Systematic error
Encyclopedia
Systematic errors are bias
es in measurement
which lead to the situation where the mean
of many separate measurements differs significantly from the actual value of the measured attribute. All measurements are prone to systematic errors, often of several different types. 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
can be either zero error or percentage error. For example, consider an experimenter taking a reading of the time period of a pendulum swinging past a fiducial mark:
If his stop-watch or timer starts with 1 second on the clock then all of his 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 his 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 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
based on 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.
Systematic errors can be either constant, or be 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 environment temperature). When they are constant, they are simply due to incorrect zeroing of the instrument. When they are not constant, they can change 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.
Constant systematic errors are very difficult to deal with, because 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.
In a statistical
context, the term systematic error usually arises where the sizes and directions of possible errors are unknown.
) are easier to detect. Measurements show trends with time rather than varying randomly about a mean
.
Drift is evident if a measurement of a constant quantity is repeated several times and the measurements drift one way during the experiment, for example if each measurement is higher than the previous measurement which could perhaps occur if an instrument becomes warmer during the experiment. If the measured quantity is variable, 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, for instance by resetting the instrument immediately before the experiment, it needs to be allowed for by subtracting its (possibly time-varying) value from the readings, and by taking it into account in 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, suppose the timing of a pendulum using an accurate stopwatch
several times gives readings randomly distributed about the mean. A 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 600nm 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.
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 cannot be discovered this way because it always pushes the results in the same direction. If the cause of a systematic error can be identified, then it can usually be eliminated.
Because random errors are reduced by re-measurement (making n times as many measurements will usually reduce random errors by a factor of ), it is worth repeating an experiment until random errors are similar in size to systematic errors. Additional measurements will be of little benefit, because the overall error cannot be reduced below the systematic error.
Bias
Bias is an inclination to present or hold a partial perspective at the expense of alternatives. Bias can come in many forms.-In judgement and decision making:...
es in measurement
Measurement
Measurement is the process or the result of determining the ratio of a physical quantity, such as a length, time, temperature etc., to a unit of measurement, such as the metre, second or degree Celsius...
which lead to the situation where the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of many separate measurements differs significantly from the actual value of the measured attribute. All measurements are prone to systematic errors, often of several different types. Sources of systematic error may be imperfect calibration of measurement instruments (zero error), changes in the environment
Surroundings
Surroundings are the area around a given physical or geographical point or place. The exact definition depends on the field. Surroundings can also be used in geography and mathematics, as well as philosophy, with the literal or metaphorically extended definition.In thermodynamics, the term is used...
which interfere with the measurement process and sometimes imperfect methods of observation
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...
can be either zero error or percentage error. For example, consider an experimenter taking a reading of the time period of a pendulum swinging past a fiducial mark:
If his stop-watch or timer starts with 1 second on the clock then all of his 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 his results; the final result will be slightly larger than the true period.
Distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
measured by radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
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
Computational mechanics
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely...
based on a mathematical model or physical law. For instance, the estimated oscillation frequency of a pendulum
Pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
will be systematically in error if slight movement of the support is not accounted for.
Systematic errors can be either constant, or be 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
Ruler
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
can be affected by environment temperature). When they are constant, they are simply due to incorrect zeroing of the instrument. When they are not constant, they can change 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.
Constant systematic errors are very difficult to deal with, because 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
Calibration
Calibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....
of the measurement instrument.
In a statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
context, the term systematic error usually arises where the sizes and directions of possible errors are unknown.
Drift
Systematic errors which change during an experiment (driftDrift
- Film and literature :* Drift , a 2002 Doctor Who novel* Drift , a series of Japanese films written and directed by Futoshi Jinno* Drift, 2007 experimental short film by Max Hattler* Drift , a fictional character...
) are easier to detect. Measurements show trends with time rather than varying randomly about a mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
.
Drift is evident if a measurement of a constant quantity is repeated several times and the measurements drift one way during the experiment, for example if each measurement is higher than the previous measurement which could perhaps occur if an instrument becomes warmer during the experiment. If the measured quantity is variable, 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, for instance by resetting the instrument immediately before the experiment, it needs to be allowed for by subtracting its (possibly time-varying) value from the readings, and by taking it into account in 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, suppose the timing of a pendulum using an accurate stopwatch
Stopwatch
A stopwatch is a handheld timepiece designed to measure the amount of time elapsed from a particular time when activated to when the piece is deactivated. A large digital version of a stopwatch designed for viewing at a distance, as in a sports stadium, is called a stopclock.The timing functions...
several times gives readings randomly distributed about the mean. A systematic error is present if the stopwatch is checked against the 'speaking clock
Speaking clock
A speaking clock service is a recorded or simulated 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 Observatory on 14 February 1933.The format of the service is...
' 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
Ammeter
An ammeter is a measuring instrument used to measure the electric current in a circuit. Electric currents are measured in amperes , hence the name. Instruments used to measure smaller currents, in the milliampere or microampere range, are designated as milliammeters or microammeters...
s and voltmeter
Voltmeter
A voltmeter is an instrument used for measuring electrical potential difference between two points in an electric circuit. Analog voltmeters move a pointer across a scale in proportion to the voltage of the circuit; digital voltmeters give a numerical display of voltage by use of an analog to...
s need to be checked periodically against known standards.
Systematic errors can also be detected by measuring already known quantities. For example, a spectrometer
Spectrometer
A spectrometer is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify materials. The variable measured is most often the light's intensity but could also, for instance, be the polarization...
fitted with a diffraction grating
Diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure, which splits and diffracts light into several beams travelling in different directions. The directions of these beams depend on the spacing of the grating and the wavelength of the light so that the grating acts as...
may be checked by using it to measure the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the D-lines of the sodium
Sodium
Sodium is a chemical element with the symbol Na and atomic number 11. It is a soft, silvery-white, highly reactive metal and is a member of the alkali metals; its only stable isotope is 23Na. It is an abundant element that exists in numerous minerals, most commonly as sodium chloride...
electromagnetic spectrum
Electromagnetic spectrum
The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation. The "electromagnetic spectrum" of an object is the characteristic distribution of electromagnetic radiation emitted or absorbed by that particular object....
which are at 600nm 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.
Systematic versus random error
Measurement errors can be divided into two components: random errorRandom error
Random errors are errors in measurement that lead to measurable values being inconsistent when repeated measures of a constant attribute or quantity are taken...
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 cannot be discovered this way because it always pushes the results in the same direction. If the cause of a systematic error can be identified, then it can usually be eliminated.
Because random errors are reduced by re-measurement (making n times as many measurements will usually reduce random errors by a factor of ), it is worth repeating an experiment until random errors are similar in size to systematic errors. Additional measurements will be of little benefit, because the overall error cannot be reduced below the systematic error.
See also
- Experimental uncertainty analysisExperimental uncertainty analysisThe purpose of this introductory article is to discuss the experimental uncertainty analysis of a derived quantity, based on the uncertainties in the experimentally measured quantities that are used in some form of mathematical relationship to calculate that derived quantity...
- Biased sampleBiased sampleIn statistics, sampling bias is when a sample is collected in such a way that some members of the intended population are less likely to be included than others. It results in a biased sample, a non-random sample of a population in which all individuals, or instances, were not equally likely to...
- Errors and residuals in statisticsErrors and residuals in statisticsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
- Observational errorObservational errorObservational error is the difference between a measured value of quantity and its true value. In statistics, an error is not a "mistake". Variability is an inherent part of things being measured and of the measurement process.-Science and experiments:...