Central tendency measures

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In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. The most common method is the arithmetic mean, but there are many other types of averages. (An axiomatic approach to averages is provided by John Bibby (1974) “Axiomatisations of the average and a further generalization of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63–65.)

Central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. Central tendency is a descriptive statistic analogous to center of mass in physical terms. The term 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.

There are several different kinds of calculations for central tendency, the kind of calculation depending on the type of data (level of measurement) for which the central tendency is being calculated.

A list of measures of central tendency

• 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.

See also

• Average
• Mean
• Normal distribution
• Population (statistics)
• Probability
• Standard deviation
• Standard error (statistics)
• Summary statistics
• Statistical theory
• T-test
• Variability measurement