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, which 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 most commonly used definition of the average is the arithmetic mean, which is the sum of numbers divided by how many numbers there are. For example, the average of the numbers 2, 3, 4, 7, and 9 is found by adding them together (25) and dividing by 5, giving us 5.

Sometimes, other ways to find an average are used. The median is the middle value when numbers are put in order, and it is often used when some numbers are much larger or smaller than the rest. The mode is the number that appears most often in a group. Each of these ways helps us understand what is typical in a set 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 value that represents the center or typical number in a group. There are many types of averages. One is called the arithmetic mean, where you add up all the numbers and then divide by how many there are.

Other types include the median (the middle number when they are ordered) and the mode (the number that appears most often). You can also create 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 series of data points over time, like daily stock prices or yearly temperatures, we might want to make the pattern easier to see. One way to do this is by using a moving average. We pick a number, say n, and then calculate the average of the first n values. We then move forward by one step, dropping the oldest value and adding a new one, and repeat this process. This helps to show the overall trend or repeating patterns in the data more clearly. There are also more complex ways to do this using a weighted average, where some values are given more importance than others.

History

The idea of an average has been used for a long time. The first known use of the arithmetic mean, which is a type of average, was in the sixteenth century. Scientists used it to find more accurate measurements when they had several results that were a bit off. For example, astronomers used averages to figure out the true position of planets or the size of the moon from many measurements that had small errors.

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

Because the word "average" is used in everyday talk, it can sometimes make it hard to see what data really means. Different ways of finding an average, like the arithmetic mean, median, or mode, can give different answers to the same question.

A teacher named Daniel Libertz from the University of Pittsburgh says that because of this, people often ignore statistics in arguments. But he also says we should not ignore averages and other statistics. 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, details and numbers are given to help everyone understand better.