Editing Central tendency (section)

==Measures==

The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed.

; [[Arithmetic mean]] <span style="font-weight: normal;">or simply,</span> 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. The median and the mode are the only measures of central tendency that can be used for [[Level of measurement#Ordinal scale|ordinal data]], in which values are ranked relative to each other but are not measured absolutely.
; [[Mode (statistics)|Mode]]: the most frequent value in the data set. This is the only central tendency measure that can be used with [[Level of measurement#Nominal level|nominal data]], which have purely qualitative category assignments.
; [[Generalized mean]]: A generalization of the [[Pythagorean means]], specified by an exponent.
; [[Geometric mean]]: the [[Nth root|''n''th root]] of the product of the data values, where there are ''n'' of these. This measure is valid only for data that are measured on a strictly positive scale.
; [[Harmonic mean]]: the [[Multiplicative inverse|reciprocal]] of the arithmetic mean of the reciprocals of the data values. This measure is valid only for data that are measured either on a strictly positive or a strictly negative scale.
; [[Weighted arithmetic mean]]: an arithmetic mean that incorporates weighting to certain data elements.
; [[Truncated mean]] <span style="font-weight: normal;">or</span> trimmed mean: the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded.
; [[Interquartile mean]]: a truncated mean based on data within the [[interquartile range]].
; [[Midrange]]: the arithmetic mean of the maximum and minimum values of a data set.
; [[Midhinge]]: the arithmetic mean of the first and third [[quartile]]s.
; [[Quasi-arithmetic mean]]: A generalization of the [[generalized mean]], specified by a [[continuous function|continuous]] [[injective function|injective]] [[function (mathematics)|function]].
; [[Trimean]]: the weighted arithmetic mean of the median and two quartiles.
; [[Winsorized mean]]: an arithmetic mean in which [[Outlier|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.

; [[Geometric median]]: the point minimizing the sum of distances to a set of sample 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.
; [[Quadratic mean]] <span style="font-weight: normal;">(often known as the [[root mean square]])</span>: useful in engineering, but not often used in statistics. This is because it is not a good indicator of the center of the distribution when the distribution includes negative values.
; [[Simplicial depth]]: the probability that a randomly chosen [[simplex]] with vertices from the given distribution will contain the given center
; [[Tukey median]]: a point with the property that every halfspace containing it also contains many sample points