The arithmetic mean, the geometric mean and the harmonic mean as well as the square and the cubic mean and the median are calculated with the mean value calculator.

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Arithmetic mean:

Geometric mean:

Harmonic mean:

Square mean:

Cubic mean:

Median:

Points:

Annotation:

Display standard deviation σ:

Display weight circle:

Mean Values

Standard Deviation σ

Arithmetic mean

Geometric mean

Harmonic mean

Square mean

Cubic mean

Median

The following table can be used to enter the values used for the mean value calculation. The inputs for the values and the weights are pairwise in a row: a_{1}, a_{2}, a_{3}, ... and the weights w_{1}, w_{2}, w_{3}, ...

If all weights are equal to 1, this corresponds to the unweighted mean value.

An alternative input is possible with load data from file. The values may be separated comma or space or semicolon. The values must be given pairwise x_{1}, w_{1}, x_{2}, w_{2}...

Load from file:

$${A}_{n}\left({a}_{i}\right)=\frac{1}{n}\sum _{i=1}^{n}{a}_{i}=\frac{{a}_{1}+{a}_{2}+...+{a}_{n}}{n}$$

The arithmetic means satisfies the equation n⋅A = a_{1} + a_{2} + ... + a_{n}. The arithmetic mean is also called average. The following figure shows a geometric construction of the arithmetic mean. In the example, the three sections with the total length 12 are replaced by the same number of sections with the average length A = 4.

The weighting factors w _{i} can be used to generalize the arithmetic mean to the weighted arithmetic mean.

$${A}_{n}\left({w}_{i},{a}_{i}\right)=\frac{\sum _{i=1}^{n}{w}_{i}{a}_{i}}{\sum _{i=1}^{n}{w}_{i}}=\frac{{w}_{1}{a}_{1}+{w}_{2}{a}_{2}+...+{w}_{n}{a}_{n}}{\sum _{i=1}^{n}{w}_{i}}$$

The geometric mean is an average for positive values. The geometric mean uses the product of the set opposed to the sum in the arithmetic average.

$${G}_{n}\left({a}_{i}\right)=\sqrt[n]{\underset{i=1}{\overset{n}{\Pi}}{a}_{i}}=\sqrt[n]{{a}_{1}\cdot {a}_{2}\cdot ...\cdot {a}_{n}}$$

For example for the two values 1 and 9 the arithmetic mean is 5 so the difference from the arithmetic mean to the two values is 4. The geometric mean is 3 so the distance is given by the same factor instead of the difference. 1 multiplied with 3 gives 3 and 3 multiplied with 3 results to 9.

The weighting factors w _{i} can be used to generalize the geometric mean to the weighted geometric mean.

$${G}_{n}\left({w}_{i},{a}_{i}\right)=\sqrt[\sum _{i=1}^{n}{w}_{i}]{\underset{i=1}{\overset{n}{\Pi}}{a}_{i}^{{w}_{i}}}=\sqrt[\sum _{i=1}^{n}{w}_{i}]{{a}_{1}^{{w}_{1}}\cdot {a}_{2}^{{w}_{2}}\cdot ...\cdot {a}_{n}^{{w}_{n}}}$$

The harmonic mean is used for averages rates and ratios. For instance, if a vehicle travels a certain distance d outbound at a speed x (e.g. 60 km/h) and returns the same distance at a speed y (e.g. 20 km/h), then its average speed is the harmonic mean of x and y (30 km/h) – not the arithmetic mean (40 km/h).

$${H}_{n}\left({a}_{i}\right)=\frac{n}{\sum _{i=1}^{n}\frac{1}{{a}_{i}}}=\frac{n}{\frac{1}{{a}_{1}}+\frac{1}{{a}_{2}}+...+\frac{1}{{a}_{n}}}$$

The weighting factors w_{i} can be used to generalize the harmonic mean to the weighted harmonic mean.

$${H}_{n}\left({w}_{i},{a}_{i}\right)=\frac{\sum _{i=1}^{n}{w}_{i}}{\sum _{i=1}^{n}\frac{{w}_{i}}{{a}_{i}}}=\frac{{w}_{1}+{w}_{2}+...+{w}_{n}}{\frac{{w}_{1}}{{a}_{1}}+\frac{{w}_{2}}{{a}_{2}}+...+\frac{{w}_{n}}{{a}_{n}}}$$

The root mean square (RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers). In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.

$${Q}_{n}\left({a}_{i}\right)=\sqrt{\frac{1}{n}\sum _{i=1}^{n}{a}_{i}^{2}}=\sqrt{\frac{{a}_{1}^{2}+{a}_{2}^{2}+...+{a}_{n}^{2}}{n}}$$

The weighting factors w_{i} can be used to generalize the quadratic mean to the weighted square mean.

$${Q}_{n}\left({w}_{i},{a}_{i}\right)=\sqrt{\frac{\sum _{i=1}^{n}{w}_{i}{a}_{i}^{2}}{\sum _{i=1}^{n}{w}_{i}}}=\sqrt{\frac{{w}_{1}{a}_{1}^{2}+{w}_{2}{a}_{2}^{2}+...+{w}_{n}{a}_{n}^{2}}{\sum _{i=1}^{n}{w}_{i}}}$$

The cubic mean is used for example for predicting the life expectancy of machine parts.

$${K}_{n}\left({a}_{i}\right)=\sqrt[3]{\frac{1}{n}\sum _{i=1}^{n}{a}_{i}^{3}}=\sqrt[3]{\frac{{a}_{1}^{3}+{a}_{2}^{3}+...+{a}_{n}^{3}}{n}}$$

The weighting factors w_{i} can be used to generalize the cubic mean to the weighted cubic mean.

$${Q}_{n}\left({w}_{i},{a}_{i}\right)=\sqrt[3]{\frac{\sum _{i=1}^{n}{w}_{i}{a}_{i}^{3}}{\sum _{i=1}^{n}{w}_{i}}}=\sqrt[3]{\frac{{w}_{1}{a}_{1}^{3}+{w}_{2}{a}_{2}^{3}+...+{w}_{n}{a}_{n}^{3}}{\sum _{i=1}^{n}{w}_{i}}}$$

For ascending sorted values a_{i}, the median is defined by:

$${a}_{1}\le {a}_{2}\le ...\le {a}_{n}$$

$${M}_{n}\left({a}_{i}\right)=\left\{\begin{array}{cc}{a}_{\frac{n+1}{2}}& \text{n : odd}\\ \frac{{a}_{\frac{n}{2}}+{a}_{\frac{n}{2}+1}}{2}& \text{n : even}\end{array}\right.$$

For values sorted ascending (a_{i}, w_{i}) sorted according to a_{i}, with the weights w_{i}∈ℝ_{+}, the weighted median is defined as follows:

$${a}_{1}\le {a}_{2}\le ...\le {a}_{n}$$

$$W=\frac{\sum _{i=1}^{n}{w}_{i}+1}{2}$$

The following two conditions mean that the sum of the weight up to the index i_{u} is less than W, and the sum of the weights from index i_{o} to the end of the sequence is also smaller as W.

$$\sum _{i=1}^{{i}_{u}}{w}_{i}<W$$

$$\sum _{i={i}_{o}}^{N}{w}_{i}<W$$

If the difference of the indices i_{u} and i_{o} is equal to 1, the median of a_{iu} and a_{io} is arithmetically averaged. If the difference is larger, the median results from the averaging of the indices i_{u} and i_{o}.

The following is an example of the weighted median calculation.

$$\sigma =\sqrt{\frac{1}{n}\sum _{i=1}^{n}{\left({a}_{i}-\mathrm{Mean}\right)}^{2}}$$

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