Average

An average is a way to find a middle value in a group of numbers. It helps us understand what is typical or common for that group. In mathematics, the average is often called the arithmetic mean. This means adding all the numbers together and then dividing by how many numbers there are.

Averages are useful in many areas of life. For example, they can tell us the typical temperature of a city, the average height of students in a class, or the average score on a test. By using averages, we can make sense of lots of data by finding one number that represents the whole group.

Besides the arithmetic mean, there are other ways to find an average, such as the median and the mode. The median is the middle number when numbers are arranged in order, and the mode is the number that appears most often. All these ways help us find a central value, but they do it a little differently.

Averages help us compare different groups and make decisions. They are important in science, sports, business, and many other fields. Understanding averages gives us a clear picture of what is normal or usual in a set of data.

Definitions

The average is usually found using the arithmetic mean. This is done by adding up all the numbers and then dividing by how many numbers there are. For example, the average of 2, 3, 4, 7, and 9 is found by adding them (25) and dividing by 5. The result is 5.

Sometimes, other ways to find an average are used. The median is the middle number when the numbers are put in order. It is useful when some numbers are much bigger or smaller than the others. The mode is the number that appears most often. These different ways help us see what is usual in a group of numbers.

General properties

All averages of a group are somewhere between the smallest and largest numbers in that group. If every number in the group is the same, then the average will also be that same number.

Most averages change in a steady way. If you make one number in the group larger or smaller, the average moves in the same direction. Also, if you multiply every number in the group by the same amount, the average will also be multiplied by that same amount. Usually, the average stays the same no matter what order the numbers are in.

List of possible averages

See also: Mean § Other means, and Central tendency § Solutions to variational problems

In math, an average is a number that shows what is typical for a group. There are many kinds of averages.

One kind is called the arithmetic mean. You add up all the numbers and then divide by how many there are.

Other kinds include the median. This is the middle number when the numbers are put in order. The mode is the number that appears most often. You can also make new kinds of averages using special math rules. For example, the harmonic mean and the geometric mean are two more types of averages.

Name	Equation or description	As solution to optimization problem
Arithmetic mean	x ¯ = 1 n ∑ i = 1 n x i = 1 n ( x 1 + ⋯ + x n ) {\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {1}{n}}(x_{1}+\cdots +x_{n})}	argmin x ∈ R ∑ i = 1 n ( x − x i ) 2 {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x-x_{i})^{2}}
Median	A middle value that separates the higher half from the lower half of the data set; may not be unique if the data set contains an even number of points	argmin x ∈ R ∑ i = 1 n \| x − x i \| {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}\|x-x_{i}\|}
Geometric median	A rotation invariant extension of the median for points in R d {\displaystyle \mathbb {R} ^{d}}	argmin x → ∈ R d ∑ i = 1 n \| \| x → − x → i \| \| 2 {\displaystyle {\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}\|\|{\vec {x}}-{\vec {x}}_{i}\|\|_{2}}
Tukey median	Another rotation invariant extension of the median for points in R d {\displaystyle \mathbb {R} ^{d}} —a point that maximizes the Tukey depth	argmax x → ∈ R d min u → ∈ R d ∑ i = 1 n ( { 1 , if ( x → i − x → ) ⋅ u → ≥ 0 0 , otherwise ) {\displaystyle {\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmax} }}\,{\underset {{\vec {u}}\in \mathbb {R} ^{d}}{\operatorname {min} }}\,\sum _{i=1}^{n}\left({\begin{cases}1,{\text{ if }}({\vec {x}}_{i}-{\vec {x}})\cdot {\vec {u}}\geq 0\\0,{\text{ otherwise}}\end{cases}}\right)}
Mode	The most frequent value in the data set	argmax x ∈ R ∑ i = 1 n ( { 1 , if x = x i 0 , if x ≠ x i ) {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmax} }}\,\sum _{i=1}^{n}\left({\begin{cases}1,{\text{ if }}x=x_{i}\\0,{\text{ if }}x\neq x_{i}\end{cases}}\right)}
Geometric mean	∏ i = 1 n x i n = x 1 ⋅ x 2 ⋯ x n n {\displaystyle {\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}={\sqrt[{n}]{x_{1}\cdot x_{2}\dotsb x_{n}}}}	argmin x ∈ R > 0 ∑ i = 1 n ( ln ⁡ ( x ) − ln ⁡ ( x i ) ) 2 , if x i > 0 ∀ i ∈ { 1 , … , n } {\displaystyle {\underset {x\in \mathbb {R} _{>0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(\ln(x)-\ln(x_{i}))^{2},\qquad {\text{if }}x_{i}>0\,\forall \,i\in \{1,\dots ,n\}}
Harmonic mean	n 1 x 1 + 1 x 2 + ⋯ + 1 x n {\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}}	argmin x ∈ R ≠ 0 ∑ i = 1 n ( 1 x − 1 x i ) 2 {\displaystyle {\underset {x\in \mathbb {R} _{\neq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}\left({\frac {1}{x}}-{\frac {1}{x_{i}}}\right)^{2}}
Contraharmonic mean	x 1 2 + x 2 2 + ⋯ + x n 2 x 1 + x 2 + ⋯ + x n {\displaystyle {\frac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{{x_{1}}+{x_{2}}+\cdots +{x_{n}}}}}	argmin x ∈ R ∑ i = 1 n x i ( x − x i ) 2 {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}x_{i}(x-x_{i})^{2}}
Lehmer mean	∑ i = 1 n x i p ∑ i = 1 n x i p − 1 {\displaystyle {\frac {\sum _{i=1}^{n}x_{i}^{p}}{\sum _{i=1}^{n}x_{i}^{p-1}}}}	argmin x ∈ R ∑ i = 1 n x i p − 1 ( x − x i ) 2 {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}x_{i}^{p-1}(x-x_{i})^{2}}
Quadratic mean (or RMS)	1 n ∑ i = 1 n x i 2 = 1 n ( x 1 2 + x 2 2 + ⋯ + x n 2 ) {\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}}	argmin x ∈ R ≥ 0 ∑ i = 1 n ( x 2 − x i 2 ) 2 {\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{2}-x_{i}^{2})^{2}}
Cubic mean	1 n ∑ i = 1 n x i 3 3 = 1 n ( x 1 3 + x 2 3 + ⋯ + x n 3 ) 3 {\displaystyle {\sqrt[{3}]{{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{3}}}={\sqrt[{3}]{{\frac {1}{n}}\left(x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}\right)}}}	argmin x ∈ R ≥ 0 ∑ i = 1 n ( x 3 − x i 3 ) 2 , if x i ≥ 0 ∀ i ∈ { 1 , … , n } {\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{3}-x_{i}^{3})^{2},\qquad {\text{if }}x_{i}\geq 0\,\forall \,i\in \{1,\dots ,n\}}
Generalized mean	1 n ⋅ ∑ i = 1 n x i p p {\displaystyle {\sqrt[{p}]{{\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}}}}	argmin x ∈ R ≥ 0 ∑ i = 1 n ( x p − x i p ) 2 , if x i ≥ 0 ∀ i ∈ { 1 , … , n } {\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{p}-x_{i}^{p})^{2},\qquad {\text{if }}x_{i}\geq 0\,\forall \,i\in \{1,\dots ,n\}}
Quasi-arithmetic mean	f − 1 ( 1 n ∑ k = 1 n f ( x k ) ) {\displaystyle f^{-1}\left({\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\right)}	argmin x ∈ dom ⁡ ( f ) ∑ i = 1 n ( f ( x ) − f ( x i ) ) 2 , if f {\displaystyle {\underset {x\in \operatorname {dom} (f)}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(f(x)-f(x_{i}))^{2},\qquad {\text{if }}f} is monotonic
Weighted mean	∑ i = 1 n w i x i ∑ i = 1 n w i = w 1 x 1 + w 2 x 2 + ⋯ + w n x n w 1 + w 2 + ⋯ + w n {\displaystyle {\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}}	argmin x ∈ R ∑ i = 1 n w i ( x − x i ) 2 {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}w_{i}(x-x_{i})^{2}}
Truncated 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 special case of the truncated mean, using the interquartile range. A special case of the inter-quantile truncated mean, which operates on quantiles (often deciles or percentiles) that are equidistant but on opposite sides of the median.
Midrange	1 2 ( max x + min x ) {\displaystyle {\frac {1}{2}}\left(\max x+\min x\right)}	argmin x ∈ R max i ∈ { 1 , … , n } \| x − x i \| {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,{\underset {i\in \{1,\dots ,n\}}{\operatorname {max} }}\,\|x-x_{i}\|}
Winsorized mean	Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain
Medoid	A representative object of a set X {\displaystyle {\mathcal {X}}} of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function d {\displaystyle d} .	argmin y ∈ X ∑ i = 1 n d ( y , x i ) {\displaystyle {\underset {y\in {\mathcal {X}}}{\operatorname {argmin} }}\sum _{i=1}^{n}d(y,x_{i})}

Moving average

Main article: Moving average

When we have a list of numbers over time, like prices or temperatures, it can be hard to see the pattern. A moving average helps make the pattern clearer. We pick a number, say n, and find the average of the first n numbers. Then we move one step forward, drop the oldest number, and add a new one. We repeat this again and again. This shows the overall trend in the numbers more clearly. There are also more complex ways to do this using a weighted average, where some numbers matter more than others.

History

The idea of an average has been used for a long time. The first known use of the arithmetic mean, a type of average, was in the sixteenth century. Scientists used it to get better measurements when they had several results that were not exact. For example, astronomers used averages to find the true position of planets or the size of the moon from many measurements.

The word "average" comes from old trading practices. Long ago, when ships faced storms and had to throw goods overboard to stay afloat, the loss was shared equally among all merchants on the ship. This idea of sharing losses equally helped create the meaning of "average" as we know it today.

Averages as a rhetorical tool

The word "average" is used a lot in everyday talk. This can sometimes make it hard to understand what numbers really mean. There are different ways to find an average, like the arithmetic mean, median, or mode. These ways can give different answers to the same question.

A teacher named Daniel Libertz from the University of Pittsburgh says that because of this, people sometimes ignore statistics when they argue. But he also says we should not ignore averages and other numbers. Instead, we should use them carefully and think about what they mean. He suggests that when we see statistics, we should talk about what they might mean together, instead of just saying one answer is right. Often, numbers are given to help everyone understand better.