Editing Observational error

{{Short description|Difference between a measured value of a quantity and its true value}}
{{redirect|Systematic bias|the sociological and organizational phenomenon|Systemic bias}}
{{More citations needed|date=September 2016}}

'''Observational error''' (or '''measurement error''') is the difference between a [[measurement|measured]] value of a [[physical quantity|quantity]] and its unknown [[true value]].<ref name="Dodge">Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. {{ISBN|978-0-19-920613-1}}</ref> Such errors are inherent in the measurement process; for example lengths measured with a ruler calibrated in whole centimeters will have a measurement error of several millimeters. The error or uncertainty of a measurement can be estimated, and is specified with the measurement as, for example, 32.3 ± 0.5 cm. 

Scientific observations are marred by two distinct types of errors, systematic errors on the one hand, and [[Statistical randomness|random]], on the other hand. The effects of '''random errors''' can be mitigated by the repeated measurements. Constant or '''systematic errors''' on the contrary must be carefully avoided, because they arise from one or more causes which constantly act in the same way, and have the effect of always altering the result of the experiment in the same direction. They therefore alter the value observed and repeated identical measurements do not reduce such errors.<ref name="Taylor">{{cite book  |title=An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements |author=John Robert Taylor |url=https://books.google.com/books?id=giFQcZub80oC&pg=PA94 |page=94, §4.1 |isbn=978-0-935702-75-0 |year=1999 |publisher=University Science Books}}</ref> 

Measurement errors can be summarized in terms of [[accuracy and precision]].
For example, length measurements with a ruler accurately calibrated in whole centimeters will be subject to random error with each use on the same distance giving a slightly different value resulting limited precision; a metallic ruler the [[temperature]] of which is not controlled will be affected by [[thermal expansion]] causing an additional systematic error resulting in limited accuracy.<ref>{{cite book |last1=Ritter |first1=Elie |title=Manuel théorique et pratique de l'application de la méthode des moindres carrés au calcul des observations |publisher=Mallet-Bachelier |page=7 |url=https://play.google.com/books/reader?id=IVEDAAAAQAAJ&pg=GBS.PA7&hl=fr |access-date=16 February 2025}}</ref>

==Science and experiments==
When either [[random variable|randomness]] or uncertainty modeled by [[probability theory]] is attributed to such errors, they are "errors" in the sense in which that term is used in [[statistics]]; see [[errors and residuals in statistics]].

[[File:Measurement distribution with systematic and random errors.svg|thumb|Distribution of measurements of known true value, with both constant systematic error and normally distributed random error.]]

Every time a measurement is repeated, slightly different results are obtained. The common [[statistical model]] used is that the error has two additive parts:<ref name=Heinrich-2007>{{Cite journal |last=Heinrich |first=Joel |last2=Lyons |first2=Louis |date=2007-11-01 |title=Systematic Errors |url=https://www.annualreviews.org/doi/10.1146/annurev.nucl.57.090506.123052 |journal=Annual Review of Nuclear and Particle Science |language=en |volume=57 |issue=1 |pages=145–169 |doi=10.1146/annurev.nucl.57.090506.123052 |issn=0163-8998|url-access=subscription }}</ref>
#'''Random error''' which may vary from observation to another.
#'''Systematic error''' which always occurs, with the same value, when we use the instrument in the same way and in the same case.
Some errors are not clearly random or systematic such as the uncertainty in the calibration of an instrument.<ref name=Heinrich-2007/>

Random errors or statistical errors in measurement lead to measurable values being inconsistent between repeated measurements of a [[time-invariant|constant]] attribute or [[physical quantity|quantity]] are taken. Random errors create [[measurement uncertainty]]. These errors are [[correlation (statistics)|uncorrelated]] between measurements. Repeated measurements will fall in a pattern and in a large set of such measurements a [[standard deviation]] can be calculated as a estimate of the amount of statistical error.<ref name=Heinrich-2007/>{{rp|147}}

Systematic errors are errors that are not determined by chance but are introduced by repeatable processes inherent to the [[system]].<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/systematic%20error |title=Systematic error |website=Merriam-webster.com |access-date=2016-09-10}}</ref> Sources of systematic errors include errors in equipment calibration, uncertainty in correction terms applied during experimental analysis, errors due the use of approximate theoretical models.<ref name=Heinrich-2007/>{{rp|loc=supl}}
Systematic error is sometimes called '''statistical bias'''. It may often be reduced with standardized procedures. 

Part of the learning process in the various [[science]]s is learning how to use standard instruments and protocols so as to minimize systematic error.
Over a long period of time, systematic errors in science can be resolved and become a form of "negative knowledge": scientist build up an understanding of how to avoid specific kinds of systematic errors.<ref>{{Cite journal |last=Allchin |first=Douglas |date=March 2001 |title=Error Types |url=https://direct.mit.edu/posc/article/9/1/38-58/15089 |journal=Perspectives on Science |language=en |volume=9 |issue=1 |pages=38–58 |doi=10.1162/10636140152947786 |issn=1063-6145|url-access=subscription }}</ref>

== Propagation of errors ==
{{main|Propagation of uncertainty}}
When two or more observations or two or more instruments are combined, the errors in each combine. Estimates of the error in the result of such combinations depend upon the statistical characteristics of each individual measurement and on the possible statistical correlation between them.<ref>{{Cite book |last=Young |first=Hugh D. |title=Statistical treatment of experimental data: an introduction to statistical methods |date=1996 |publisher=Waveland Press |isbn=978-0-88133-913-0 |edition=Repr |location=Long Grove, Ill |url=https://archive.org/details/H_D_Young__Statistical_Treatment_of_Experimental_Data/page/n53/mode/2up}}</ref>{{rp|92}}

== Characterization ==
{{main|Accuracy and precision}}
Measurement errors can be divided into two components: random error and systematic error.<ref name="Taylor"/>

'''Random error''' is always present in a measurement. It is caused by inherently unpredictable fluctuations in the readings of a measurement apparatus or in the experimenter's interpretation of the instrumental reading. Random errors show up as different results for ostensibly the same repeated measurement. They can be estimated by comparing multiple measurements and reduced by averaging multiple measurements.

'''Systematic error''' is predictable and typically constant or proportional to the true value. If the cause of the systematic error can be identified, then it usually can be eliminated. Systematic errors are caused by imperfect calibration of measurement instruments or imperfect methods of [[observation]], or interference of the [[surroundings|environment]] with the measurement process, and always affect the results of an [[experiment]] in a predictable direction. Incorrect zeroing of an instrument is an example of systematic error in instrumentation.

The Performance Test Standard PTC 19.1-2005 "Test Uncertainty", published by the [[American Society of Mechanical Engineers]] (ASME), discusses systematic and random errors in considerable detail. In fact, it conceptualizes its basic uncertainty categories in these terms.

Random error can be caused by unpredictable fluctuations in the readings of a measurement apparatus, or in the experimenter's interpretation of the instrumental reading; these fluctuations may be in part due to interference of the environment with the measurement process. The concept of random error is closely related to the concept of [[accuracy and precision|precision]]. The higher the precision of a measurement instrument, the smaller the variability ([[standard deviation]]) of the fluctuations in its readings.

==Sources ==
===Sources of systematic error{{anchor|Systematic}}===
====Imperfect calibration====
Sources of systematic error may be imperfect calibration of measurement instruments (zero error), changes in the [[Biophysical environment|environment]] which interfere with the measurement process and sometimes imperfect methods of [[observation]] can be either zero error or percentage error. If you consider an experimenter taking a reading of the time period of a pendulum swinging past a [[fiducial marker]]:  If their stop-watch or timer starts with 1 second on the clock then all of their results will be off by 1 second (zero error). If the experimenter repeats this experiment twenty times (starting at 1 second each time), then there will be a [[percentage error]] in the calculated average of their results; the final result will be slightly larger than the true period.

[[Distance]] measured by [[radar]] will be systematically overestimated if the slight slowing down of the waves in air is not accounted for. Incorrect zeroing of an instrument is an example of systematic error in instrumentation.

Systematic errors may also be present in the result of an [[Computational mechanics|estimate]] based upon a [[mathematical model]] or [[physical law]]. For instance, the estimated [[oscillation frequency]] of a [[pendulum]] will be systematically in error if slight movement of the support is not accounted for.

====Quantity====
Systematic errors can be either constant, or related (e.g. proportional or a percentage) to the actual value of the measured quantity, or even to the value of a different quantity (the reading of a [[ruler]] can be affected by environmental temperature). When it is constant, it is simply due to incorrect zeroing of the instrument. When it is not constant, it can change its sign. For instance, if a thermometer is affected by a proportional systematic error equal to 2% of the actual temperature, and the actual temperature is 200°, 0°, or −100°, the measured temperature will be 204° (systematic error = +4°), 0° (null systematic error) or −102° (systematic error = −2°), respectively. Thus the temperature will be overestimated when it will be above zero and underestimated when it will be below zero.

====Drift====
Systematic errors which change during an experiment ([[Stochastic drift|drift]]) are easier to detect. Measurements indicate trends with time rather than varying randomly about a [[mean]]. Drift is evident if a measurement of a [[constant (mathematics)|constant]] quantity is repeated several times and the measurements drift one way during the experiment. If the next measurement is higher than the previous measurement as may occur if an instrument becomes warmer during the experiment then the measured quantity is variable and it is possible to detect a drift by checking the zero reading during the experiment as well as at the start of the experiment (indeed, the [[Vernier scale|zero reading]] is a measurement of a constant quantity). If the zero reading is consistently above or below zero, a systematic error is present. If this cannot be eliminated, potentially by resetting the instrument immediately before the experiment then it needs to be allowed by subtracting its (possibly time-varying) value from the readings, and by taking it into account while assessing the accuracy of the measurement.

If no pattern in a series of repeated measurements is evident, the presence of fixed systematic errors can only be found if the measurements are checked, either by measuring a known quantity or by comparing the readings with readings made using a different apparatus, known to be more accurate. For example, if you think of the timing of a pendulum using an accurate [[stopwatch]] several times you are given readings randomly distributed about the mean. Hopings systematic error is present if the stopwatch is checked against the '[[speaking clock]]' of the telephone system and found to be running slow or fast. Clearly, the pendulum timings need to be corrected according to how fast or slow the stopwatch was found to be running.

Measuring instruments such as [[ammeter]]s and [[voltmeter]]s need to be checked periodically against known standards.

Systematic errors can also be detected by measuring already known quantities. For example, a [[spectrometer]] fitted with a [[diffraction grating]] may be checked by using it to measure the [[wavelength]] of the D-lines of the [[sodium]] [[electromagnetic spectrum]] which are at 600&nbsp;nm and 589.6&nbsp;nm. The measurements may be used to determine the number of lines per millimetre of the diffraction grating, which can then be used to measure the wavelength of any other spectral line.

Constant systematic errors are very difficult to deal with as their effects are only observable if they can be removed. Such errors cannot be removed by repeating measurements or averaging large numbers of results. A common method to remove systematic error is through [[calibration]] of the measurement instrument.

===Sources of random error{{anchor|Random}}===
The random or stochastic error in a measurement is the error that is random from one measurement to the next. Stochastic errors tend to be [[normal distribution|normally distributed]] when the stochastic error is the sum of many independent random errors because of the [[central limit theorem]]. Stochastic errors added to a regression equation account for the variation in ''Y'' that cannot be explained by the included ''X''s.

==Surveys==
The term "observational error" is also sometimes used to refer to response errors and some other types of [[non-sampling error]].<ref name="Dodge"/> In survey-type situations, these errors can be mistakes in the collection of data, including both the incorrect recording of a response and the correct recording of a respondent's inaccurate response. These sources of non-sampling error are discussed in Salant and Dillman (1994) and Bland and Altman (1996).<ref>{{cite book |last1=Salant |first1=P. |first2=D. A. |last2=Dillman |title=How to conduct your survey |url=https://archive.org/details/howtoconductyour00sala |url-access=registration |location=New York |publisher=John Wiley & Sons |year=1994 |isbn=0-471-01273-4 }}</ref><ref>{{cite journal |last1=Bland |first1=J. Martin |first2=Douglas G. |last2=Altman |title=Statistics Notes: Measurement Error |journal=BMJ |volume=313 |issue=7059 |year=1996 |pages=744 |doi=10.1136/bmj.313.7059.744 |pmid=8819450 |pmc=2352101 }}</ref>

These errors can be random or systematic. Random errors are caused by unintended mistakes by respondents, interviewers and/or coders. Systematic error can occur if there is a systematic reaction of the respondents to the method used to formulate the survey question. Thus, the exact formulation of a survey question is crucial, since it affects the level of measurement error.<ref>{{cite book |last1=Saris |first1=W. E. |last2=Gallhofer |first2=I. N. |year=2014 |title=Design, Evaluation and Analysis of Questionnaires for Survey Research |edition=Second |location=Hoboken |publisher=Wiley |isbn=978-1-118-63461-5 }}</ref> Different tools are available for the researchers to help them decide about this exact formulation of their questions, for instance estimating the quality of a question using [[Multitrait-multimethod matrix|MTMM experiments]]. This information about the quality can also be used in order to [[Correction for attenuation|correct for measurement error]].<ref>DeCastellarnau, A. and Saris, W. E. (2014). A simple procedure to correct for measurement errors in survey research. European Social Survey Education Net (ESS EduNet). Available at:  [http://essedunet.nsd.uib.no/cms/topics/measurement/ http://essedunet.nsd.uib.no/cms/topics/measurement] {{Webarchive|url=https://web.archive.org/web/20190915033804/http://essedunet.nsd.uib.no/cms/topics/measurement |date=2019-09-15 }}</ref><ref>{{cite journal | last1 = Saris | first1 = W. E. | last2 = Revilla | first2 = M. | year = 2015 | title = Correction for measurement errors in survey research: necessary and possible | url =http://repositori.upf.edu/bitstream/10230/28341/1/RECSM-WP-031-Correction%20for%20measurement%20errors%20in%20survey%20research.pdf | journal = Social Indicators Research | volume =  127| issue = 3| pages =  1005–1020| doi = 10.1007/s11205-015-1002-x | hdl = 10230/28341 | s2cid = 146550566 | hdl-access = free }}</ref>

==Effect on regression analysis==
If the [[dependent variable]] in a regression is measured with error, regression analysis and associated hypothesis testing are unaffected, except that the [[coefficient of determination|R<sup>2</sup>]] will be lower than it would be with perfect measurement.

However, if one or more [[independent variable]]s is measured with error, then the regression coefficients and standard [[hypothesis test]]s are invalid.<ref>{{cite book
  | last = Hayashi | first = Fumio
  | title = Econometrics
  | year = 2000
  | publisher = Princeton University Press
  | isbn = 978-0-691-01018-2
  | page = 187
  }}</ref> This is known as [[Regression dilution|attenuation bias]].<ref>{{Cite book|last1=Angrist|first1=Joshua David | url=https://www.worldcat.org/oclc/877846199|title=Mastering 'metrics : the path from cause to effect| last2=Pischke|first2=Jörn-Steffen | year=2015|isbn=978-0-691-15283-7| publisher=Princeton University Press|location=Princeton, New Jersey | pages=221 | oclc=877846199 | quote=The bias generated by this sort of measurement error in regressors is called attenuation bias.}}</ref>

==See also==
{{Div col|colwidth=20em}}
*[[Bias (statistics)]]
*[[Cognitive bias]]
*[[Correction for attenuation|Correction for measurement error]] (for Pearson correlations)
*[[Error]]
*[[Errors and residuals in statistics]]
*[[Errors-in-variables models]]
*[[Instrument error]]
*[[Measurement uncertainty]]
*[[Metrology]]
*[[Outlier]]
*[[Propagation of uncertainty]]
*[[Regression dilution]]
*[[Replication (statistics)]]
*[[Statistical theory]]
*[[Systemic bias]]
*[[Test method]]
{{Div col end}}

==References==
{{Reflist}}

==Further reading==
*{{cite journal|jstor=1267450|title=Errors of Measurement in Statistics |first=W. G. |last=Cochran |journal=[[Technometrics]] |volume=10 |issue=4 |date=1968 |pages=637–666 |doi=10.2307/1267450 |s2cid=120645541 }}

{{Biases}}
{{Authority control}}

[[Category:Accuracy and precision]]
[[Category:Errors and residuals]]
[[Category:Statistical reliability]]