Systematic error can be difficult to identify and correct. Given
a particular experimental procedure and setup, it doesn't matter
how many times you repeat and average your measurements; the error
remains unchanged. No statistical analysis of the data set will
eliminate a systematic error, or even alert you to its presence.
Systematic error can be located and minimized with careful analysis
and design of the test conditions and procedure; by comparing your
results to other results obtained independently, using different
equipment or techniques; or by trying out an experimental procedure
on a known reference value, and adjusting the procedure until the
desired result is obtained (this is called calibration). A few items
- What are the characteristics of your test equipment, and of
the item you are testing? Under what conditions will the instrument
distort or change the physical quantity you are trying to measure?
For example, a voltmeter seems straightforward enough. You hook
it up to two points in a circuit and it gives you the voltage
between them. Under conditions of very low current or high voltage,
however, the voltmeter itself becomes a significant part of the
circuit, and the measured voltage may be significantly altered.
Similarly, a large temperature probe touched to a small object
may significantly affect its temperature, and distort the reading.
- Check that any equations or computer programs you are using
to process data behave in the way you expect. Sometimes it is
wise to try a program out on a set of values for which the correct
results are known in advance, much like the calibration of equipment
- It is unusual to make a direct measurement of the quantity you
are interested in. Most often, you will be making measurements
of a related physical quantity, often several times removed, and
at each stage some kind of assumption must be made about the relationship
between the data you obtain and the quantity you are actually
trying to measure. Sometimes this is a straightforward conversion
process; other cases may be more subtle. For example, gluing on
a strain gauge is a common way to measure the strain (amount of
stretch) in a machine part. However, a typical strain gauge gives
the average strain along one axis in one particular small area.
If it is installed at an angle to the actual strain, or if there
is significant strain along more than one axis, the reading from
the gauge can be misleading unless properly interpreted.
- Calibration: Sometimes systematic error can be tracked down
by comparing the results of your experiment to someone else's
results, or to results from a theoretical model. However, it may
not be clear which of the sets of data is accurate. Calibration,
when feasible, is the most reliable way to reduce systematic errors.
To calibrate your experimental procedure, you perform it upon
a reference quantity for which the correct result is already known.
When possible, calibrate the whole apparatus and procedure in
one test, on a known quantity similar in size and type to your
EXAMPLE: Suppose that you want to calibrate a standard mechanical
bathroom scale to be as accurate as possible. It has lines marked
on the dial every two pounds, and a small knob for zero adjustment.
You can get a fairly good idea of its precision by stepping on and
off of it several times , and looking at the variation between measurements.
If the measured weight varies between 149 and 151 pounds, for example,
the precision is about one pound. The accuracy cannot be any better
than this, but it can certainly be worse, particularly if the scale
has not been calibrated recently. If the scale is linear, a plot
of the actual weight vs. the weight as measured by the scale (the
calibration curve) will be a straight line, and can be determined
by establishing two calibration points. For convenience, the first
reference weight is usually zero, though it need not be.
The first step in calibrating the scale, therefore, is to adjust
the scale to read zero when there is nothing on it. The second step
is to see what it reads with a known weight on it. This second calibration
point should be as far from the first as feasible, to establish
an accurate calibration curve. This known weight could be obtained
by weighing yourself on a scale known to be highly accurate (in
a doctor's office, for example), and then immediately weighing yourself
on the bathroom scale. Suppose that the true weight is known to
be 160 pounds, and the scale reading averages 150 pounds. Some instruments
have a range adjustment to correct this error, but bathroom scales
generally don't. Instead, you would note that the true weight is
6.7% higher than what the scale reads, and the calibration would
be complete. If the scale read 75 pounds, you'd know that the true
weight was 80 pounds, and so forth.
If the scale was not linear, you would have to use many different
calibration weights to produce a well-defined calibration curve.
Not surprisingly, engineers use linear measurement equipment whenever
possible. Even if the scale were somewhat nonlinear, you could still
get good accuracy in the region of your weight with only two calibration
points. For example, if you weigh 160 pounds, you could calibrate
the scale at 155 and 165 pounds, and be reasonably certain that
the slope of the calibration curve would not change significantly
over a 10 lb. interval. Far outside that interval, though, the scale
could be quite inaccurate.