Mean of Numbers

For a list of $ n $ values $ X = \{x_1, x_2, \dots, x_n \} $. The arithmetic mean has for definition the sum of all the values divided by the number count of values $ n $. $$ \bar{x} = {1 \over n} \ sum_{i=1}^n{x_i} $$

When values are associaed with coefficients (digits or numbers), then use the weighted arithmetic mean.

For a list of $ n $ values $ X = \{x_1, x_2, \dots, x_n \} $. The geometric mean has for definition the $ n $-th root of the product of values. $$ \bar{x}_{geom} = \sqrt[n]{\prod_{i=1}^n{x_i}} $$

The geometric mean is often used to calculate an average interest rate.

For a list of n values $ X = \{x_1, x_2, \dots, x_n \} $. The harmonic mean has for definition the ratio of n to the sum of the inverse of the values. $$ \bar{x}_{harm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} $$

For a list of n values $ X = \{x_1, x_2, \dots, x_n \} $. The root mean square (or quadratic mean) has for definition the root of the sum of each value squared, divided by root of n: $$ \bar{x}_{quad} = \sqrt{\frac{1}{n}\sum_{i=1}^n{x_i^2}} $$

It is impossible to find the original numbers from the mean value. There are endless lists of possible numbers with the same mean value.