Systematic error and random error

While measuring a physical quantity, we do not expect the value obtained to be the exact true value. It is important to give some sort of indication of how close the result is likely to be the true value, that is to say, some indication of the precision of reliability of the measurements. In physics, we do this by including an estimate of the error along with the result value. When analyzing the results, it is important to consider the sources of error and how these sources affected the results. The errors and uncertainty of measurements are always estimated in an indirect way and the calculations include some assumptions. Errors may be divided into two primary kinds, systematic and random errors. A systematic error is the one that remains constant or changes in a regular fashion in repeated measurements of one and the same quantity. On the contrary, a random error is the one that varies and which is likely to be positive or negative. Let’s take a look at some key differences between the two.

What is Systematic Error?

Systematic error is the one that occurs in the same direction each time and it remains constant or changes in a regular fashion in repeated measurements of one and the same quantity. A systematic error remains constant throughout a set of readings and causes the measured quantity to be shifted away from the accepted or predicted value. Systematic errors occur because the experimental arrangement is different from that assumed in the theory and the correction factor which takes account of this difference is ignored. In many cases, such errors are caused by some flaw in the experimental apparatus. Systematic error can be eliminated by using proper technique, calibrating equipment and employing standards.

What is Random Error?

As its name implies, random error is the one that varies in a random manner and which is produced by unpredictable and unknown variations in the total experimental process. Any type of error that is inconsistent and does not repeat in the same magnitude or direction except by chance is considered to be a random error. Gulliksen defines random error in a statistical sense in terms of the mean error, the correlation between the error and the true score, and correlation between errors being zero. For example, wind speed may drop and pick up at different points in time resulting in variations in the results. Random error is discovered by performing measurements of same quantity repeatedly under the same conditions.

Difference between Systematic and Random Error

Meaning of  Systematic vs.  Random Error

Errors can be divided into two primary kinds, systematic and random errors. Systematic error, as the name implies, is a consistent, repeatable error that deviates from the true value of measurement by a fixed amount. Systematic error is the one that occurs in the same direction each time due to the fault of the measuring device. On the contrary, any type of error that is inconsistent and does not repeat in the same magnitude or direction except by chance is considered to be a random error. Random errors are sometimes called statistical errors.

Nature of Systematic vs. Random Error

Random errors are discovered by performing measurements of the same quantity number of times under the same conditions and they involve the variability inherent in the natural world and in making any measurement. Systematic errors, on the other hand, can be discovered experimentally by comparing a given result with a measurement of the same quantity performed using a different method or by using a more accurate measuring instrument. Systematic errors give results that are either consistently above the true value or consistently below the true value.

Cause of Systematic vs. Random Error

Systematic errors are consistent and are caused by some flaw in the experimental apparatus or a flawed experimental design. Such errors are caused by faulty measuring devices that are either used incorrectly by individuals while taking the measurement or instruments that are imperfectly calibrated. Systematic errors are believed to be more dangerous than random errors. Random errors, on the other hand, are caused by unpredictable variations in the readings of a measurement device or by an observer’s inability to interpret the instrumental reading.

Elimination

Systematic errors can be eliminated by using proper technique, calibrating equipment and employing standards. Systematic errors are usually produced by faulty human interpretations or changes in environment during the experiments, which are difficult to eliminate completely. Repeated measurements with the same instrument neither reveal nor do they eliminate a systematic error. In principal, all systematic errors can be eliminated, but there will always remain some random errors in any measurement. Random errors, however, can be reduced by taking average of a large number of observations.

Systematic vs. Random Error: Comparison Chart

Summary of Systematic vs. Random Error

In principal, all systematic errors can be eliminated, but there will always remain some random errors in any measurement. Random errors, however, can be reduced by taking average of a large number of observations. Systematic errors are usually produced by faulty human interpretations or changes in environment during the experiments, which are difficult to eliminate completely. This is why systematic errors are potentially more dangerous than random errors. However, systematic errors can be eliminated by using proper technique, calibrating equipment and employing standards.

No matter how careful you are when conducting experiments, there will likely be an experimental error. Whether through the challenges inherent taking the measurements accurately or problems with your equipment, avoiding error altogether is next to impossible. To counteract this issue, scientists do their best to categorize errors and quantify any uncertainty in measurements they make. Finding out the difference between systematic and random errors is a key part of learning to design better experiments and to minimize any errors that do creep through.

What Is Random Error?

Random error describes errors that fluctuate due to the unpredictability or uncertainty inherent in your measuring process, or the variation in the quantity you’re trying to measure.

A scientist measuring an insect, for example, would try to position the insect at the zero point of a ruler or measuring stick, and read the value at the other end. The ruler itself will probably only measure down to the nearest millimeter, and reading this with precision can be difficult. You may underestimate the true size of the insect or overestimate it, based on how well you read the scale and your judgment as to where the head of the insect stops. The insect might also move ever so slightly from the zero position without you realizing. Repeating the measurement multiple times yields many different results because of this, but they would likely cluster around the true value.

Similarly, taking measurements of a quantity that changes from moment to moment leads to random error. Wind speed, for example, may pick up and fall off at different points in time. If you take a measurement one minute, it probably won’t be exactly the same a minute later. Again, repeated measurements will lead to results that fluctuate but cluster around the true value.

What Is Systematic Error?

A systematic error is one that results from a persistent issue and leads to a consistent error in your measurements. For example, if your measuring tape has been stretched out, your results will always be lower than the true value. Similarly, if you’re using scales that haven’t been set to zero beforehand, there will be a systematic error resulting from the mistake in the calibration (e.g., if a true weight of 0 reads as 5 grams, 10 grams will read as 15 and 15 grams will read as 20).

Other Differences Between Systematic and Random Errors

The main difference between systematic and random errors is that random errors lead to fluctuations around the true value as a result of difficulty taking measurements, whereas systematic errors lead to predictable and consistent departures from the true value due to problems with the calibration of your equipment. This leads to two extra differences that are worth noting.

Random errors are essentially unavoidable, while systematic errors are not. Scientists can’t take perfect measurements, no matter how skilled they are. If the quantity you’re measuring varies from moment to moment, you can’t make it stop changing while you take the measurement, and no matter how detailed your scale, reading it accurately still poses a challenge. The good news is that repeating your measurement multiple times and taking the average effectively minimizes this issue.

Systematic errors may be difficult to spot. This is because everything you measure will be wrong by the same (or a similar) amount and you may not realize there is an issue at all. However, unlike random errors they can often be avoided altogether. Calibrate your equipment properly prior to using it, and systematic errors will be much less likely.

From Wikipedia, the free encyclopedia

«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 experiments[edit]

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:

1. Systematic error which always occurs, with the same value, when we use the instrument in the same way and in the same case.
2. 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.

Characterization[edit]

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.

Sources[edit]

Sources of systematic error[edit]

Imperfect calibration[edit]

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.

Quantity[edit]

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[edit]

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 error[edit]

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.

Surveys[edit]

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 analysis[edit]

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]

See also[edit]

References[edit]

Further reading[edit]

• Cochran, W. G. (1968). «Errors of Measurement in Statistics». Technometrics. 10 (4): 637–666. doi:10.2307/1267450. JSTOR 1267450.