Central tendency

# Central tendency

Overview
In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value. In practical statistical analysis, the terms 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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Encyclopedia
In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value. In practical statistical analysis, the terms 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 measure of central tendency is an average
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

of a set of measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...

s, the word average being variously construed as mean, median, or other measure of location, depending on the context. However, the term is applied to multidimensional data as well as to univariate
Univariate
In mathematics, univariate refers to an expression, equation, function or polynomial of only one variable. Objects of any of these types but involving more than one variable may be called multivariate...

data and in situations where a transformation
Data transformation (statistics)
In statistics, data transformation refers to the application of a deterministic mathematical function to each point in a data set — that is, each data point zi is replaced with the transformed value yi = f, where f is a function...

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 an "average" computed for the transformed data back to the original units. In addition, there are several different kinds of calculations for central tendency, where the kind of calculation depends on the type of data (level of measurement
Level of measurement
The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement"...

).

Both "central tendency" and "measure of central tendency" apply to either statistical population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

s or to samples from a population.

## Basic measures of central tendency

The following may be applied to individual dimensions of multidimensional data, after transformation, although some of these involve their own implicit transformation of the data.
• Arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

- the sum of all measurements divided by the number of observations in the data set
• Median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

- the middle value that separates the higher half from the lower half of the data set
• Mode
Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....

- the most frequent value in the data set
• Geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

- the nth root
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

of the product of the data values
• Harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

- the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

of the arithmetic mean of the reciprocals of the data values
• Weighted mean
Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...

- an arithmetic mean that incorporates weighting to certain data elements
• Truncated mean
Truncated mean
A truncated mean or trimmed mean is a statistical measure of central tendency, much like the mean and median. It involves the calculation of the mean after discarding given parts of a probability distribution or sample at the high and low end, and typically discarding an equal amount of both.For...

- 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
Midhinge
In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.Equivalently, it is the 25% trimmed mid-range; it is an L-estimator....

- the arithmetic mean of the two quartiles.
• Trimean
Trimean
In statistics the trimean , or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles:This is equivalent to the average of the median and the midhinge:...

- the weighted arithmetic mean of the median and two quartiles.
• Winsorized mean
Winsorized mean
A Winsorized mean is a Winsorized statistical measure of central tendency, much like the mean and median, and even more similar to the truncated mean...

- an arithmetic mean in which extreme values are replaced by values closer to the median.