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Encyclopedia > Measures of central tendency

In mathematics, an average or central tendency of a set (list) of data refers to a measure of the "middle" of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency. The most common method, and the one generally referred to simply as the average, is the arithmetic mean. Please see the table of mathematical symbols for explanations of the symbols used. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Data is the plural of datum. ... Descriptive statistics is a branch of statistics that denotes any of the many techniques used to summarize a set of data. ... In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). ... The following table lists many specialized symbols commonly used in mathematics. ...

In statistics, the term central tendency is used in some fields of empirical research to refer to what statisticians sometimes call "location". A "measure of central tendency" is either a location parameter or a statistic used to estimate a location parameter. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Empirical research is any activity that uses direct or indirect observation as its test of reality. ... In statistics, if a family of probabiblity densities parametrized by a scalar- or vector-valued parameter μ is of the form fμ(x) = f(x − μ) then μ is called a location parameter, since its value determines the location of the probability distribution. ... A statistic (singular) is the result of applying a statistical algorithm to a set of data. ...

A list of measures of central tendency

There are several different kinds of calculations for central tendency, the kind of calculation that should be used depends on the type of data (level of measurement) and purpose for which the central tendency is being calculated. The level of measurement of a variable in mathematics and statistics is a classification that was proposed in order to describe the nature of information contained within numbers assigned to objects and, therefore, within the variable. ...

  • Arithmetic mean - the sum of all measurements divided by the number of observations in the data set
  • Median - the middle value that separates the higher half from the lower half of the data set
  • Mode - the most frequent value in the data set
  • Geometric mean - the nth root of the product of the data values
  • Harmonic mean - the reciprocal of the arithmetic mean of the reciprocals of the data values
  • Generalized mean - the nth root of the arithmetic mean of the nth powers of the data values
  • Weighted mean - an arithmetic mean that incorporates weighting to certain data elements
  • Truncated mean - the arithmetic mean of data values after a certain number or proportion of the highest and lowers data values have been discarded
  • Interquartile mean - a special case of the truncated mean
  • Midrange - the arithmetic mean of the highest and lowest values of the data or distribution.

In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set (cardinality). ... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... In mathematics, the nth root or radical of the non-negative real number a, written as , is the unique non-negative real number b such that bn = a. ... In mathematics, the harmonic mean is one of several methods of calculating an average. ... In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... A generalized mean or power mean is an abstraction of the arithmetic, geometric and harmonic means. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic mean. ... A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. ... The interquartile mean (IQM) is a statistical measure of central tendency, much like the mean (in more popular terms called the average), the median, and the mode. ... The midrange of a set of statistical data values is the arithmetic mean of the smallest and largest values in the set. ...

Other averages

Other more sophisticated averages are: trimean, trimedian, and normalised mean, to name a few. These are usually more representative of the whole dataset.

One can create one's own average metric using the generic formula y = f -1((f(x1)+f(x2)+...+f(xn))/n) where f is any invertible function. For example, expmean (exponential mean) is a mean using the function f(x) = e^x and due to its nature, it is biased towards the higher values.

The only significant reason why the arithmetic mean (classical average) is generally used in scientific papers is that there are various (statistical) tests which can be applied to test the statistical significance of the results, as well as the correlations that are explored through these metrics.

See also

  Results from FactBites:
Measures of Central Tendency: Mean, Median & Mode (1170 words)
Notice that not all measures of central tendency are appropriate for all kinds of variables.
The rule is that, as the symmetry of a distribution increases, the three measures of central tendency converge on the same value.
As the asymmetry or skewness of a distribution increases, the three measures of central tendency diverge systematically.
  More results at FactBites »



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