Definition:Measure of Central Tendency

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Definition

A measure of central tendency is a central or typical value for a probability distribution or set of sample data.


The most important examples are often introduced at elementary school level conveniently alliterated as mean, mode and median:


Mean

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers.

The arithmetic mean of $x_1, x_2, \ldots, x_n$ is defined as:

\(\ds A_n\) \(:=\) \(\ds \dfrac 1 n \sum_{k \mathop = 1}^n x_k\)
\(\ds \) \(=\) \(\ds \dfrac {x_1 + x_2 + \cdots + x_n} n\)

That is, to find out the arithmetic mean of a set of numbers, add them all up and divide by how many there are.


Mode

Let $S$ be a set of quantitative data.

The mode of $S$ is the element of $S$ which occurs most often in $S$.

If there is more than one such element of $S$ which occurs equally most often, it is then understood that each of these is a mode.

If there is no element of $S$ which occurs more often (in the extreme case, all elements are equal) then $S$ has no mode.


Median

Let $S$ be a set of ordinal data.

Let $S$ be arranged in order of size.

The median is the element of $S$ that is in the middle of that ordered set.


Suppose there are an odd number of elements of $S$ such that $S$ has cardinality $2 n - 1$.

The median of $S$ in that case is the $n$th element of $S$.


Suppose there are an even number of elements of $S$ such that $S$ has cardinality $2 n$.

Then the middle of $S$ is not well-defined, and so the median of $S$ in that case is the arithmetic mean of the $n$th and $n + 1$th elements of $S$.


Also known as

Some sources refer to central tendency as centrality.

The term location can also be seen.


Also see

  • Results about measures of central tendency can be found here.


Sources

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