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Mean of Numbers

Tool to calculate the different means of a number list. The mathematic mean of a list of numbers is one of the statistical representations that can illustrate the distribution of the numbers in the list.

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Mean of Numbers -

Tag(s) : Statistics, Data Processing

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# Mean of Numbers

## Mean Calculator

 Compute Arithmetic Mean (Classic) Harmonic Mean Quadratic Mean

### How to compute an arithmetic mean?

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

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

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

### How to compute a geometric 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.

Example: The list of 3 values 1, 1.5, 2 has for geometric mean $\sqrt{ 1 \times 1.5 \times 2 } \approx 1.4422$

### How to compute an harmonic mean?

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. $$\bar{x}_{harm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}$$

The harmonic mean is often used to compute a speed average.

Example: The list of speed values 50 and 100 has for harmonic mean $2/(1/50+1/100) = 66.67$

### How to compute an quadratic mean (Root Mean Square)?

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: $$\bar{x}_{quad} = \sqrt{\frac{1}{n}\sum_{i=1}^n{x_i^2}}$$

The RMS is used in electricity to calculate the effective value.

Example: The list of 3 values 4,5 and 6, this distribution has for quadratic mean $\sqrt{\frac{4^2+5^2+6^2}{3}} = \approx 5.06$

### How to find back numbers from the mean value?

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

Example: 10,20,30 has the same arithmetic mean as -100,0,1,99,100

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