Systematic errors may be also present in the result of a 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) 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 100°, 0°, or -100°, the measured temperature will be 102° (systematic error = +2°), 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.
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 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 589.0 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.
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