The Nature of Measurement: Part III: Dealing With Errors

The last part in this series defined "error" and made a differentiation between mistake and error, then separated errors between "systematic" and "random." This part of the series further explains errors and how to deal with them.

Dealing With Systematic Errors

Once discovered and quantified, systematic errors can be essentially compensated or corrected by either observing (measuring) in such a way that the error is canceled in the mechanical process of measuring or by calculating a correction and adding this to the observation (reading).

The first requirement is to recognize and accept the possible existence of errors. Next, identify the various sources that might be affecting a reading systematically. Then, the analyst must determine what the "system" is. That is, what is the relationship between the changes in readings and the size of the quantity, as affected by a particular systematic error? Is it a constant, unchanging with the size of the quantity? Is it linear, changing in direct proportion to the change in the variable causing the error, and/or in proportion to the size of the quantity being measured? Or, does it follow some other mathematical relationship? Is there some physics involved?

As an example, we discovered many years ago that a material will stretch when tension is applied to it. The thicker the material, the less it will stretch, per unit of length, given any particular tension. We have tested the surveyor's steel tape and determined that the expansion caused by tension is fairly linear for this ductile material. Thus, we can use a formula for the systematic error that considers the tension applied, the length over which it is applied, the cross-sectional area of the tape, and the elasticity of steel. The stretch obeys known mathematical and physical laws.

All instruments must be calibrated; that is, compared with a standard, under known (measurable) conditions. For example, we calibrate a tape by comparing its length with a standard known to be "accurate" (another subject in this series), under known tension, temperature (since the tapes expand and contract with changes in temperature), and "support condition" (whether laying fully supported or supported only between the points of reading).

The general formula for systematic errors is T = R + C, where T = the "true" value, R = the reading, and C = the total of all observable corrections. On the cover of my book, entitled Surveying Measurements and their Analysis, and on my seminar brochures for Surveyor's Educational Seminars, is the caption, "The truth is equal to one's initial observations plus the corrections discovered through added experience or knowledge." You see, error theory can be applied to affairs outside the world of measurement. A person often is far from possessing the truth of a perceived matter until that person is willing to get rid of biases or prejudices affecting such observation. As has been said, measurement theory is somewhat of a mirror of many things in life. Understanding these concepts helps a person to deal with life and its pitfalls.

Examples of Systematic Errors

To illustrate systematic errors, I'll use my digital, battery-operated bathroom scale as an example. One might think it is exact, since it is electronic (a common fallacy, even among some surveyors). To check the scale, I placed 25-pound barbell plates on it, one at a time. Before I did this, I noted that it read -1.1 pounds (actually 998.9) with nothing on it. Then, when adding the barbell plates one at a time, I observed readings of 24.2, 49.5, 74.9, and 100.1 lbs.

The -1.1 pound indexing error represents what is called a constant error. I must add 1.1 pounds to anything I weigh, regardless of its mass. I can use it to correct the readings for the barbell plates. The readings, corrected for this constant error are 25.3, 50.6, 76.0 and 101.2.

At this point the observer must begin to use what we call "measurement analysis" in surveying. The data must be interpreted so that it can be applied to our benefit. If I divide 50.6 by two, I get 25.3, the same number I observed for the one plate. If I divide 76.0 by three, I get 25.33, almost exactly the 25.3 for one or two plates. And, 101.2 divided by four is also 25.3. If the plates were manufactured accurately at 25.0 pounds each (a faith assumption I chose to make here), then I have a "systematic error" of +0.3 pounds per each 25 pounds, or +1.2 for each 100 lbs. It is "systematic" because it follows a "system." It is predictable. It obeys mathematical laws. There is a linear relationship between the weight and the error.

To apply this new knowledge to my body weight, I would immediately add 1.1 pounds to what I observe, then subtract 0.012 pounds times the reading. An algebraic equation is True Weight = Observed Weight + Constant Correction + Variable Correction. In my case for one weighing, TW = 144.5 +1.1 -0.012(144.5) = 143.9 lbs. (to the nearest tenth).

It is observed that, to apply a correction, I simply change the sign of the error. That is, since the constant error was negative, the correction for it is positive. Likewise, since the variable error was positive (the reading was high for a "known" weight), the correction to compensate for it is negative in sign.

Example One

As an illustration of the mistakes that can be made in neglecting systematic errors such as these, I will recall an experience in accompanying my wife to the doctor a few times several years ago. They weighed her each time on the same scale. The nurse recorded the weight for the doctor to use in his professional decisions. I distinctly remember that my wife was wearing light clothing one warm day and heavier clothes on a cooler day. I knew that both weights were inaccurate, as they did not account for the clothing at all, and that the discrepancy between the two weight readings was also inaccurate because the difference in bulk of the clothing was ignored. Yet, this $60-per-hour physician was making judgments about my wife's health and progress, partly based on this data! He was making comments and recommendations affected by just a couple of pounds, and the systematic errors exceeded this amount. By the way, my wife needed to gain a little, not lose weight. She'd want me to say that.

Example Two

I used to do a lot of measuring of road race courses, when I was running marathons and other distance races. To check the accuracy of a course, I would calibrate my automobile odometer, using the Interstate Highway mile markers. On a long trip, I would glance at the odometer the instant I crossed one of those little green mile posts and estimate the odometer reading to hundredths of a mile. After 10 or 20 miles, a pattern would become apparent, much like what happened with the barbell plate weighing. One such odometer calibration yielded readings of 1.02, 2.02, 4.06, 5.08, 7.11, 10.16, 12.20, 15.25, 18.29, and 20.32. A little bit of analysis here will reveal that every time I had an accurate mile (applying faith again to the accuracy of the mile posts and intervals between), the reading averaged 1.016 miles. Thus, in the future, in checking a 10- mile race course, marked by the race director, I would "lay out" readings of 1.02, 2.03, 3.05, 4.06, 5.08,….., 10.16 miles. In 1973, I found that the Athens Marathon (that's in Ohio, not Greece) was about 0.75 miles short of the regulation distance of 26.21875 miles, and in 1974, it was approximately 1.5 miles long! I used to be unpopular with both race directors and the runners. Those runners in 1973 thought they had all set "PRs" (personal records), since the course was so short. They wanted to deny my findings. They were on a "high" from their apparent success, and the race director wanted me to shut up and "just come and run" like everybody else (his exact words, in fact). But, in 1974, some of the runners were willing to listen to me. The race director really hated me after what I publicized about the long course.

Several years later, a professional runner was denied a $50,000 bonus when he missed breaking the world- best time for a marathon by less than five seconds. I am convinced to this day that the course was long, considering the methods used at the time for the measuring and subsequent calibration/certification of the course. The man was probably cheated out of a bonus because of misunderstandings about measurement science.

Consequences of Systematic Errors

The example about my wife's weight when visiting the doctor, the checking and calibration of road race courses and other such examples help to illustrate how misunderstandings about measurement can hurt somebody. The results can cause undeserved celebrations, undeserved disappointments, unnecessary cost, lost rewards, lost time and inconvenience.

In surveying, when systematic errors occur and are not corrected or compensated, surveyors often have unnecessarily large discrepancies between their measurements and those of others, and even between duplications of their own measurements. This causes embarrassment for the profession and confusion for the users of the data. Systematic errors that go uncorrected make us look silly and incompetent.

Analysis of Random Errors

Random errors follow statistical behavioral laws, such as probability and compensation, which dictate a tendency of plus and minus errors to cancel, predict that large errors occur less frequently than small ones, and say that an error of any particular size is as likely to be positive as negative. A characteristic theoretical pattern of error distribution occurs upon analysis of a large number of repeated measurements of a quantity. Thus, the mathematics disciplines associated with random errors are statistics and probability. It is not geometry, although geometry may affect the results. For example, I have determined, through statistical analysis, that the standard deviation in reading a one-second optical theodolite is consistently between 0.6 and 1.0 seconds. The value varies with skill of the observer, quality and cleanliness of the optics of the instrument, lighting and probably other factors. Statistical analysis of random errors is the subject of another in this series on measurement.

Two Types of Measurements

Measurements can be direct or indirect. A direct measurement is one where the reading observed represents the quantity measured, without a need to add, take averages or use geometric formulas to compute the value desired. An indirect measurement requires calculation. After a little experience, surveyors begin to realize that few, if any, measurements are truly direct. For one reason, averages are usually taken to arrive at a final reading, or readings are often combined in some other way, such as when tape lengths are added together. Also, many values are indirectly determined from other measured values, such as areas or unknown sides of a triangle. But, the most important reason why measurements are generally indirect is that calculation must be performed to correct for systematic errors. "What you see" is not what you have. It is the indirect nature of measurements that forces the need to often apply some rather sophisticated mathematical procedures to analysis of errors and thus determine a "best value" to represent the size of the quantity. Some of these refinements will be discussed later in this series.

Learning the Difference Is Important

Learning the difference between a measurement and a count, between a mistake and an error, and between a systematic and a random error is extremely important in coping with numerical data of any kind. Similarly, the difference between precision and accuracy must be understood. These terms and their meanings are the subject of the next article. Gradually, the full meaning of this very enlightening science will be revealed.

Ben Buckner is an educator, author, seminar presenter with Surveyors' Educational Seminars and a Contributing Editor for the magazine.

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