# Mean Value Calculator

## Online calculator for calculating weighted and unweighted mean values

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.

Scale:
Number of digits=
Arithmetic mean:
Geometric mean:
Harmonic mean:
Square mean:
Cubic mean:
Median:
Points:
Annotation:
Display standard deviation σ:
Display weight circle:

### Calculated mean values

Mean Values
Standard Deviation σ
Arithmetic mean
Geometric mean
Harmonic mean
Square mean
Cubic mean
Median

### Data points

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: a1, a2, a3, ... and the weights w1, w2, w3, ...

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

Number of measuring points

n=

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 x1, w1, x2, w2...

## Mean Value Definitions

### Arithmetic mean

$Anai= 1 n ∑ i = 1 n a i =a1+a2+...+ann$

The arithmetic means satisfies the equation n⋅A = a1 + a2 + ... + an. 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. ### Weighted arithmetic mean

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

$Anwiai= ∑ i = 1 n w i a i ∑ i = 1 n w i =w1a1+w2a2+...+wnan ∑ i = 1 n w i$

### Geometric mean

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.

$Gnai= Π i = 1 n a i n =a1⋅a2⋅...⋅ann$

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.

### Weighted geometric means

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

$Gnwiai= Π i = 1 n a i wi ∑ i = 1 n w i =a1w1⋅a2w2⋅...⋅anwn ∑ i = 1 n w i$

### Harmonic mean

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).

$Hnai= n ∑ i = 1 n 1 a i =n1a1+1a2+...+1an$

### Weighted harmonic mean

The weighting factors wi can be used to generalize the harmonic mean to the weighted harmonic mean.

$Hnwiai= ∑ i = 1 n w i ∑ i = 1 n wi a i = w 1 + w 2 + ... + w n w1a1+w2a2+...+wnan$

### Square mean

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.

$Qnai= 1 n ∑ i = 1 n a i 2 =a12+a22+...+an2n$

### Weighted square mean

The weighting factors wi can be used to generalize the quadratic mean to the weighted square mean.

$Qnwiai= ∑ i = 1 n w i a i 2 ∑ i = 1 n w i =w1a12+w2a22+...+wnan2 ∑ i = 1 n w i$

### Cubic mean

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

$Knai= 1 n ∑ i = 1 n a i 3 3 =a13+a23+...+an3n3$

### Weighted cubic mean

The weighting factors wi can be used to generalize the cubic mean to the weighted cubic mean.

$Qnwiai= ∑ i = 1 n w i a i 3 ∑ i = 1 n w i 3 =w1a13+w2a23+...+wnan3 ∑ i = 1 n w i 3$

### Median

For ascending sorted values ai, the median is defined by:

$a1≤a2≤...≤an$

$Mnai= a n+12 n : odd a n2 + a n2+1 2 n : even$

### Weighted median

For values sorted ascending (ai, wi) sorted according to ai, with the weights wi∈ℝ+, the weighted median is defined as follows:

$a1≤a2≤...≤an$

$W= ∑ i = 1 n w i + 1 2$

The following two conditions mean that the sum of the weight up to the index iu is less than W, and the sum of the weights from index io to the end of the sequence is also smaller as W.

$∑ i = 1 iu w i < W$

$∑ i = io N w i < W$

If the difference of the indices iu and io is equal to 1, the median of aiu and aio is arithmetically averaged. If the difference is larger, the median results from the averaging of the indices iu and io.

The following is an example of the weighted median calculation. ### Standard deviation from mean value

$σ = 1 n ∑ i = 1 n a i - Mean 2$

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## More Calculators

Here is a list of of further useful calculators and sites:

Index Regression Percentage calculator Normal Distribution Plot Calculator NxN Gauss method Matrix Determinant Number Systems