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.

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.

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 $

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:

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

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
}