# Definition:Measure of Central Tendency

## 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:

- $\ds A_n := \dfrac 1 n \sum_{k \mathop = 1}^n x_k$

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 quantitative 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 element 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 element 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**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**centrality** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**central tendency** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**centrality** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**central tendency**

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