# Systematic error

"Systematic bias" redirects here. It is not to be confused with Systemic bias.

Systematic error is an error that is not determined by chance but is introduced by an inaccuracy (as of observation or measurement) inherent in the system.[1]

Systematic error may also be an error having a nonzero mean, so that its effect is not reduced when observations are averaged.[2]

## Systematic Errors in the Sciences

### 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 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

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.

### Removing Systematic Errors

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.

In a statistical 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 (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. 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 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 error and systematic error.[3] 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 independent measurements will usually reduce random errors by a factor of n), 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.

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.