# Online calculator for calculating weighted and unweighted mean values.

Number of digits =

##### Line style

The setting is accepted with recalculation of the mean values.

Arithmetic mean

Geometric mean

Harmonic mean

Square mean

Cubic mean

Median

## Mean Value Calculator

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.

Number of measuring points n=

#### Enter the measured values: a1, a2, a3, ... and the weights w1, w2, w3, ...

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

An alternative entry is possible via the following text field. The decimal separator is the point (.). The values can be separated by comma or space or semicolon. It is important that the last number is terminated with a semicolon.

For the input of value and weight in pairs, the first character must be #. E.g. #1 2 2 1 3 2; correspond a1=1, w1=2, a2=2, w2=1, a3=3, w3=2,

### 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

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

### 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

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

### Weighted harmonic agent

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

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

### 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

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

### Weighted cubic agent

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:

$a1≤a2≤...≤an$

$Mnai= a n+12 n : ungerade a n2 + a n2+1 2 n : gerade$

### Weighted median

For values sorted ascending (a i , w i ) sorted according to a i , 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 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.

$∑ i = 1 iu w i < W$

$∑ i = io N w i < W$

If the difference of the indices i u and i o is equal to 1, the median of a i u i o 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.

### Standard deviation from mean value

$σ = 1 n ∑ i = 1 n a i - Mean 2$