# True value

The true value of a quantity can be characterized from different points of view. The following three definitions are helpful in understanding this important basic term; they formulate differently without contradicting each other.

• In measurement technology, the true value (of a measured variable ) is explained as the end of a path: “Value of the measured variable as the goal of evaluating measurements of the measured variable”.
• The international Joint Committee for Guides in Metrology understands its goal and formulates it in the International Vocabulary of Metrology as follows: “A value that corresponds to the definition of a variable”.
• The following can be found in quality assurance and statistics: “Actual characteristic value under the conditions prevailing during the determination”.

## Statistics / data analysis

Determining the true value of a quantity is one of the tasks of statistics / data analysis . The collection of data can be viewed as a measurement process. Examples for the subsequent reduction and presentation of the data can be found in descriptive data analysis and in exploratory data analysis .

Examples of different problem areas for determining the true value of a quantity:

• The inferential data analysis deduces the characteristics of the non-surveyed population from the surveyed sample. Example: The audience rating for TV shows.
• Measurement of natural constants . Example: The best possible value for the magnetic field constant including measurement uncertainty.${\ displaystyle \ mu _ {0}}$
• Measurement of relationships. Example: The value of the mass involved in an oscillation of a helical spring, determined from the relationship between the period of oscillation of the helical spring and the attached mass.

Other methods are used to verify the type of model assumptions, for example:

• Is the assumption that the data are normally distributed (roughly) appropriate. ( Quantile-quantile diagram , probability paper )
• or about symmetries in the distribution and possible outliers (quantile diagram).
• The analysis and interpretation of residuals in a regression analysis , etc. a. the existence of a systematic error and the distribution of the random errors.
• The analysis and interpretation of intercepts and zeros of a function graph determined from contexts, etc. a. also for indications of a systematic error.

Empirically collected data differ from the true value by the systematic and random deviations. One can see the connection through the equation

Measured value = true value + systematic deviation + random deviation

express. Random deviations scatter around the true value in terms of amount and sign; a systematic deviation can be positive or negative in some cases. This thought model describes the situation clearly and precisely, but in this form is unsuitable for practice; because in practice only the data (the measured values ​​are data in the sense of statistics) and not the exact value of the other quantities are known.

In practice, therefore, this thought model has to be modified. An estimated value (forecast value ) for the true value is determined from the data and then the differences between the estimated value and the data are calculated, which are called residuals in this model to distinguish them from the deviations. The following applies to data evaluation

Metric = estimate of true value + residual

Residuals can be both positive and negative, depending on whether the metric is greater or lesser than the guess for the true value. Two particular estimates should be mentioned:

• The median of the data. It is robust against outliers and minimizes the sum of the distances between the estimated value and the measured value.
• The arithmetic mean of the data. It is sensitive to outliers, for this estimate the sum of the residuals is zero and the sum of the residual squares is minimal. By means of the error calculation it can be shown that in a measurement series that does not contain any outliers and with no systematic deviation, the arithmetic mean is the best estimate for the true value.

An analysis and interpretation of the residuals as well as other methods mentioned above should provide information about the accuracy of the statement of the true value and about the appropriateness of model assumptions. At the end of the calculation an interval is given which, due to its construction, contains the true value as often as possible. However, there is no guarantee that the true value lies in this interval.

## Physics / measurement technology

The true value of a physical quantity “is an ideal value that is estimated from measurements . Exceptions are defined values ​​of measured variables (e.g. angle of the full circle, speed of light in a vacuum) or the determinable finite number of elements of a specified set of objects ”.

The theories of classical physics allow a meaningful understanding of the true value of a measurand. On the other hand has in the field of atoms and subatomic particles , the quantum mechanics are applied; in this theory it depends on the chosen interpretation or formulation (see interpretations of quantum mechanics ) whether one accepts or denies the existence of true values.

In general, the true value follows from a theory or a model : The mathematical formulas describing the physical relationships work with the true values ​​of the quantities. “The true value of a mathematical-theoretical characteristic is also called the exact value . With a numerical calculation method, however, the exact value will not always be the result of the determination ”. A “known value for comparison purposes, the deviation of which from the true value is considered negligible for the purpose of comparison” is designated as a correct value .

An example is the case law :

${\ displaystyle s = {\ frac {1} {2}} \, gt ^ {2}.}$

The mathematical formulation sets true values ​​for the path , the gravitational acceleration and the time span , i.e. H. error-free values ​​ahead. If read in measured values, the relationship would not be exactly fulfilled. But the true value of the gravitational acceleration can be approximated from measured values ​​/ data . That makes this equation and the evaluation process interesting in this context: A physical law is usually first formulated on a trial basis and then checked with the help of experiments . ${\ displaystyle s}$ ${\ displaystyle g}$ ${\ displaystyle t}$${\ displaystyle g}$

True values ​​cannot be measured directly, but can only be limited to value intervals. A confirmation of natural laws is therefore only possible within the scope of the measurement uncertainty . The metrology has central importance to the analysis and processing of measurement deviations.

The true value should not be confused with the actual value of a controlled variable.

## Individual evidence

1. http://www.qz-online.de/service/qm-glossar/artikel/wahrer-wert-168381.html
2. a b DIN 1319–1: 1995 Basic concepts in measurement technology - Part 1: Basic concepts, No. 1.3
3. JCGM 200: 2012 International Dictionary of Metrology http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf International vocabulary of metrology - Basic and general concepts and associated terms (VIM) , No. 2.11 ( PDF; 3.8 MB; accessed on February 28, 2016)
4. Burghart Brinkmann: International Dictionary of Metrology: Basic and General Terms and Associated Terms (VIM) German-English version ISO / IEC Guide 99: 2007 . 4th edition. Beuth, Berlin 2012, ISBN 978-3-410-22472-3 , pp. 34 ( limited preview in Google Book search).
5. a b DIN 55350-13: 1987 Terms of quality assurance and statistics; Terms for the accuracy of investigative procedures and investigation results , No. 1.3
6. P. Zöfel: Statistics in Practice , Gustav Fischer Verlag Stuttgart 1992, ISBN 3-8252-1293-9 , pp. 73ff.
7. J. Bortz: Statistics for Social Scientists , Springer Verlag Berlin 1999, p. 17ff.
8. A.Büchter / W.Henn: Elementare Stochastik ; Springer Verlag Berlin 2005, p. 23ff.
9. CODATA Recommended Values. National Institute of Standards and Technology, accessed May 22, 2019 . .
10. ILMES - Internet Lexicon of Methods in Empirical Social Research ( Memento from November 2, 2013 in the Internet Archive )
11. DIN 1319-1, No. 3.2
12. M. Stockhausen: Mathematical treatment of scientific phenomena, volume 1 Treatment of measured values , UTB Steinkopff Darmstadt, ISBN 3-7985-0549-7
13. M. Borovcnik / G.Ossimitz: Materials for Descriptive Statistics and Exploratory Data Analysis , Hölder-Pichler-Tempsky Vienna 1987, p. 97
14. DIN 1319-1, No. 1.4