# Weight function

Underweight (even weighting, the weighting scheme ) refers to the assessment of individual factors of a mathematical model, for example, in terms of their importance or reliability. You will cause more important or more reliable elements have greater influence on the result.

- 5.1 Weighting of statistically stray sizes
- 5.2 Weighting of measured variables
- 5.3 Economics
- 5.4 tests

## Example

For entry into a technical school, the score is more important than the score in history in mathematics. Now, if the average is determined, the two scores are not simply added together and divided by 2, but first both scores are multiplied by a weighting factor (in short: weight) multiplied, and then added together and divided by the sum of the weights.

For example, the score is multiplied in mathematics with weight 2 for the technical high school, the score in history with the weight 1

If you were not overweight mathematics with a factor of 2, but with the factor 1 as history, then both would have the same opportunities, namely ( 80 40) 2 = 60 points.

## Determining the weighting factor

Decisive for the quality of the weighted value is the adequacy of the weighting factor. This can be (as in the classic example ) set arbitrarily: If history has a weight of 1, and mathematics a weight of 2 - what weight should then obtain the subject geometry? more 1.8 or more 2.2? Or when comparing electricity from the nuclear power plant and electricity from coal-fired power plant: what weight to get the values " current price" or " exhaust " or " nuclear waste "?

Depending on the political and economic interest individual values are weighted differently. This completely different overall results are generated. Weighted results are understandable and assessable only with knowledge of the underlying political and economic interests. This also applies to weighted values behind which stuck complicated statistical calculations.

## Calculation

The weighted average is calculated as follows:

So the weighted average is calculated as

With the standard deviation.

Example: A teacher weights the third double of 4 classwork.

By weighting the grade 3 with a higher value than the other grades, the mean shifts upward ( the " bad" touch down ).

See also: Weighted arithmetic mean

## Types of weights

One can distinguish between different types of weights:

### Empirical distinction

- Design Weighting: Figure disproportional stratified sampling procedures.
- Redressment (also post-weighting ): Subsequent adjustment to known marginal distributions, eg systematic distortions in the sample by non- random failures.

### Mathematical distinction

- Frequency weights ( Frequency Weights ): weights that specify how often an observation ( characteristic) appears in the record.
- Analytical weights ( Analytic Weights ): weights that indicate how many cases are attributable to an aggregate feature. These are frequency weights with normalization to the sample size.
- Probability weights ( Probability Weights ): weights that take into account what has selection probability of an observation. Is the inverse of the probability of selection
- Importance Weights

## Application

### Weighting of statistically stray sizes

Is in physical quantities, the scattering of each value is known, then it is appropriate to weight in the calculation of the mean values according to their scattering. Does the th value of the scattering, as is the associated weight, the standard deviation simplifies to.

### Weighting of measured variables

In metrology, it may be appropriate to weight different readings with the reciprocals of their uncertainties. This ensures that in further calculations values are weighted according stronger with smaller uncertainties.

### Economy

In the economic area weighting schemes find particular application in the calculation of shopping carts (and thus price indices ) and effective exchange rates.

### Tests

If an examination consists of several compartments, and that an overall result of the test shall be made , the individual results of the subjects are often combined with a certain weighting. For final exams in recognized training occupations are mostly the training regulations for the profession before the weighting factors, in some cases also attacks the respective examination regulations.