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Observational error (or measurement error) is the difference between a measured value of a
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a uni ...
and its
true value In statistics, as opposed to its general use in mathematics, a parameter is any measured quantity of a statistical population that summarises or describes an aspect of the population, such as a mean or a standard deviation. If a population ex ...
.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. In
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, an error is not necessarily a " mistake". Variability is an inherent part of the results of measurements and of the measurement process. Measurement errors can be divided into two components: ''
random In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...
'' and ''systematic''. Random errors are
error An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistic ...
s in measurement that lead to measurable values being inconsistent when repeated measurements of a constant attribute or
quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a uni ...
are taken. Systematic errors are errors that are not determined by chance but are introduced by repeatable processes inherent to the
system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and expresse ...
. Systematic error may also refer to an error with a non-zero
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arith ...
, the
effect Effect may refer to: * A result or change of something ** List of effects ** Cause and effect, an idiom describing causality Pharmacy and pharmacology * Drug effect, a change resulting from the administration of a drug ** Therapeutic effect, a ...
of which is not reduced when observations are averaged. Measurement errors can be summarized in terms of
accuracy and precision Accuracy and precision are two measures of ''observational error''. ''Accuracy'' is how close a given set of measurements (observations or readings) are to their '' true value'', while ''precision'' is how close the measurements are to each oth ...
. Measurement error should not be confused with
measurement uncertainty In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by ...
.


Science and experiments

When either
randomness In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...
or uncertainty modeled by
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
is attributed to such errors, they are "errors" in the sense in which that term is used in
statistics Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
; see
errors and residuals in statistics In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its " true value" (not necessarily observable). The er ...
. Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results. The common
statistical model A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form ...
used is that the error has two additive parts: #Systematic error which always occurs, with the same value, when we use the instrument in the same way and in the same case. #Random error which may vary from observation to another. Systematic error is sometimes called statistical bias. It may often be reduced with standardized procedures. Part of the learning process in the various
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence ...
s is learning how to use standard instruments and protocols so as to minimize systematic error. Random error (or
random variation A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
) is due to factors that cannot or will not be controlled. One possible reason to forgo controlling for these random errors is that it may be too expensive to control them each time the experiment is conducted or the measurements are made. Other reasons may be that whatever we are trying to measure is changing in time (see dynamic models), or is fundamentally probabilistic (as is the case in quantum mechanics — see
Measurement in quantum mechanics In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what m ...
). Random error often occurs when instruments are pushed to the extremes of their operating limits. For example, it is common for digital balances to exhibit random error in their least significant digit. Three measurements of a single object might read something like 0.9111g, 0.9110g, and 0.9112g.


Characterization

Measurement errors can be divided into two components: random error and systematic error. 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 Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. ...
, or interference of the environment with the measurement process, and always affect the results of an
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
in a predictable direction. Incorrect zeroing of an instrument leading to a zero error 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 The American Society of Mechanical Engineers (ASME) is an American professional association that, in its own words, "promotes the art, science, and practice of multidisciplinary engineering and allied sciences around the globe" via " continuing ...
(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
precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...
. The higher the precision of a measurement instrument, the smaller the variability (
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, w ...
) of the fluctuations in its readings.


Sources


Sources of systematic error


Imperfect calibration

Sources of systematic error may be imperfect calibration of measurement instruments (zero error), changes in the environment which interfere with the measurement process and sometimes imperfect methods of
observation Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. ...
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 Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over") ...
measured by
radar Radar is a detection system that uses radio waves to determine the distance (''ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, ...
will be systematically overestimated if the slight slowing down of the waves in air is not accounted for. Incorrect zeroing of an instrument leading to a zero error is an example of systematic error in instrumentation. Systematic errors may also be present in the result of an
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is der ...
based upon a
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics ...
or
physical law Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) ...
. For instance, the estimated oscillation frequency of a
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
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 A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...
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 ( drift) are easier to detect. Measurements indicate trends with time rather than varying randomly about a
mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arith ...
. Drift is evident if a measurement of a 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 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 A stopwatch is a timepiece designed to measure the amount of time that elapses between its activation and deactivation. A large digital version of a stopwatch designed for viewing at a distance, as in a sports stadium, is called a stop clock. ...
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 A speaking clock or talking clock is a live or recorded human voice service, usually accessed by telephone, that gives the correct time. The first telephone speaking clock service was introduced in France, in association with the Paris Observato ...
' 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 An ammeter (abbreviation of ''Ampere meter'') is an instrument used to measure the current in a circuit. Electric currents are measured in amperes (A), hence the name. For direct measurement, the ammeter is connected in series with the circuit ...
s and
voltmeter A voltmeter is an instrument used for measuring electric potential difference between two points in an electric circuit. It is connected in parallel. It usually has a high resistance so that it takes negligible current from the circuit. ...
s need to be checked periodically against known standards. Systematic errors can also be detected by measuring already known quantities. For example, a
spectrometer A spectrometer () is a scientific instrument used to separate and measure spectral components of a physical phenomenon. Spectrometer is a broad term often used to describe instruments that measure a continuous variable of a phenomenon where th ...
fitted with a
diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffraction, diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form ...
may be checked by using it to measure the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
of the D-lines of the
sodium Sodium is a chemical element with the symbol Na (from Latin ''natrium'') and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable ...
electromagnetic spectrum The electromagnetic spectrum is the range of frequencies (the spectrum) of electromagnetic radiation and their respective wavelengths and photon energies. The electromagnetic spectrum covers electromagnetic waves with frequencies ranging fro ...
which are at 600 nm and 589.6 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 In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of know ...
of the measurement instrument.


Sources of random error

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 normally distributed when the stochastic error is the sum of many independent random errors because of the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themse ...
. 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. 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). 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. 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 MTMM experiments. This information about the quality can also be used in order to correct for measurement error.


Effect on regression analysis

If the
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
in a regression is measured with error, regression analysis and associated hypothesis testing are unaffected, except that the R2 will be lower than it would be with perfect measurement. However, if one or more
independent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
s is measured with error, then the regression coefficients and standard
hypothesis test A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
s are invalid. This is known as attenuation bias.


See also

*
Bias (statistics) Statistical bias is a systematic tendency which causes differences between results and facts. The bias exists in numbers of the process of data analysis, including the source of the data, the estimator chosen, and the ways the data was analyzed. ...
*
Cognitive bias A cognitive bias is a systematic pattern of deviation from norm or rationality in judgment. Individuals create their own "subjective reality" from their perception of the input. An individual's construction of reality, not the objective input, ...
* Correction for measurement error (for Pearson correlations) *
Errors and residuals in statistics In statistics and optimization, errors and residuals are two closely related and easily confused measures of the deviation of an observed value of an element of a statistical sample from its " true value" (not necessarily observable). The er ...
*
Error An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistic ...
*
Replication (statistics) In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "... the repetition of the ...
*
Statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistica ...
*
Metrology Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fran ...
*
Regression dilution Regression dilution, also known as regression attenuation, is the biasing of the linear regression slope towards zero (the underestimation of its absolute value), caused by errors in the independent variable. Consider fitting a straight line fo ...
*
Test method A test method is a method for a test in science or engineering, such as a physical test, chemical test, or statistical test. It is a definitive procedure that produces a test result. In order to ensure accurate and relevant test results, a test m ...
*
Propagation of uncertainty In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of ex ...
* Instrument error *
Measurement uncertainty In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. All measurements are subject to uncertainty and a measurement result is complete only when it is accompanied by ...
*
Errors-in-variables models In statistics, errors-in-variables models or measurement error models are regression models that account for measurement errors in the independent variables. In contrast, standard regression models assume that those regressors have been measured ...
*
Systemic bias Systemic bias, also called institutional bias, and related to structural bias, is the inherent tendency of a process to support particular outcomes. The term generally refers to human systems such as institutions. Institutional bias and structur ...


References


Further reading

* {{Biases Accuracy and precision Errors and residuals Statistical reliability