Weighted average - how to calculate? Weighted average arithmetic, geometric, harmonic, power

The weighted average is one of the mathematical problems that causes considerable difficulties in calculation. In this article, you will learn how to calculate it correctly, as well as the difference between an arithmetic weighted average and a geometric, harmonic and power average, and with the help of which formulas you can calculate them.

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1. Weighted average - definition

Let's start by explaining what this mathematical concept means.

The weighted average is the average of the components that we assign different meanings so that those items that have more weight have a greater impact on the overall average.

If all available elements have the same weight, and therefore the same meaning, then the weighted average is equal to the starting average (otherwise known as the base average).

The weighted average can be calculated in various ways (e.g. as a geometric or arithmetic average), hence the formula for its calculation depends on its type.

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It is important to remember that a weighted average can only give a correct result if the weights are uncorrelated with each other and therefore not interdependent.

Such a problem may arise when calculating the measurement uncertainty.

Then we calculate the average of the M series of values ​​Yi = f (X1, X2 ... XN).

The arithmetic mean of Yi (i = 1,2, ..., M) and the weighted mean with weights that are equal to the partial uncertainties u (Yi) in the -1 power may give different results.

Weighted average is best used to calculate the mean value and its uncertainty where all Xij are independent, e.g. the quantities Yi have been measured on different equipment, in a different laboratory and under different conditions. If we do not have such independence, we should use a different average.

2. Arithmetic weighted average - formula

To calculate the arithmetic mean, use the following formula:

The formula for the arithmetic mean

Important!

Data with more weight are more important in determining the weighted average than data with less weight. But if the weights are equal, the weighted average is equal to the arithmetic mean. Note that the weighted average has similar characteristics to the arithmetic mean, but has several conflicting properties (e.g., the Simpson paradox).

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3. Weighted geometric mean - formula

We can also calculate the geometric weighted average. We calculate it from the formula:

Geometric weighted average

When all our weights are equal, the geometric weighted mean is equal to the geometric mean.

4. Harmonic weighted average - formula

The weighted average harmonic is calculated from the formula:

Weighted harmonic average

When the weights are equal, the weighted harmonic mean equals the harmonic mean.

5. Power weighted average - formula

To calculate the weighted variant for the power mean of any real nonzero q order, we need to use the formula:

Power weighted average

The power weighted mean for order 0 is described above the weighted geometric mean. On the other hand, for the +/- ∞ rows, entering the weights does not matter for the average value.

For rank -1, the mean is the harmonic weighted mean, while for rank 2, the mean is the squared weighted mean.

See also: Whole Numbers - Which Is What? Examples

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