# Arithmetic Mean
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https://algebrica.org/arithmetic-mean/
Fetched from algebrica.org post 17885; source modified 2025-10-29T19:04:36.
## What is the arithmetic mean?
The **arithmetic mean** is the most common and intuitive form of average. As a special case within the broader family of power means it expresses the representative value of a data set by dividing the total sum of all observations by their [number](../types-of-numbers.md). Since it is grounded on additive aggregation, the arithmetic mean is ideal for describing quantities that combine linearly (for example, raw measurements or values that do not evolve proportionally or [exponentially](../exponential-function.md)). In essence, it identifies the equilibrium point of the distribution, the value around which the data tend to balance.
* * *
In general form, the arithmetic mean is expressed as:
\\[
M\_a = \\frac{1}{n}\\sum\_{i=1}^{n} x\_i
\\]
where \\( x\_1, x\_2, \\dots, x\_n \\) are the observed values and \\( n \\) is the total number of elements.
- The arithmetic mean can be applied to any set of [real numbers](../types-of-numbers.md), including negative and zero values.
- Because it is sensitive to extreme values, the arithmetic mean can be distorted by outliers, making other means, like the median or [geometric mean](../geometric-mean.md), more appropriate in some cases.
- The arithmetic mean is always greater than or equal to the geometric mean.
## Example 1
Let’s consider the following data set of five numerical values and let’s calculate the arithmetic mean:
\\(\\mathbf{xᵢ}\\)
**Values**
x₁
7.2
x₂
4.8
x₃
9.1
x₄
5.5
x₅
6.4
In this case, \\( n = 5 \\). Substituting the values, we get:
\\[
M\_a = \\frac{7.2 + 4.8 + 9.1 + 5.5 + 6.4}{5} = \\frac{33.0}{5} = 6.6
\\]
Hence, the arithmetic mean of the series is approximately:
\\[
M\_a = 6.6
\\]
## Weighted arithmetic mean
In some cases, not all data points contribute equally to the overall result. The weighted arithmetic mean extends the idea of the simple arithmetic mean by assigning a weight \\( w\_i \\) to each observation \\( x\_i \\), reflecting its relative importance or frequency within the dataset. It is defined as:
\\[
M\_{aw} = \\frac{\\sum\_{i=1}^{n} w\_i x\_i}{\\sum\_{i=1}^{n} w\_i}
\\]
where \\( x\_i \\) are the observed values and \\( w\_i > 0 \\) are their associated weights.
- The weighted arithmetic mean generalizes the simple arithmetic mean by introducing importance factors \\( w\_i \\).
- It ensures that larger or more relevant observations have a stronger influence on the final result.
- When all weights are equal, the weighted arithmetic mean reduces to the standard arithmetic mean.
## Example 2
Let’s consider a business case where a company wants to calculate the weighted arithmetic mean of its monthly sales. Each month has a different number of working days, which serve as the weights for the calculation.
**Month**
\\(x\_i\\) **\= daily sales in $**
\\(w\_i\\) **\= working days**
January
420
20
February
380
22
March
460
18
April
400
21
May
440
19
By applying the formula of the weighted arithmetic mean, we obtain:
\\[
\\begin{align} M\_{aw} &= \\frac{(420 \\times 20) + (380 \\times 22) + (460 \\times 18) + (400 \\times 21) + (440 \\times 19)}{20 + 22 + 18 + 21 + 19} \\[10pt] &= \\frac{8400 + 8360 + 8280 + 8400 + 8360}{100} \\[3pt] &= \\frac{41800}{100} \\[10pt] &= 418 \\end{align}
\\]
Hence, the weighted arithmetic mean of the company’s sales is \\(M\_{aw} = 418\\) $ per day.
