The Nature of Measurement: Part IV-Precision and Accuracy

In this series we have discussed the difference between numbers as measurements and numbers as counts, the difference between errors and mistakes, and the various types and categories of measurement errors. Counts, we learned, can be exact whereas measurements cannot. Mistakes and errors are also different concepts, and occur for fundamentally different reasons, despite the fact that we have been conditioned to think of them as synonymous. Mistakes should be seen as misjudgments caused by carelessness and similar causes. Errors, on the other hand, arise from imperfections in instruments and the perceptions of people estimating readings and other variables, as well as fluctuations in the environmental conditions surrounding the measuring operations. Using the concepts as defined in measurement science, we found that errors cannot occur in counts. Errors can occur only in measurements. Mistakes can occur in both counts and measurements. We also defined two types of errors: systematic and random. Systematic errors obey mathematical and/or physical laws, and thus are predictable, correctable or avoidable. Random errors, by contrast, were explained as being unavoidable, because of imperfections in measurement systems (people, instruments, and nature). Such errors can be controlled, minimized, investigated and estimated, but never eliminated. They follow statistical laws of probability, and in this sense do have some predictability when studies are made to evaluate them.

With this as background, we can now look at two other very important concepts in measurement: precision and accuracy. Confusing these two concepts can lead to incorrect analysis of both measurement data and computed results, and either incorrectly written standards or mis-application of theoretically correct standards. Precision and accuracy relate directly to random and systematic errors, respectively.

Definitions of Terms

As with mistakes and errors, precision and accuracy have unfortunately been treated as synonyms among the general populace. However, in the world of measurement (and, in reality, everywhere else, whether it is recognized or not) there is an important difference between these concepts, regardless of words used to describe them.

Precision is in one sense related to the care and refinement of the measuring process, including instrument quality, attentiveness of the observer, stability of the environment in which the measurements are executed and overall design of the survey. In another sense, it is the degree of numerical agreement among measurements of the same quantity, or the repeatability of the readings. In still a third sense, it is simply the number of decimal places expressed in a measured quantity. All three meanings are consistent, since the care and refinement of the method directly affects the agreement among repeated measurements. This agreement reveals the decimal place, which has meaning. Precision relates to the method of measurement and the expression of the measured value.

Accuracy has only one meaning: conformity with the truth. The "truth" in measurement science is defined by one or more of the following:

• An adopted physical or other standard (on distance, weight, volume, time, etc.)

• Geometric laws

• A system decided as correct by some recognized authority

Conformity with the truth is decided by the stability and precision in defining these standards, laws and systems, by calibration of instruments, avoiding or removing mistakes, and by detecting and removing systematic errors caused by the environment or instrument adjustments.

It can be realized that conformity with a standard, and thus accuracy, can change as the definition of the standard changes. The rod, for example, certainly varied when it was defined by the toe-to-heel length of the first 16 men coming out of church. As modern definitions of our official unit (the meter) become more refined, the standard, and thus accuracy as related to the standard, becomes more stable. However, there will be variations caused by historical differences in definitions.

Geometric laws are the exception as to the unknown nature of the "true value." For example, the sum of the angles of a closed figure is (n-2)180º. Accuracy of the sum of the angles can thus be checked. But, the accuracy of any particular angle cannot.

Besides adopted legal standards and geometric truth, surveyors or government authorities sometimes also choose to select something as "true," such as a system (datum) of monuments of high "order of accuracy," and make subsequent measurements fit such "control." But, as with "situation ethics" in the real world, or other arbitrary philosophies purportedly embodying truth, using such systems to decide accuracy works only until a "better" standard comes along. Rejection of the NAD 27 datum in favor of the NAD 83 datum, then almost immediately rejecting this in favor of the High Accuracy Reference Network (HARN) adjusted control are examples of what happens when standards are set that have inherent errors in them.

Theoretically the true value exists, but it is elusive because of errors and variations in the standards and systems used to determine "truth." If stable standards and systems of control are more accurate than the ability of the surveyor to measure, accuracy reduces to the control of errors and mistakes in individual measurements.

Precision Further Explained

Precision is usually viewed in a relative or comparative sense. Precision and random errors are directly related. Precision increases when random errors decrease, and the reverse is also true. Statistical measures such as scatter, range and standard deviation are smaller when precision is higher because of the better quality control of random errors.

Precision can be observed in a limited sense simply by duplicating a measurement. One surveyor might observe 1,248.54 feet on one occasion and 1,248.59 feet another time. The 0.05-foot discrepancy is an indication of the precision. If another surveyor measures the distance as 1,249.32 feet and repeats it as 1,249.34 feet, the second surveyor has better precision than the first surveyor because the latter's two readings agree better. Of course, to better study precision of one measuring procedure, or to compare two different procedures, more than one repetition of each procedure is desirable. When the number of repetitions, using a specific method are sufficiently high, a statistical measure of precision (called standard deviation) can be computed. How to do this, and the value of this precision index in determining and comparing precision, will be the subject of a future article.

Precision can be determined, also in a limited way, by observing the number of digits expressed in a measurement. A level instrument operator may be able to observe a reading on a conventional level rod to the thousandths of a foot at a sighting distance of 20 or 30 feet, to the hundredths at 50-to-150 feet, and to the tenths for distances of 300-to-400 feet. Rod readings of 3.423 feet, 3.42 feet, and 3.4 feet are different in precision. The limitation of using this means to evaluate precision is that observers and computational personnel do not always express numbers to the correct precision. For example a reading using a method yielding hundredths might be recorded as 23.4 when it actually should be 23.40, and the opposite also occurs. Sometimes computational personnel simply record whatever comes out of the calculator, according to the arbitrary setting of the "FIX" on the calculator, without evaluating the precision of the numbers used in the computational steps. All of this relates to significant figures, which will be discussed in a later article.

Another way to evaluate precision, again in a limited sense, is to simply consider the instrumentation. A surveyor using a 1" theodolite is using a more precise instrument than one using a 1' vernier transit. However, the total methodology, including the geometric aspects of the survey, must always be considered when evaluating precision, not just the "least count" of the instrument. Suppose, for example, the transit surveyor used geodetic targets on tripods and optical plummets for centering over the ground stations, and had very long lines in the traverse. And, suppose the surveyor with the precise theodolite employed range poles for sightings and plumb bobs for centering the instrument, and had short sight distances. The overall method of the transit surveyor might be considered more precise, despite the fact that the reading circle is less precise than that of the theodolite.

There are many variables affecting precision in the field: geometric aspects of the survey; distances; repetitions made; slope angles; several variables related to the refinement of the instrument; skill of the observer; general care taken in centering and aligning; and general quality control over other aspects of the observations. In fact, everything affecting random errors affects precision, since reducing random errors improves it. The concept of precision and what affects it is more complex than what many realize. Complete evaluation of it requires a thorough understanding of the errors and, preferably, conducting controlled statistical tests of the measurement method.

Accuracy Further Explained

Suppose two surveyors measured a distance. One determined it to be 1,248.725 feet and the other surveyor reports it as 1,248.852 feet. Which is right? This question is answered when we identify which surveyor paid more attention to detection and removal of systematic errors and mistakes. Suppose both surveyors used electronic distance instruments. The first surveyor calibrated the instrument using a National Geodetic Survey calibration baseline and found the constant and PPM corrections; performed tests to determine accurate values for the reflector constants; checked and adjusted optical plummets of the instruments; and carefully observed the temperature and atmospheric pressure, keying these values into the instrument so as to compensate for these natural errors. The second surveyor did none of these calibration and systematic error tests or compensation procedures, and had left last seasons's temperature and pressure values in the instrument. Several applicable systematic errors would occur in the second surveyor's measurements from the sources indicated here, and possibly others, that, when combined, could easily account for the approximate 0.13-foot discrepancy.

Remember, the true value is always equal to the reading plus the sum of the applicable corrections. There are no exceptions to this. No surveyor is exempt from the need to correct for systematic errors in the quest for accuracy. Just as a person cannot find truth in any situation until all biases and prejudices have been compensated or removed from his or her thinking, all of the facts are known, and actions are taken consistent with the facts, accuracy in measured data is impossible without detecting and removing systematic errors, and knowing the error sources in the first place.

Comparing and Contrasting Precision and Accuracy

A measurement can be precise but inaccurate, as well as accurate but imprecise. For example, if a measurement was made with much care using a highly refined instrument, repeated readings of the same quantity would agree closely and thus precision would exist. But if the instrument contained one or more undetected, uncorrected systematic errors, the results would be inaccurate. In contrast, it is possible that the mean of several repeated measurements of this same quantity, using a less refined (but calibrated) method, could be closer to the true value and thus this procedure would yield more accurate results even though there was less agreement among the readings.

An example that explains the difference between precision and accuracy better than any other in surveying has to do with error of closure in traversing. Many surveyors seem to think that error of closure checks the accuracy of the work. Wrong! Error of closure primarily checks the precision, not the accuracy. It checks accuracy only in that it can find blunders. But, since it cannot detect systematic errors in the distances, it cannot fully check accuracy. I have always been dismayed (nothing new for me) that we still have so many written and adopted surveying measurement standards that dictate using error of closure as a means to check accuracy. When surveyors speak of a "1 part in 5,000 survey," I have found that many are only talking about the actual error of closure, not predicted relative errors, which consider the design of the procedures and control of systematic errors. You will get the same error of closure in the field whether systematic errors in distances were corrected or not!

Another surveying example relates to leveling. Let us say that an enterprising leveling crew discovers they can make better time by using a very long level rod, lengthening back-sight distances, as they level up a slope. They set a bench mark on top of a hill, after dozens of turns in the level line. Then they level back down the hill and "close" on the initial bench mark, and observe a nearly perfect closure. So, they conclude that their work was "good" and they go have a beer to celebrate their "results." Job well done? Look again. Since they had longer backsight distances going up the hill (made possible by use of the long rod), they had cumulative systematic errors from at least four sources:

• Any line-of-sight (collimation) error in the instrument, however small

• Earth's curvature and atmospheric re- fraction

• Any systematic error in the graduations (length) of the level rod

• Rod being slightly out of plumb each time

But, you say "aren't these errors compensated?" Yes, when the leveling crew went back down the hill the accumulated errors going up the hill were taken out of the circuit, resulting in apparently good results. But the error is still in the bench mark on top of the hill.

We must realize that precision is only a measure of control of random errors, nothing more. You cannot achieve accuracy in measurements merely by controlling random errors, either in the field or in the computational process. The most sophisticated instruments or least squares adjustment software, complete with error ellipses and pretty graphics, will do relatively little to achieve accuracy in your final values.

It is also possible to have accuracy without precision. For example, my automobile odometer is less precise than a surveying steel tape because the reading to tenths (with estimates to hundredths) of a mile is not as precise as the taped readings, even when the various random errors in taping are considered. However, if I calibrate the odometer and apply correction factors to establish a long distance, I am probably more accurate than the surveyor who applied no correction or calibration factors to a taped measurement. This statement becomes more valid as the distance becomes longer, because eventually the inaccuracy that comes from lack of correcting for systematic errors in the taping will exceed the imprecision in estimating the odometer reading to only hundredths of a mile at the two end points.

Closing Remarks

The caption and logo in my textbook, Surveying Measurements and their Analysis reads, "The truth is equal to one's initial observations plus the corrections discovered through added experience or knowledge." It is intended to apply to measurements, and to anything else where truth is being sought through observations. It is universal in its applications. Let us be reminded that truth is never found until biases are discovered and appropriate corrections applied. Measurements or opinions are but perceptions, as they stand alone. They approach truth only when compared with an acceptable standard.

The basic problem with precision, whether it pertains to measurement or opinion, is that it does not check anything outside of its own closed system. Isn't it rather foolish to allow repeatability to provide confidence regarding observations, or to check anything only against itself or something or someone with similar biases? Learning to make comparisons and weigh things against standards is probably one of the hardest things for anybody to learn and to do consistently. Perhaps this is why precision is so often confused with accuracy when it comes to evaluating measurement quality in surveying.

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

» Back to our July/August 1997 Issue