The Nature of Measurement: Part II, Mistakes and Errors

In the first part of this series, a distinction was drawn between numbers as counts versus numbers as measurements. It was explained that counts can be exact whereas measurements are always inexact. We ended with the thought that counts are susceptible to mistakes, whereas measurements are susceptible to both mistakes and errors.

As with counts and measurements, I maintain that most of us have also been conditioned to think that error and mistake are the same things. Our education, the media, literature and even dictionaries mostly use the terms as synonyms. As a result, most of the public, and perhaps even some in the sciences and engineering, may have trouble analyzing numbers which derive from measurements. The difference between a mistake and an error is important to understand if you are to analyze measured data properly. If we accept the proposition that counts are susceptible to mistakes only, and measurements are susceptible to both mistakes and errors (as we define the terms), analysis of numerical data must follow different procedures, depending on the nature of the data (whether count or measurement.)

Cause and Prevention of Mistakes

A mistake is a "blunder," a "goof," a "slip-up." Mistakes occur in both counting and measuring, and in a lot of other things. Mistakes in measurement can be traced to carelessness, inattention, improper training, bad habits, lack of innate ability, poor judgment, adverse measuring or observing conditions, and various negative attitudes, emotions and perceptions that plague humans.

Mistakes can be caused by making a decision without sufficient information or evidence. People often try to use their intellect, experience or precedent to decide something, without having gathered relevant evidence concerning the specific matter under investigation. Land surveyors, for example, often make decisions about the location of property corners without sufficient evidence. This failure falls under the category of poor judgment or bad habits.

We make mistakes through transposing numbers, striking wrong keys on calculators, misreading scales, etc. In everyday affairs, people make mistakes in a similar manner, perhaps mis-dialing a phone number or using the wrong date on a letter.

I feel that there are few true "accidents." Most should be classified as mistakes, and someone should take responsibility for them. The fender benders I observe on a section of curved road near where I live on frosty mornings are caused by lack of proper training, bad habits, negative emotions and attitudes such as impatience and selfishness, or just plain carelessness and lack of common sense. Human inability to confront and control individual oversights and poor judgments lurk behind most "accidents" or mistakes.

Many mistakes are easily recognized and realized. Others may have such small effects that they go unnoticed. An example in measurement would be transposing the last two digits in a large number like 1,834.65 feet. Others, based more on judgment or acting without complete information, may not have immediate, or consistent, consequences. Driving too fast for conditions may not always result in disaster, for example. Voting for the crook instead of the honest person or buying a faulty product may not have immediate consequences. The mistake is realized later, as "hindsight" reveals it.

Mistakes will never be completely eliminated from measurements, but they can be reduced in most cases by developing the measurer in a way that he or she learns to be more careful, attentive, conscientious and proud of the work being done. Through proper training and development of good work habits, development and maintenance of positive attitudes, and understanding the theory and practice involved with the variable being measured, mistakes can be controlled and practically eliminated. We will always need to confront them, however, because human imperfections make them inevitable.

Errors and Their Sources

When properly defined, "error" pertains only to measurements—that is, to estimating anything where exactness is not possible. It does not apply to counts, where exactness is possible. Errors are unavoidable even for the most thoroughly trained and motivated measurer. They occur to some extent in virtually every measurement because of imperfections of instruments and people, as well as influences of the natural environment. There are basically two types of errors in measurements: systematic and random. I will explain them later. First, let's examine the sources of errors. There are four of them:

Natural errors. Measurements are usually made in an environment that is essentially uncontrollable (outdoors). Effects on instruments and processes from such factors as temperature, atmospheric pressure, atmospheric refraction, humidity, solar and other heat, wind, gravity, and earth's curvature must be measured, and readings must be corrected for these variables if accurate results are to be expected.

Instrumental errors. All measurements employ instruments, from the simple plumb line to the most sophisticated electronic apparatus. Some error is always present in the measurements due to imperfection in manufacture, adjustment or basic characteristics of the instrument. Even when "in adjustment" there is error, since the adjusting process usually must involve human manipulation and judgment with no perfection in the procedure used.

Personal errors. Since humans are directly involved with all measurements, and since humans are imperfect, errors are inevitable in measurements. Automation and electronics have reduced personal errors in measurements, but not eliminated them. People still perform centering and alignment judgments, for example, even when readings are digital.

Calculation errors. Unless sufficient digits are recorded and carried through all computation steps, and unless conversion factors and constants contain sufficient digits, round-off errors occur. Significant figures in measurements directly affect the significant figures in computed results. Significant figures and round-off errors, and the broader subject of precision, are subjects for other parts in this series.

Systematic Errors

Systematic errors are those that generally obey or follow some mathematical or physical law. The cause of such errors can usually be traced to instrument maladjustment, lack of calibration, or the environment. If they are discovered, they can be quantified through instrument tests or calibration and through understanding the various effects of nature. Since they are discoverable, they essentially can be corrected. Systematic errors occur when the cross-hairs of a surveying instrument get out of adjustment, or when a surveyor's steel tape or a tailor's cloth tape become stretched. They occur when the tires on a vehicle are not of a diameter which would yield a true distance consistent with what the odometer reads. They occur in electronic distance measurements because the measurements are affected by changes in the temperature and atmospheric pressure. They occur because of manufacturing errors in graduations of any type of scale.

Random Errors

Random errors are unavoidable. They follow random patterns, or the laws of "chance." They have unknown signs; thus, they are expressed as "plus or minus." The magnitude of such an error is unknown, but it can be estimated. These errors are caused by human and instrument imperfections as well as uncertainties in determining the effects of the environment on measurements.

Personal errors are nearly all random in nature. People cannot perceive anything with exactness. In surveying, this refers to the alignment of cross-hairs on targets, centering of instruments over ground points, reading rods and scales, centering level bubbles, etc. Random errors are small misjudgments, not mistakes. They are what happens when a person tries to "do it right," but misses the mark by a small amount due to imperfections in the system (which includes himself).

Furthermore, all instruments, even so-called automatic systems with precise digital readings, have certain imperfections in all their components, be it the optics, electronics, or mechanical features. Unpredictable changes in adjustment of instruments, or loose fitting parts can cause random errors.

There will inevitably be uncertainties in determining all variables affecting instruments that are used in a natural environment. In fact, some influences are essentially unmeasurable, such as constantly varying wind speed and changing radiant heat from the sun. Calculation errors (round-off errors) have a random effect on calculated results. In a long series of calculations, such an error "propagates" and affects both intermediate and final results. Error propagation will be discussed in another part of this series.

Understanding the nature of random errors helps to understand why systematic errors are never really fully corrected, since the observation of the physical phenomena causing the error, or the aligning and calibration of instruments in itself contains personal, random errors. Thus, measurements have "uncertainties" or random errors which remain unquantifiable. Random errors are dealt with by controlling or managing them. It is a quality control process. They cannot be corrected or eliminated, only minimized and controlled.

Mistakes vs. Random Errors

Many people, even professionals who use measurements regularly have some difficulty differentiating between mistakes and random errors when applying the concepts in practice. The difference deserves a little more explanation here, since the misconceptions are so widespread.

Mistakes occur because of negligence, while random errors occur due to imperfection. Here we will define negligence as either deliberate or wilful deviation from accepted practices or adopted standards (whether what is accepted or adopted is known to the individual or not), or an occurrence caused by carelessness, acting with insufficient or faulty information, etc.

Mistakes are either deliberate, as in fraud or tampering with data, or they occur because someone is unwilling to study, learn and employ correct procedures, to control emotions, to keep in practice, to focus on the task at hand, or to think. By contrast, random errors occur naturally, even when the individual is attempting to perform the procedure correctly.

It is true that there is a "gray area" between random errors and mistakes. If a person is too hasty in some mechanical measuring procedure, for example, the large random errors that occur start to look a lot like mistakes. Since it is poor judgment to hurry in such an instance, the results have a lot of scatter, and some might say that they contain small mistakes. Whether they are small mistakes or large random errors does not matter as much as knowing how to deal with the data.

Mostly, however, there is a clear distinction between a mistake and a random error. If a person observed 1,874.56 feet from a scale and recorded it as 1,874.65 feet, that is a mistake. Let us say, however, that one person observed the reading as 1,874.56 feet and another person observed it as 1,874.65 feet, and neither person made a mistake in either observing or recording the numbers as they appeared to that person. If the range of 0.09 feet is acceptable according to standards and specifications, we can say that the difference between the readings was caused by random errors.

We have all learned that a miscount or any mistake is generally frowned upon, and reflects upon us in a negative way. People sometimes get punished for mistakes, and rewarded for consistently avoiding them. This is as it should be. However, it has also seemingly taught us to think of everything as either "right" or "wrong." We have grown to feel that someone must be blamed for the "wrongs." We cannot apply this process to inexact, perceived things. Two people can conscientiously measure the same quantity and get different values, even after applying careful measurement analysis and correcting for systematic errors. What is wonderful about the science of measurement is that both these people can be equally "right." Isn't that a mirror of life itself? Two people can perceive a situation differently and neither is considered "wrong" (unless, of course, one of them has not corrected initial perceptions for bias or prejudice.) The key to "getting it right," in measurement or anything else, is to recognize, then either avoid or remove, both mistakes and errors.

A point we will develop further in this series is that, truly, "to err is human." But that does not apply in the same sense to mistakes. Although it is "human" to do both, a person should never rationalize mistakes away by citing this old quotation. To do so is a "cop-out" if it applies to mistakes. It avoids taking responsibility and indicates an unwillingness to be held accountable. However, there is no need for condemnation, criticism, blame or apology where random errors are concerned (assuming they are controlled and managed).

To cope with numerical data of any kind, it is extremely important to learn the difference between a measurement and a count, between a mistake and an error, and between a systematic and a random error.

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

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