# Extrapolation

A projection is an estimated extrapolation of an overall result from a partial result. It is used when all the information for the overall result is not yet available, the amount of information is too large to cope with with reasonable effort, or a full survey is not possible at all. In order to obtain the most precise projection possible, the partial result used must take into account all conceivable aspects and be numerically large enough. One then speaks of a representative sample or a random sample .

## Examples

• In elections , election research institutes try to predict the final result from the first election results. For this z. For example, the results from voting districts that have already been counted are compared with the results from the previous election in these voting districts and their deviation from the overall final result at that time. If only a few constituencies have been counted, the results of the post-election survey are also included in the extrapolation . The election forecast must be distinguished from the election projection , which is mainly based on polling voters after they cast their votes in individual, mostly particularly representative constituencies, the results of which in the previous election were particularly close to the official final result ( post-election survey or exit poll ). The respondents are asked about their choice of choice, but also about social structural characteristics such as age, gender, denomination, school leaving certificate or occupational group in order to obtain a cross-section of the total population that is as characteristic as possible. Therefore, in contrast to the extrapolation, a forecast can be published when the polling stations close (in Germany usually at 6 p.m.), i.e. before any votes have been counted.

Other examples of the application of an extrapolation are as a basis

• the partial counts of the constituents of the blood for diagnostic support or
• the counting of different areas of our or a foreign galaxy to roughly determine the total number of stars that make it up .

## Bound and free extrapolation

Free extrapolation: According to the general and classical understanding, extrapolations are tasks of three . Example: 10 pupils surveyed had a total pocket money (sum of all 10) of 100 euros. How much total pocket money does the entire class of 30 students have? Here, the estimation functions for a characteristic X (e.g. for the total value) are considered. Only the observed characteristic values, the sample size (possibly separated into layers) and the total amount (possibly separated into layers) were used.

Bound extrapolation: The bound extrapolation uses prior information. If external information is used, one speaks of bound extrapolation ( using auxiliary information at the estimation stage ).

Example: In Belgium all pig farm owners were asked how many pigs they owned. In addition, the number of a sample of 100 farms was checked. It was to be assumed that the farmers systematically gave too little information. From the difference that was measured in the sample farms by questioning and control, an improved estimate could be derived for all farms as a whole. (Source: Heinrich Strecker : Modern Methods in Agricultural Statistics. 1957.)

Historically, difference estimators (variable in the Y-axis intercept) and ratio estimators (variable in the slope) were used here. Today, regression estimators are used that combine both properties. However, with small samples, regression estimates are more prone to outliers .