Minimizing Systematic Error

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 to consider:

  1. 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.
  2. 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 described below.
  3. 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.
  4. 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 unknown quantities.

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.

Introduction

Main Body
•Experimental Error

•Minimizing Systematic Error

•Minimizing Random Error

•Propagation of Error

•Significant Figures

Questions