## accuracy

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# Accuracy

Accuracy in measurements

In calculation

Rounding

Accuracy is the degree to which an experimental reading or a calculation approximates the true value. Lack of accuracy may be due to measurement error or due to numerical approximation. The less error in an experiment or calculation means the more accurate the results. Error analysis can provide information about the accuracy of a result.

## Accuracy in measurements

Errors in experiments stem from incorrect design, inexact equipment, noise from random sources, and approximation. Imperfections in equipment are a fact of life; sometimes design imperfections are unavoidable as well. Approximation is also unavoidable and depends on the quality and correctness of the measuring equipment. For example, if the finest marks on a ruler are in centimeters, then the final measurement made using such a device cannot have an accuracy of more than half a centimeter. But if the ruler has millimeter markings, then the measurements can be an order of magnitude (i.e., a factor of 10) more accurate.

Accuracy differs from precision. Precision deals with how repeatable measurements areif multiple measurements return numbers that are close to each other, then the experimental results are precise. The results may, however, not be accurate. For instance, if one measured a three-inch piece of string five times and found every time that its length was 4.001 inches, one would have performed a series of measurements with low accuracy and high precision.

## In calculation

Approximations are also unavoidable in calculation. A perfectly accurate decimal or binary representation of 1/3, pi (π), and many other numbers cannot be written down, because it would require an infinite number of digits; calculations made using such numbers cannot, therefore, generally be done without some loss of information. Fortunately, the accuracy required from a given calculation usually does not require perfection. The person designing the calculation must, however, make sure that the approximations used are sufficiently close that they do not endanger the usefulness of the result. Accuracy is also an issue in computation because rounding errors accumulate, series expansions are truncated, and other numerical issues arise.

## Rounding

If you buy several items, all of which are subject to a sales tax, then you can calculate the total tax by summing the tax on each item. For the total tax to be accurate to the penny, you must do all the calculations to an accuracy of tenths of a penny (in other words, to three significant digits), then round the sum to the nearest penny (two significant digits). If you calculated only to the penny, then each measurement might be off by as much as half a penny (\$0.005) and the total possible error would be this amount multiplied by the number of items bought. If you bought three items, then the error could be as large as 1.5 cents; if you bought 10 items, then your total tax could be off by as much as 5 cents.

As another example, if you want to represent the value of π to an accuracy of two decimal places, then you express it as 3.14. This can also be expressed as 3.14 +/0.005, since any number from 3.135 to 3.145 would be expressed the same way if truncated (cut down) to two significant decimal places. Calculations using numbers with two decimal places can only be accurate, at best, to one decimal place; for example, adding two measurements of π that are both accurate only to the second place would involve adding their uncertainties as well, namely, 3.14 +/0.005 + 3.14 +/0.005 = 6.28 +/0.01. The true value of 2π, according to this calculation, might be anything from 6.27 to 6.29.

In scientific and engineering calculations, the issue of how many digits are both needed and meaningful arises constantly. Calculating a result that has more digits than measurement accuracy can justify is meaningless and produces the dangerous illusion that the result is more accurate than it really is; yet using too few digits to overcome all possible numerical error in ones calculations will cause information to be lost. One approach to this problem is to perform calculations using more digits than the accuracy of ones data can justify, then rounding the result to the correct number of significant digits. Mathematical techniques, not guesswork, are used for dealing with numerical errors in computation.

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# Accuracy

Accuracy is how close an experimental reading or calculation is to the true value. Lack of accuracy may be due to error or due to approximation . The less total error in an experiment or calculation, the more accurate the results. Error analysis can provide information about the accuracy of a result.

## Accuracy in measurements

Errors in experiments stem from incorrect design, inexact equipment, and approximations in measurement. Imperfections in equipment are a fact of life, and sometimes design imperfections are unavoidable as well. Approximation is also unavoidable and depends on the fineness and correctness of the measuring equipment. For example, if the finest marks on a ruler are in centimeters, then the final measurement is not likely to have an accuracy of more than half a centimeter. But if the ruler includes millimeter markings, then the measurements can be an order of magnitude more accurate.

Accuracy differs from precision. Precision deals with how repeatable measurements are—if multiple measurements return numbers that are close to each other, then the experimental results are precise. The results may be far from accurate, but they will be precise.

## In calculations

Approximations are also unavoidable in calculations. Neither people nor computers can provide a totally accurate number for 1/3 or pi or any of several other numbers, and often the desired accuracy does not require it. The person calculating does, however make sure that the approximations are sufficiently small that they do not endanger the useful accuracy of the result. Accuracy becomes an issue in computations because rounding errors accumulate, series expansions are attenuated, and other methods that are not analytical tend to include errors.

## Rounding

If you buy several items, all of which are subject to a sales tax, then you can calculate the total tax by summing the tax on each item. However, for the total tax to be accurate to the penny, you must do all the calculations to an accuracy of tenths of a penny (in other words, to three significant digits), then round the sum to the nearest penny (two significant digits). If you calculated only to the penny, then each measurement might be off by as much as half a penny (\$0.005) and the total possible error would be this amount multiplied by the number of items bought. If you bought three items, then the error could be as large as 1.5 cents; if you bought 10 items then your total tax could be off by as much as five cents.

As another example, if you want to know the value of pi to an accuracy of two decimal places, then you could express it as 3.14. This could also be expressed as 3.14 +/- 0.005 since any number from 3.135 to 3.145 could be expressed the same way—to two significant decimal points. Any calculations using a number accurate to two decimal places are only accurate to one decimal place. In a similar example, the accuracy of a table can either refer to the number of significant digits of the numbers in a table or the number of significant digits in computations made from the table.

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# Accuracy

Accuracy is the closeness of an experimental measurement to the "true value" (i.e., actual or specified) of a measured quantity. A "true value" can determined by an experienced analytical scientist who performs repeated analyses of a sample of known purity and/or concentration using reliable, well-tested methods.

Measurement is inexact, and the magnitude of that exactness is referred to as the error. Error is inherent in measurement and is a result of such factors as the precision of the measuring tools, their proper adjustment, the method, and competency of the analytical scientist.

Statistical methods are used to evaluate accuracy by predicting the likelihood that a result varies from the "true value." The analysis of probable error is also used to examine the suitability of methods or equipment used to obtain, portray, and utilize an acceptable result. Highly accurate data can be difficult to obtain and costly to produce. However, different applications can require lower levels of accuracy that are adequate for a particular study.

[Judith L. Sims ]

## RESOURCES

### BOOKS

Jaisingh, Lloyd R. Statistics for the Utterly Confused. New York, NY: McGraw-Hill Professional, 2000.

Salkind, Neil J. Statistics for People Who (Think They) Hate Statistics. Thousand Oaks, CA: Sage Publications, Inc., 2000.

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ac·cu·ra·cy / ˈakyərəsē/ • n. (pl. -cies) the quality or state of being correct or precise. ∎  the ability to perform a task with precision. ∎  technical the degree to which the result of a measurement, calculation, or specification conforms to the correct value or a standard: the accuracy of radiocarbon dating. Compare with precision.

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accuracy See precision.