Central tendency

In statistics, a central tendency (or, more commonly, a measure of central tendency) is a central or typical value for a probability distribution.^[1] 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.^[2]

The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value." ^[2][3]

The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysts may judge whether data has a strong or a weak central tendency based on its dispersion.

Measures of central tendency

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 (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. The median and the mode are the only measures of central tendency that can be used for ordinal data, in which values are ranked relative to each other but are not measured absolutely.
• Mode – the most frequent value in the data set. This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments.
• Geometric mean – the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale.
• Harmonic mean – the reciprocal of the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale.
• Weighted mean – an arithmetic mean that incorporates weighting to certain data elements.
• Truncated mean (or 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 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.

The Quadratic mean (often known as the root mean square) is useful in engineering, but is 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.

Solutions to variational problems

Several measures of central tendency can be characterized as solving a variational problem, in the sense of the calculus of variations, namely minimizing variation from the center. That is, given a measure of statistical dispersion, one asks for a measure of central tendency that minimizes variation: such that variation from the center is minimal among all choices of center. In a quip, "dispersion precedes location". In the sense of L^p spaces, the correspondence is:

L^p	dispersion	central tendency
L^1	average absolute deviation	median
L^2	standard deviation	mean
L^∞	maximum deviation	midrange

Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any other point. The uniqueness of this characterization of mean follows from convex optimization. Indeed, for a given (fixed) data set x, the function

$f_2(c) = \|x-c\|_2$

represents the dispersion about a constant value c relative to the L² norm. Because the function ƒ₂ is a strictly convex coercive function, the minimizer exists and is unique.

Note that the median in this sense is not in general unique, and in fact any point between the two central points of a discrete distribution minimizes average absolute deviation. The dispersion in the L¹ norm, given by

$f_1(c) = \|x-c\|_1$

is not strictly convex, whereas strict convexity is needed to ensure uniqueness of the minimizer. In spite of this, the minimizer is unique for the L^∞ norm.

Relationships between the mean, median and mode

For unimodal distributions the following bounds are known and are sharp:^[4]

$\frac{| \theta - \mu |}{ \sigma } \le \sqrt{ 3 } ,$

$\frac{| \nu - \mu |}{ \sigma } \le \sqrt{ 0.6 } ,$

$\frac{| \theta - \nu |}{ \sigma } \le \sqrt{ 3 } ,$

where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.

For every distribution,^[5][6]

$\frac{| \nu - \mu |}{ \sigma } \le 1.$

See also

• Expected value
• Location parameter

References

1. Weisberg H.F (1992) Central Tendency and Variability, Sage University Paper Series on Quantitative Applications in the Social Sciences, ISBN 0-8039-4007-6 p.2
2. Upton, G.; Cook, I. (2008) Oxford Dictionary of Statistics, OUP ISBN 978-0-19-954145-4 (entry for "central tendency")
3. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP for International Statistical Institute. ISBN 0-19-920613-9 (entry for "central tendency")
4. Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". Annals of Mathematical Statistics, 22 (3) 433–439
5. Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114
6. Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142