Errors and Calibration, Page 1
Errors and Calibration Author: John M. Cimbala, Penn State University
Latest revision: 27 August 2009
Random vs. Systematic Errors
• There are two general categories of error: systematic (or bias) errors and random (or precision) errors.
• Systematic errors (also called bias errors) are consistent, repeatable errors. For example, suppose the first
two millimeters of a ruler are broken off, and the user is not aware of it. Everything he or she measures will
be too short by two millimeters – a systematic error.
• Random errors (also called precision errors) are caused by a lack of repeatability in the output of the
measuring system. The most common sign of random errors is scatter in the measured data. For example,
background electrical noise often results in small random errors in the measured output.
Systematic (or Bias) Errors
• Systematic errors are consistent, repeatable errors in a set of measurements
• The systematic error is defined as the average of measured values minus the true value.
• Systematic errors arise for many reasons. Here are just a few:
o calibration errors – perhaps due to nonlinearity or errors in the calibration method.
o loading or intrusion errors – the sensor may actually change the very thing it is trying to measure.
o spatial errors – these arise when a quantity varies in space, but a measurement is taken only at one
location (e.g. temperature in a room - usually the top of a room is warmer than the bottom).
o human errors – these can arise if a person consistently reads a scale on the low side, for example.
o defective equipment errors – these arise if the instrument consistently reads too high or too low due to
some internal problem or damage (such as our defective ruler example above).
• A nondimensional form of bias error is the mean bias error, defined as MBE = systematic error / true value.
Random (or Precision) Errors
• Random errors are unrepeatable, inconsistent errors in the measurements, resulting in scatter in the data.
• The random error of one data point is defined as the reading minus the average of readings.
• Example:
Given: Five temperature readings are taken of some snow: -1.30, -1.50, -1.20, -1.60, and -1.50oC.
To do: Find the maximum magnitude of random error in oC.
Solution: The mean (average) of the five temperature readings is -1.42oC. The largest deviation from this is
the reading of -1.20oC. The random (or precision) error for this data point is defined as the reading minus
the average of readings, or -1.20 - (-1.42) = 0.22oC. Thus, the maximum absolute value of random error
is 0.22oC. You can verify that the magnitude of the random error for any of the other data points is less
than this.
Comment: Five readings are not sufficient to measure a good average reading, especially in this example
since the individual readings vary so widely.
Accuracy
• Accuracy is the closeness of agreement between a measured value and the true value.
• Accuracy error is formally defined as the measured value minus the true value.
A
• The accuracy error of a reading (which may also be called inaccuracy or
uncertainty) represents a combination of bias and precision errors.
• The overall accuracy error (or the overall inaccuracy) of a set of readings is
defined as the average of all readings minus the true value. Thus, overall accuracy
error is identical to systematic or bias error.
Precision
• Precision characterizes the random error of the instrument’s output.
• Precision error (of one reading) is defined as the reading minus the average of
readings. Thus, precision error is identical to random error. B
• The drawings to the right show the difference between precision and accuracy. Two
people, A and B, shoot guns at targets. Both people shoot eight times. Each plus
sign marks the spot where a bullet hits the target. It is proper to say:
A is more precise than B, but not as accurate.
On the other hand, B is more accurate than A, but not as precise.
• Instrument precision is often associated with instrument resolution, but these are not
the same thing. An instrument can have great resolution, but poor precision.
