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Arithmetic Mean

Tool to compute a mean from numbers. The arithmetic mean (or commonly the mean/average) of a list of numbers is a statistical representation showing the distribution of the numbers in the list.

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Arithmetic Mean -

Tag(s) : Statistics, Data Processing

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# Arithmetic Mean

## Weighted (Arithmetic) Mean Calculator

### How to compute an arithmetic mean?

The arithmetic mean is the mathematical term to define what is commonly called mean or >average.

Take a list of $n$ values (digits, numbers, natural or real numbers) $X = \{x_1, x_2, \dots, x_n \}$. The arithmetic mean is defined by the sum of the values divided by the number of values $n$.

$$\bar{x} = {1 \over n} \ sum_{i=1}^n{x_i}$$

The arithmetic mean is generally used to give a general trend to a set of homogeneous data, possibly bounded, for example school marks between 0 and 20.

Example: The list of $4$ numbers $\{ 12, 14, 18, 13 \}$ its average value is $(12+14+18+13)/4 = 14.25$

This definition can be extended to a function, see function mean calculator.

To find the central value of a list, see median calculator.

When the values are assigned coefficients, see the weighted mean calculator.

### How to change of scale an arithmetic mean?

For a list of $n$ values $X = \{x_1, x_2, \dots, x_n\}$ having a mean $\bar {x} = m$, two possible scale changes:

If all $x_i$ are increased by $a$ then the arithmetic mean is also increased by $a$ and becomes $\bar{x} = m + a$

If all $x_i$ are multiplied by $a$ then the arithmetic mean is also multiplied by $a$ and becomes $\bar{x} = m \times a$

### How to calculate the average of 2 lists knowing their means?

To perform a sort of addition of an average of 2 lists: a list $A$ of $n_1$ values and average $\bar {A} = m_1$ and a list $B$ of $n_2$ values and average $\bar {B} = m_2$, then the average of the 2 lists is given by the formula:

$$\overline{A+B} = \frac{n_1 m_1 + n_2 m_2}{n_1+n_2}$$

### How to code an arithmetic mean (source code)?

Classic method: // pseudocode function mean(array[N]) { sum = 0 for i = 0; i < N ; i++ { sum += array[i] } return sum / N }

Optimized method for floats (avoid big values) : // pseudocode function mean(array[N]) { m = 0 for i = 0; i < N; i++) { m += (array[i] - m)/(i+1); } return m }

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