Systematic error formula

«Systematic bias» redirects here. For the sociological and organizational phenomenon, see Systemic bias.

Observational error (or measurement error) is the difference between a measured value of a quantity and its true value.^[1] In statistics, 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 and systematic.^[2]Random errors are errors in measurement that lead to measurable values being inconsistent when repeated measurements of a constant attribute or quantity are taken. Systematic errors are errors that are not determined by chance but are introduced by repeatable processes inherent to the system.^[3] Systematic error may also refer to an error with a non-zero mean, the effect of which is not reduced when observations are averaged.^{[citation needed]}

Measurement errors can be summarized in terms of accuracy and precision.
Measurement error should not be confused with measurement uncertainty.

Science and experimentsEdit

When either 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.

Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results. The common statistical model 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 sciences is learning how to use standard instruments and protocols so as to minimize systematic error.

Random error (or random variation) 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). 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.

CharacterizationEdit

Measurement errors can be divided into two components: random error and systematic error.^[2]

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 environment with the measurement process, and always affect the results of an experiment 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 (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. The higher the precision of a measurement instrument, the smaller the variability (standard deviation) of the fluctuations in its readings.

SourcesEdit

Sources of systematic errorEdit

Imperfect calibrationEdit

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. 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 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 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.

QuantityEdit

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.

DriftEdit

Systematic errors which change during an experiment (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 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 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 ammeters and voltmeters 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 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 of the measurement instrument.

Sources of random errorEdit

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. Stochastic errors added to a regression equation account for the variation in Y that cannot be explained by the included Xs.

SurveysEdit

The term «observational error» is also sometimes used to refer to response errors and some other types of non-sampling error.^[1] 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).^[4]^[5]

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.^[6] 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.^[7]^[8]

Effect on regression analysisEdit

If the dependent variable 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 variables is measured with error, then the regression coefficients and standard hypothesis tests are invalid.^[9]^{: p. 187} This is known as attenuation bias.^[10]

ReferencesEdit

^ ^a ^b Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 978-0-19-920613-1
^ ^a ^b John Robert Taylor (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books. p. 94, §4.1. ISBN 978-0-935702-75-0.
^ «Systematic error». Merriam-webster.com. Retrieved 2016-09-10.
^ Salant, P.; Dillman, D. A. (1994). How to conduct your survey. New York: John Wiley & Sons. ISBN 0-471-01273-4.
^ Bland, J. Martin; Altman, Douglas G. (1996). «Statistics Notes: Measurement Error». BMJ. 313 (7059): 744. doi:10.1136/bmj.313.7059.744. PMC 2352101. PMID 8819450.
^ Saris, W. E.; Gallhofer, I. N. (2014). Design, Evaluation and Analysis of Questionnaires for Survey Research (Second ed.). Hoboken: Wiley. ISBN 978-1-118-63461-5.
^ 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 Archived 2019-09-15 at the Wayback Machine
^ Saris, W. E.; Revilla, M. (2015). «Correction for measurement errors in survey research: necessary and possible» (PDF). Social Indicators Research. 127 (3): 1005–1020. doi:10.1007/s11205-015-1002-x. hdl:10230/28341. S2CID 146550566.
^ Hayashi, Fumio (2000). Econometrics. Princeton University Press. ISBN 978-0-691-01018-2.
^ Angrist, Joshua David; Pischke, Jörn-Steffen (2015). Mastering ‘metrics : the path from cause to effect. Princeton, New Jersey. p. 221. ISBN 978-0-691-15283-7. OCLC 877846199. The bias generated by this sort of measurement error in regressors is called attenuation bias.

Two Types of Experimental Error

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No matter how careful you are, there is always error in a measurement. Error is not a «mistake»—it’s part of the measuring process. In science, measurement error is called experimental error or observational error.

There are two broad classes of observational errors: random error and systematic error. Random error varies unpredictably from one measurement to another, while systematic error has the same value or proportion for every measurement. Random errors are unavoidable, but cluster around the true value. Systematic error can often be avoided by calibrating equipment, but if left uncorrected, can lead to measurements far from the true value.

Key Takeaways

Random error causes one measurement to differ slightly from the next. It comes from unpredictable changes during an experiment.
Systematic error always affects measurements the same amount or by the same proportion, provided that a reading is taken the same way each time. It is predictable.
Random errors cannot be eliminated from an experiment, but most systematic errors can be reduced.

Random Error Example and Causes

If you take multiple measurements, the values cluster around the true value. Thus, random error primarily affects precision. Typically, random error affects the last significant digit of a measurement.

The main reasons for random error are limitations of instruments, environmental factors, and slight variations in procedure. For example:

When weighing yourself on a scale, you position yourself slightly differently each time.
When taking a volume reading in a flask, you may read the value from a different angle each time.
Measuring the mass of a sample on an analytical balance may produce different values as air currents affect the balance or as water enters and leaves the specimen.
Measuring your height is affected by minor posture changes.
Measuring wind velocity depends on the height and time at which a measurement is taken. Multiple readings must be taken and averaged because gusts and changes in direction affect the value.
Readings must be estimated when they fall between marks on a scale or when the thickness of a measurement marking is taken into account.

Because random error always occurs and cannot be predicted, it’s important to take multiple data points and average them to get a sense of the amount of variation and estimate the true value.

Systematic Error Example and Causes

Systematic error is predictable and either constant or else proportional to the measurement. Systematic errors primarily influence a measurement’s accuracy.

Typical causes of systematic error include observational error, imperfect instrument calibration, and environmental interference. For example:

Forgetting to tare or zero a balance produces mass measurements that are always «off» by the same amount. An error caused by not setting an instrument to zero prior to its use is called an offset error.
Not reading the meniscus at eye level for a volume measurement will always result in an inaccurate reading. The value will be consistently low or high, depending on whether the reading is taken from above or below the mark.
Measuring length with a metal ruler will give a different result at a cold temperature than at a hot temperature, due to thermal expansion of the material.
An improperly calibrated thermometer may give accurate readings within a certain temperature range, but become inaccurate at higher or lower temperatures.
Measured distance is different using a new cloth measuring tape versus an older, stretched one. Proportional errors of this type are called scale factor errors.
Drift occurs when successive readings become consistently lower or higher over time. Electronic equipment tends to be susceptible to drift. Many other instruments are affected by (usually positive) drift, as the device warms up.

Once its cause is identified, systematic error may be reduced to an extent. Systematic error can be minimized by routinely calibrating equipment, using controls in experiments, warming up instruments prior to taking readings, and comparing values against standards.

While random errors can be minimized by increasing sample size and averaging data, it’s harder to compensate for systematic error. The best way to avoid systematic error is to be familiar with the limitations of instruments and experienced with their correct use.

Key Takeaways: Random Error vs. Systematic Error

The two main types of measurement error are random error and systematic error.
Random error causes one measurement to differ slightly from the next. It comes from unpredictable changes during an experiment.
Systematic error always affects measurements the same amount or by the same proportion, provided that a reading is taken the same way each time. It is predictable.
Random errors cannot be eliminated from an experiment, but most systematic errors may be reduced.

Sources

Bland, J. Martin, and Douglas G. Altman (1996). «Statistics Notes: Measurement Error.» BMJ 313.7059: 744.
Cochran, W. G. (1968). «Errors of Measurement in Statistics». Technometrics. Taylor & Francis, Ltd. on behalf of American Statistical Association and American Society for Quality. 10: 637–666. doi:10.2307/1267450
Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9.
Taylor, J. R. (1999). An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books. p. 94. ISBN 0-935702-75-X.

Источник

Errors using inadequate data are much less than those
using no data at all.

C.
Babbage]

No measurement of a physical quantity can be entirely accurate. It is
important to know, therefore, just how much the measured value is likely
to deviate from the unknown, true, value of the quantity. The art of
estimating these deviations should probably be called uncertainty analysis,
but for historical reasons is referred to as error analysis. This document
contains brief discussions about how errors are reported, the kinds of
errors that can occur, how to estimate random errors, and how to carry
error estimates into calculated results. We are not, and will not be,
concerned with the “percent error” exercises common in high
school, where the student is content with calculating the deviation from
some allegedly authoritative number.

You might also be interested in our tutorial
on using figures (Graphs).

Significant figures

Whenever you make a measurement, the number of meaningful digits that
you write down implies the error in the measurement. For example if you
say that the length of an object is 0.428 m, you imply an uncertainty
of about 0.001 m. To record this measurement as either 0.4 or 0.42819667
would imply that you only know it to 0.1 m in the first case or to 0.00000001
m in the second. You should only report as many significant figures as
are consistent with the estimated error. The quantity 0.428 m is said
to have three significant figures, that is, three digits that make sense
in terms of the measurement. Notice that this has nothing to do with
the «number of decimal places». The same measurement in centimeters
would be 42.8 cm and still be a three significant figure number. The
accepted convention is that only one uncertain digit is to be reported
for a measurement. In the example if the estimated error is 0.02 m you
would report a result of 0.43 ± 0.02 m, not 0.428 ± 0.02
m.

Students frequently are confused about when to count a zero as a significant
figure. The rule is: If the zero has a non-zero digit anywhere to its left,
then the zero is significant, otherwise it is not. For example 5.00 has
3 significant figures; the number 0.0005 has only one significant figure,
and 1.0005 has 5 significant figures. A number like 300 is not well defined.
Rather one should write 3 x 10², one significant figure, or
3.00 x 10², 3 significant figures.

Absolute and relative errors

The absolute error in a measured quantity is the uncertainty in the
quantity and has the same units as the quantity itself. For example if
you know a length is 0.428 m ± 0.002 m, the 0.002 m is an absolute
error. The relative error (also called the fractional error) is obtained
by dividing the absolute error in the quantity by the quantity itself.
The relative error is usually more significant than the absolute error.
For example a 1 mm error in the diameter of a skate wheel is probably
more serious than a 1 mm error in a truck tire. Note that relative errors
are dimensionless. When reporting relative errors it is usual to multiply
the fractional error by 100 and report it as a percentage.

Systematic errors

Systematic errors arise from a flaw in the measurement scheme which
is repeated each time a measurement is made. If you do the same thing
wrong each time you make the measurement, your measurement will differ
systematically (that is, in the same direction each time) from the correct
result. Some sources of systematic error are:

Errors in the calibration of the measuring instruments.
Incorrect measuring technique: For example, one might make an incorrect
scale reading because of parallax error.
Bias of the experimenter. The experimenter might consistently read
an instrument incorrectly, or might let knowledge of the expected value
of a result influence the measurements.

It is clear that systematic errors do not average to zero if you average
many measurements. If a systematic error is discovered, a correction
can be made to the data for this error. If you measure a voltage with
a meter that later turns out to have a 0.2 V offset, you can correct
the originally determined voltages by this amount and eliminate the error.
Although random errors can be handled more or less routinely, there is
no prescribed way to find systematic errors. One must simply sit down
and think about all of the possible sources of error in a given measurement,
and then do small experiments to see if these sources are active. The
goal of a good experiment is to reduce the systematic errors to a value
smaller than the random errors. For example a meter stick should have
been manufactured such that the millimeter markings are positioned much
more accurately than one millimeter.

Random errors

Random errors arise from the fluctuations that are most easily observed
by making multiple trials of a given measurement. For example, if you
were to measure the period of a pendulum many times with a stop watch,
you would find that your measurements were not always the same. The main
source of these fluctuations would probably be the difficulty of judging
exactly when the pendulum came to a given point in its motion, and in
starting and stopping the stop watch at the time that you judge. Since
you would not get the same value of the period each time that you try
to measure it, your result is obviously uncertain. There are several
common sources of such random uncertainties in the type of experiments
that you are likely to perform:

Uncontrollable fluctuations in initial conditions in the measurements.
Such fluctuations are the main reason why, no matter how skilled the
player, no individual can toss a basketball from the free throw line
through the hoop each and every time, guaranteed. Small variations
in launch conditions or air motion cause the trajectory to vary and
the ball misses the hoop.
Limitations imposed by the precision of your measuring apparatus,
and the uncertainty in interpolating between the smallest divisions.
The precision simply means the smallest amount that can be measured
directly. A typical meter stick is subdivided into millimeters and
its precision is thus one millimeter.
Lack of precise definition of the quantity being measured. The length
of a table in the laboratory is not well defined after it has suffered
years of use. You would find different lengths if you measured at different
points on the table. Another possibility is that the quantity being
measured also depends on an uncontrolled variable. (The temperature
of the object for example).
Sometimes the quantity you measure is well defined but is subject
to inherent random fluctuations. Such fluctuations may be of a quantum
nature or arise from the fact that the values of the quantity being
measured are determined by the statistical behavior of a large number
of particles. Another example is AC noise causing the needle of a voltmeter
to fluctuate.

No matter what the source of the uncertainty, to be labeled «random» an
uncertainty must have the property that the fluctuations from some «true» value
are equally likely to be positive or negative. This fact gives us a key
for understanding what to do about random errors. You could make a large
number of measurements, and average the result. If the uncertainties
are really equally likely to be positive or negative, you would expect
that the average of a large number of measurements would be very near
to the correct value of the quantity measured, since positive and negative
fluctuations would tend to cancel each other.

Estimating random errors

There are several ways to make a reasonable estimate of the random error
in a particular measurement. The best way is to make a series of measurements
of a given quantity (say, x) and calculate the mean ,
and the standard deviation from
this data. The mean is defined as

where x_i is the result of the i^th measurement
and N is the number of measurements. The standard deviation
is given by

If a measurement (which is subject only to random fluctuations) is repeated
many times, approximately 68% of the measured valves will fall in the
range .

We become more certain that ,
is an accurate representation of the true value of the quantity x the
more we repeat the measurement. A useful quantity is therefore the standard
deviation of the mean defined
as . The quantity is
a good estimate of our uncertainty in .
Notice that the measurement precision increases in proportion to as
we increase the number of measurements. Not only have you made a more
accurate determination of the value, you also have a set of data that
will allow you to estimate the uncertainty in your measurement.

The following example will clarify these ideas. Assume you made the
following five measurements of a length:

	Length (mm)	Deviation from the mean
	22.8	0.0
	23.1	0.3
	22.7	0.1
	22.6	0.2
	23.0	0.2
sum	114.2	0.18	sum of the squared deviations
	divide by 5	divide by 5 and take the square root	(N = number data points = 5)
mean	22.8	0.19	standard deviation
		divide by
		0.08	standard deviation of the mean

Thus the result is 22.84 ± .08 mm. (Notice the use of significant
figures).

In some cases, it is scarcely worthwhile to repeat a measurement several
times. In such situations, you often can estimate the error by taking
account of the least count or smallest division of the measuring device.
For example, when using a meter stick, one can measure to perhaps a half
or sometimes even a fifth of a millimeter. So the absolute error would
be estimated to be 0.5 mm or 0.2 mm.

In principle, you should by one means or another estimate the uncertainty
in each measurement that you make. But don’t make a big production out
of it. The essential idea is this: Is the measurement good to about 10%
or to about 5% or 1%, or even 0.1%? When you have estimated the error,
you will know how many significant figures to use in reporting your result.

Propagation of errors

Once you have some experimental measurements, you usually combine them
according to some formula to arrive at a desired quantity. To find the
estimated error (uncertainty) for a calculated result one must know how
to combine the errors in the input quantities. The simplest procedure
would be to add the errors. This would be a conservative assumption,
but it overestimates the uncertainty in the result. Clearly, if the errors
in the inputs are random, they will cancel each other at least some of
the time. If the errors in the measured quantities are random and if
they are independent (that is, if one quantity is measured as being,
say, larger than it really is, another quantity is still just as likely
to be smaller or larger) then error theory shows that the uncertainty
in a calculated result (the propagated error) can be obtained from a
few simple rules, some of which are listed in Table 1. For example if
two or more numbers are to be added (Table 1, #2) then the absolute error
in the result is the square root of the sum of the squares of the absolute
errors of the inputs, i.e.

then

In this and the following expressions, and are
the absolute random errors in x and y and is
the propagated uncertainty in z. The formulas do not apply to
systematic errors.

The general formula, for your information, is the following;

It is discussed in detail in many texts on the theory of errors and
the analysis of experimental data. For now, the collection of formulae
in table 1 will suffice.

Table 1: Propagated errors in z due to errors in x and y.
The errors in a, b and c are assumed to be
negligible in the following formulae.