson correlation coefficient Measure of linear correlation

In statistics, the Pearson correlation coefficient ― also known as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ― is a measure of linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus it is essentially a normalised measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationship or correlation. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1.

Moving average type of statistical measure over subsets of a dataset

In statistics, a moving average is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is a type of finite impulse response filter. Variations include: simple, cumulative, or weighted forms.

The following is a glossary of terms used in the mathematical sciences statistics and probability.

In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, , where p = 2.

In statistics, the bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator. Bias can also be measured with respect to the median, rather than the mean, in which case one distinguishes median-unbiased from the usual mean-unbiasedness property. Bias is a distinct concept from consistency. Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more.

In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because 360° equals 0° modulo a full cycle. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of circular statistics and of statistics of non-Euclidean spaces.

The sample mean and the sample covariance are statistics computed from a sample of data on one or more random variables.

References

  1. Jacobs, Harold R. (1994). Mathematics: A Human Endeavor (Third ed.). W. H. Freeman. p. 547. ISBN   0-7167-2426-X.
  2. 1 2 3 Medhi, Jyotiprasad (1992). Statistical Methods: An Introductory Text. New Age International. pp. 53–58. ISBN   9788122404197.
  3. Weisstein, Eric W. "Arithmetic Mean". mathworld.wolfram.com. Retrieved 21 August 2020.
  4. Krugman, Paul (4 June 2014) [Fall 1992]. "The Rich, the Right, and the Facts: Deconstructing the Income Distribution Debate". The American Prospect.
  5. {{Cite web|title=Mean {{!}tannica.com/science/mean|access-date=2020-08-21|website=Encyclopedia Britannica|language=en}}
  6. Thinkmap Visual Thesaurus (30 June 2010). "The Three M's of Statistics: Mode, Median, Mean June 30, 2010". www.visualthesaurus.com. Retrieved 3 December 2018.
  7. "Notes on Unicode for Stat Symbols". www.personal.psu.edu. Retrieved 14 October 2018.
  8. If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
    Using Pythagoras' theorem, QC² = QO² + OC² QC = QO² + OC² = QM.
    Using Pythagoras' theorem, OC² = OG² + GC² GC = OC² OG² = GM.
    Using similar triangles, HC/GC = GC/OC HC = GC²/OC = HM.

Further reading