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# Accuracy and Precision

Accuracy and precision both concern the quality of a measure. However, the terms have different meanings and should not be used as substitutes for one another.

## Precision

Precision depends on the unit used to obtain a measure. The smaller the unit, the more precise the measure. Consider measures of time, such as 12 seconds and 12 days. A measurement of 12 seconds implies a time between11.5 and 12.5 seconds. This measurement is precise to the nearest second, with a maximum potential error of 0.5 seconds. A time of 12 days is far less precise. Twelve days suggests a time between 11.5 and 12.5 days, yielding a potential error of 0.5 days, or 43,200 seconds! Because the potential error is greater, the measure is less precise. Thus, as the length of the unit increases, the measure becomes less precise.

The number of decimal places in a measurement also affects precision. A time of 12.1 seconds is more precise than a time of 12 seconds; it implies a measure precise to the nearest tenth of a second. The potential error in12.1 seconds is 0.05 seconds, compared with a potential error of 0.5 seconds with a measure of 12 seconds.

Although students learn that adding zeros after a decimal point is acceptable, doing so can be misleading. The measures of 12 seconds and 12.0 seconds imply a difference in precision. The first figure is measured to the nearest seconda potential error of 0.5 seconds. The second figure is measured to the nearest tentha potential error of 0.05 seconds. Therefore, a measure of 12.0 seconds is more precise than a measure of 12 seconds.

Differing levels of precision can cause problems with arithmetic operations. Suppose one wishes to add 12.1 seconds and 14.47 seconds. The sum, 26.57 seconds, is misleading. The first time is between 12.05 seconds and12.15 seconds, whereas the second is between 14.465 and 14.475 seconds. Consequently, the sum is between 26.515 seconds and 26.625 seconds. A report of 26.57 seconds suggests more precision than the actual result possesses.

The generally accepted practice is to report a sum or difference to the same precision as the least precise measure. Thus, the result in the preceding example should be reported to the nearest tenth of a second; that is, rounding the sum to 26.6 seconds. Even so, the result may not be as precise as is thought. If the total is actually closer to 26.515 seconds, the sum to the nearest tenth is 26.5 seconds. Nevertheless, this practice usually provides acceptable results.

Multiplying and dividing measures can create a different problem. Suppose one wishes to calculate the area of a rectangle that measures 3.7 centimeters (cm) by 5.6 cm. Multiplication yields an area of 20.72 square centimeters. However, because the first measure is between 3.65 and 3.75 cm, and the second measure is between 5.55 and 5.65 cm, the area is somewhere between 20.2575 and 21.1875 square centimeters. Reporting the result to the nearest hundredth of a square centimeter is misleading. The accepted practice is to report the result using the fewest number of significant digits in the original measures. Since both 3.7 and 5.6 have two significant digits, the result is rounded to two significant digits and an area of 21 square centimeters is reported. Again, while the result may not even be this precise, this practice normally produces acceptable results.

## Accuracy

Rather than the absolute error to which precision refers, accuracy refers to the relative error in a measure. For example, if one makes a mistake by 5 centimeters in measuring two objects that are actually 100 and 1,000 cm, respectively, the second measure is more accurate than the first. The first has an error of 5 percent (5 cm out of 100 cm), whereas the second has an error of only 0.5 percent (5 cm out of 1,000 cm).

How Are Precision and Accuracy Different? To illustrate the difference between precision and accuracy, suppose that a tape measure is used to measure the circumference of two circlesone small and the other large. Suppose a result of 15 cm for the first circle and 201 cm for the second circle are obtained. The two measures are equally precise; both are measures to the nearest centimeter. However, their accuracy may be quite different. Suppose the measurements are both about 0.3 cm too small. The relative errors for these measures are 0.3 cm out of 15.3 cm (about 1.96 percent) and0.3 cm out of 201.3 cm (about 0.149 percent). The second measurement is more accurate because the error is smaller when compared with the actual measurement. Consequently, for any specific measuring tool, one can be equally precise with the measures. But accuracy is often greater with larger objects than with smaller ones.

Confusion can arise when using these terms. The tools one uses affect both the precision and accuracy of one's measurements. Measuring with a millimeter tape allows greater precision than measuring with an inch tape. Because the error using the millimeter tape should be less than the inch tape, accuracy also improves; the error compared with the actual length is likely to be smaller. Despite this possible confusion and the similarity of the ideas, it is important that the distinction between precision and accuracy be understood.

Bob Horton

### Internet Resources

Foote, Kenneth E., and Donald J. Hueber. "Error, Accuracy, and Precision." In The Geographer's Craft Project, Department of Geography, The University of Colorado at Boulder, 1995. <http://www.colorado.edu/geography/gcraft/notes/error/error.html>.

Garrison, Dean H., Jr. "Precision without Accuracy in the Cruel World of Crime Scene Work." 1994. <http://www.chem.vt.edu/ethics/garrison/precision.html>.