Central tendency: Difference between revisions
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In [[statistics]], the term '''central tendency''' relates to the way in which [[quantitative data]] tend to cluster around some value.<ref name=Dodge>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP for [[International |
In [[statistics]], the term '''central tendency''' relates to the way in which [[quantitative data]] tend to cluster around some value.<ref name=Dodge>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP for [[International Statistical Institute]]. ISBN 0-19-920613-9</ref><ref name=Upton>Upton, G.; Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP ISBN 978-0-19-954145-4</ref> If a central tendency does not exist for a population, its distribution may be [[bimodal distribution|bimodal]] or U-shaped. |
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If a central tendency exists, then '''measures of central tendency''' can be used to quantify this property. Such measures consist of a pair of values, one specifying the "central value" (a measure of location), and another specifying the [[statistical dispersion]] (a measure of spread).<ref name=Dodge/><ref name=Upton/> Statistics that measure central tendency can be used in [[descriptive statistics]] as [[summary statistic]]s for a data set. They can also sometimes be used [[estimator]]s of [[location parameter|location]] and [[scale parameter]]s of a [[statistical model]]. In practical statistical analysis, the terms here are often used before one has chosen even a preliminary form of analysis: thus an initial objective might be to "choose an appropriate measure of central tendency". |
If a central tendency exists, then '''measures of central tendency''' can be used to quantify this property. Such measures consist of a pair of values, one specifying the "central value" (a measure of location), and another specifying the [[statistical dispersion]] (a measure of spread).<ref name=Dodge/><ref name=Upton/> Statistics that measure central tendency can be used in [[descriptive statistics]] as [[summary statistic]]s for a data set. They can also sometimes be used [[estimator]]s of [[location parameter|location]] and [[scale parameter]]s of a [[statistical model]]. In practical statistical analysis, the terms here are often used before one has chosen even a preliminary form of analysis: thus an initial objective might be to "choose an appropriate measure of central tendency". |
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In the simplest cases, the idea of central tendency is used for [[univariate]] data. However, the term is also applied to [[Multivariate statistics|multivariate]] data and in situations where a [[Data transformation (statistics)|transformation]] of the data values for some or all dimensions would usually be considered necessary: in the latter cases, the notion of a "central location" is retained in converting a central tendency computed for the transformed data back to the original units. |
In the simplest cases, the idea of central tendency is used for [[univariate]] data. However, the term is also applied to [[Multivariate statistics|multivariate]] data and in situations where a [[Data transformation (statistics)|transformation]] of the data values for some or all dimensions would usually be considered necessary: in the latter cases, the notion of a "central location" is retained in converting a central tendency computed for the transformed data back to the original units. |
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Both "central tendency" and "measures of central tendency" apply to either [[statistical population]]s or to samples from a population. |
Both "central tendency" and "measures of central tendency" apply to either [[statistical population]]s or to samples from a population. The term "central tendency" dates from the late 1920s.<ref name=Upton/> |
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==Measures of central tendency== |
==Measures of central tendency== |
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Measures of central tendency consist of pairs of values, a measure of location and a measure of spread. Sometimes these measures occur as natural pairs. A common piaring is the [[arithmetic mean|mean]] and [[standard deviation]].<ref name=Upton/> |
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===Measures of location=== |
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The following may be applied to one-dimensional data, after transformation, although some of these involve their own implicit transformation of the data. |
The following may be applied to one-dimensional data, after transformation, although some of these involve their own implicit transformation of the data. |
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*[[Arithmetic mean]] (or simply, mean) – the sum of all measurements divided by the number of observations in the data set |
*[[Arithmetic mean]] (or simply, mean) – the sum of all measurements divided by the number of observations in the data set |
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*[[Median]] – the middle value that separates the higher half from the lower half of the data set |
*[[Median]] – the middle value that separates the higher half from the lower half of the data set |
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*[[Mode (statistics)|Mode]] – the most frequent value in the data set |
*[[Mode (statistics)|Mode]] – the most frequent value in the data set |
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*[[Geometric mean]] – the [[Nth root|''n''th root]] of the product of the data values |
*[[Geometric mean]] – the [[Nth root|''n''th root]] of the product of the data values, where there are ''n'' of these |
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*[[Harmonic mean]] – the [[Multiplicative inverse|reciprocal]] of the arithmetic mean of the reciprocals of the data values |
*[[Harmonic mean]] – the [[Multiplicative inverse|reciprocal]] of the arithmetic mean of the reciprocals of the data values |
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*[[Weighted mean]] – an arithmetic mean that incorporates weighting to certain data elements |
*[[Weighted mean]] – an arithmetic mean that incorporates weighting to certain data elements |
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*[[Winsorized mean]] – an arithmetic mean in which extreme values are replaced by values closer to the median. |
*[[Winsorized mean]] – an arithmetic mean in which extreme values are replaced by values closer to the median. |
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Any of the above may be applied to each dimension of multi-dimensional data |
Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. In addition, there is the |
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*[[Geometric median]] - which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. |
*[[Geometric median]] - which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions. |
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===Measures of spread=== |
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*[[ |
*[[Standard deviation]] |
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*[[ |
*[[Mean deviation]] |
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*[[Median absolute deviation]] |
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==References== |
==References== |
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Revision as of 12:13, 20 April 2013
In statistics, the term central tendency relates to the way in which quantitative data tend to cluster around some value.[1][2] If a central tendency does not exist for a population, its distribution may be bimodal or U-shaped.
If a central tendency exists, then measures of central tendency can be used to quantify this property. Such measures consist of a pair of values, one specifying the "central value" (a measure of location), and another specifying the statistical dispersion (a measure of spread).[1][2] Statistics that measure central tendency can be used in descriptive statistics as summary statistics for a data set. They can also sometimes be used estimators of location and scale parameters of a statistical model. In practical statistical analysis, the terms here are often used before one has chosen even a preliminary form of analysis: thus an initial objective might be to "choose an appropriate measure of central tendency".
In the simplest cases, the idea of central tendency is used for univariate data. However, the term is also applied to multivariate data and in situations where a transformation of the data values for some or all dimensions would usually be considered necessary: in the latter cases, the notion of a "central location" is retained in converting a central tendency computed for the transformed data back to the original units.
Both "central tendency" and "measures of central tendency" apply to either statistical populations or to samples from a population. The term "central tendency" dates from the late 1920s.[2]
Measures of central tendency
Measures of central tendency consist of pairs of values, a measure of location and a measure of spread. Sometimes these measures occur as natural pairs. A common piaring is the mean and standard deviation.[2]
Measures of location
The following may be applied to one-dimensional data, after transformation, although some of these involve their own implicit transformation of the data.
- Arithmetic mean (or simply, 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, where there are n of these
- Harmonic mean – the reciprocal of the arithmetic mean of the reciprocals 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 lowest data values have been discarded.
- Midrange – the arithmetic mean of the maximum and minimum values of a data set.
- Midhinge – the arithmetic mean of the two quartiles.
- Trimean – the weighted arithmetic mean of the median and two quartiles.
- Winsorized mean – an arithmetic mean in which extreme values are replaced by values closer to the median.
Any of the above may be applied to each dimension of multi-dimensional data, but the results may not be invariant to rotations of the multi-dimensional space. In addition, there is the
- Geometric median - which minimizes the sum of distances to the data points. This is the same as the median when applied to one-dimensional data, but it is not the same as taking the median of each dimension independently. It is not invariant to different rescaling of the different dimensions.
