Mathematics
Limits of Accuracy
Limits of accuracy refer to the maximum possible error or uncertainty in a measurement or calculation. It is determined by the precision of the measuring instrument or the accuracy of the mathematical model used. The concept of limits of accuracy is important in ensuring the reliability and validity of scientific and mathematical results.
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eBook - PDFThe Uncertainty of Measurements
Physical and Chemical Metrology: Impact and Analysis
- Shri Krishna Kimothi(Author)
- 2001(Publication Date)
- ASQ Quality Press(Publisher)
163 7 Measurement Uncertainty and Its Evaluation: Evolving Concepts MEASUREMENT UNCERTAINTY: LIMITS TO ERROR A measurement result whose accuracy is entirely unknown is worth nothing. Accuracy depends upon the error of measurement. It is also accepted that the actual error of a measurement result is unknown and unknowable. This is a contradictory situation. Without knowing its error, one cannot utilize a mea- surement result confidently for making a decision, but at the same time its error is unknown and unknowable. Metrologists have found a solution to this contradiction in the application of statistical tools. If the absolute value of error cannot be determined, the “limits to error” can always be “inferred” by using rudimentary statistical techniques. Here the term limits to error means the maximum value of expected error. This is a worst-case situation in the sense that the error will not exceed the limit. The actual error will be less than the limits-to-error value. Another term used in the context of limits to error is inferred. When we infer something, we are not sure about its correctness, so any inference about limits to error always has a risk of being incorrect. If the risk of being incorrect is reasonably low, the knowledge about the limits to error is better than not knowing anything about the quantity of error at all. The “limits to error” of a measurement result is in fact the uncertainty of measurement. The uncertainty is generated by two factors: • One factor is the precision of the measurement process by which the result has been derived. This is evaluated through repeated application of the measurement process to obtain repeat measurements of the same parameter. Precision is the characteristic of the measurement process linked to the closeness of repeat measurements among themselves. Thus, a numerical index of precision should come from the variability of the repeat measurement observations.
eBook - PDFDealing with Data
The Commonwealth and International Library: Physics Division
- Arthur J. Lyon, W. Ashhurst(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
The failures which also occur, and the revisions which are constantly made as science progresses, show, on the other hand, that the limitations are real. 2. READING AND SETTING ERRORS Most measurements involve the reading of some type of scale, and the most obvious type of uncertainty in a measure-ment is that associated with the limits of the accuracy to which the scale can be read. For example, if the diameter of a cylinder is measured with vernier calipers, it will perhaps be found that several repeated measurements give exactly the same value, say 1-93 cm, and if the vernier scale enables one to distin-guish quite clearly between 1-93 and either 1-92 or 1-94, one might reasonably claim that the value 1-93 is correct to the nearest 0-01 cm, i.e. that the true value lies between the limits 1-93 ±0-005 cm. In that case the same kind of accuracy is being claimed as is normally expected in the result of a computation in pure mathematics. The limits given are supposed to be the outside limits of the possible error ; and as far as one could judge from such a statement, all values within the limits, e.g. 1-928, 4 DEALIN G WITH DATA 1-931, 1-934, etc., are equally likely, so that the central value 1-930 is no more likely—and no less likely—than values near the extremes. We say in such a case that the probability dis-tribution of the possible errors is rectangular, and it may be represented diagrammatically as in Fig. 1.1. The probability of any error between —0-005 and +0-005 is constant, and the probability of any error outside these limits iszero. This can be interpreted in the sense that the probability of the true value lying anywhere between 1-925 and 1-935 is constant, and the probability of its lying anywhere outside these limits is zero.* >OI O + C > O L _ E r r o r C c m ) —I 1 1 i 1 1-92 1-93 1*94 —*■ Measured value Ccm) FIG.
eBook - PDFUncertainty Analysis for Engineers and Scientists
A Practical Guide
- Faith A. Morrison(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
The techniques are transparent, with the worksheets keeping track of assumptions made as well as the relative impacts of the various assumptions. Keeping track of assumptions facilitates the ever-so-important process of revisiting error estimates, when, for example, seeking a more precise answer, seeking to improve a measurement process, or trying to improve the error estimates previously made. Having stated our goals, we now devote the rest of this chapter to discussing how four important concepts relate to our error-analysis system: precision and accuracy; significant figures; error limits; and types of uncertainty or error. In Chapter 2 we turn to the statistics of random errors – the statistics of random processes form the basis of all error analyses. 1.1 Precision and Accuracy In common speech the words “precision” and “accuracy” are nearly syn- onyms, but in terms of experimentally determined quantities, we use these words to describe two different types of uncertainty. Accuracy describes how close a measurement is to its true value. If your weight is 65 kg and your bathroom scale says that you weigh 65 kg, the scale is accurate. If, however, your scale reports 75 kg as your weight, it fails to reflect your true weight and the scale is inaccurate. To assess accuracy, we must know the true value of a measured quantity. To assess accuracy, we must know the true value of a measured quantity. 1.2 Significant Figures 3 Precision is also a measure of the quality of a measurement, but precision makes no reference to the true value. A measurement is precise if it may be distinguished from another, similar measurement of a quantity. For example, a bathroom scale that reports weight to one decimal place as 75.3 kg is more precise than a scale that reports no decimal places, 75 kg. Precise numbers have more digits associated with them. With a precise bathroom scale, we can distinguish between something that weighs 75.2 kg and something that weighs 75.4 kg.
eBook - PDF- Valery A. Slaev, Anna G. Chunovkina, Leonid A. Mironovsky(Authors)
- 2013(Publication Date)
- De Gruyter(Publisher)
As a measuring instrument a simple Section 3.2 Potential and limit accuracies of measurements 171 chronometer can be used (electronic wristwatch) which provides the required accuracy of measurements. At the same time the method of measurement is the method of direct readout performed by the operator, who is the person who wishes to leave from the railway station. 3) If is a question of whether or not the external conditions in moving from point B to point A accidentally influence the time of moving, then multiple measurements are carried out under different conditions , and a dispersion or root-mean-square deviation of the values measured ĩT n .i/ are found. 4) In the above various extreme situations are not taken into consideration (traf-fic congestions, unexpected ice-covered ground, absenteeism or nonfunction of trans-portation, etc.). These anomalies can be neglected if their probability . 1 0.997 / D 0.003. In the opposite case this probability has to be taken into account within a confi-dence interval, and when that increases it is necessary to correspondingly increase the time reserve for the departure from point B. 3.2 Potential and limit accuracies of measurements We need to distinguish the potential and limit accuracy of measurements. The limit accuracy is understood to be the maximum achievable accuracy at which a measure-ment of a physical quantity can be carried out at a given developmental stage of sci-ence and engineering. The limit accuracy of measurements is realized, as a rule, in national measurement standards reproducing the units of a physical quantity with the maximum accuracy within the framework of the respective country. Frequently it is identified with the ultimate sensitivity of a measuring instrument. By potential accuracy we mean the maximum accuracy at the contemporary devel-opmental stage of scientific knowledge and engineering.
- Micah Altman, Jeff Gill, Michael P. McDonald(Authors)
- 2004(Publication Date)
- Wiley-Interscience(Publisher)
2.2 FUNDAMENTAL THEORETICAL CONCEPTS A number of concepts are fundamental to the discussion of accuracy in statistical computation. Because of the multiplicity of disciplines that the subject touches on, laying out some terminology is useful. 2.2.1 Accuracy and Precision For the purposes of analyzing the numerical properties of computations, we must distinguish between precision and accuracy. Accuracy (almost) always refers to the absolute or relative error of an approximate quantity. In contrast, precision has several different meanings, even in scientific literature, depending on the context. When referring to measurement, precision refers to the degree of agree-ment among a set of measurements of the same quantity—the number of digits (possibly in binary) that are the same across repeated measurements. However, on occasion, it is also used simply to refer to the number of digits reported in an estimate. Other meanings exist that are not relevant to our discussion; for example, Bayesian statisticians use the word precision to describe the inverse of the variance. In the context of floating point arithmetic and related numerical analysis, precision has an alternative meaning: the accuracy with which basic arithmetic operations are performed or quantities are stored in memory. 2.2.2 Problems, Algorithms, and Implementations An algorithm is a set of instructions, written in an abstract computer language that when executed solves a specified problem . The problem is defined by the complete set of instances that may form the input and the properties the solution must have. For example, the algorithmic problem of computing the maximum of a set of values is defined as follows: • Problem: Find the maximum of a set of values. • Input: A sequence of n keys k 1 , . . . , k n of fixed size. • Solution: The key k ∗ , where k ∗ ≥ k i for all i ∈ n .
No longer available |Learn more- Matthew French(Author)
- 2016(Publication Date)
- Mercury Learning and Information(Publisher)
If you want to make a discovery, as emphasized earlier in this chapter you should “be awkward”! Data Analysis and Errors • 115 This chapter begins with an explanation of the terminology of errors. It then develops data and error analysis techniques with a particular focus on graphs. 4.1 ACCURACY AND PRECISION Figure 4.1: Targets illustrating the differences between the terms accurate and precise. The center of the target corresponds to the true value. The stars are the individual measurements. 116 • Physics Lab Experiments The accuracy of a measurement is how far the measurement is away from the true value of the quantity that is being measured. The precision of a measurement is the full width of the spread of data measured when taking repeat measurements under identical conditions. A measurement can be accurate and not precise, precise and not accurate, precise and accurate or neither precise nor accurate. Often either a target board (see Figure 4.1) or a distribution graph (see Figure 4.2) is used to illustrate this. For example a measurement may have a large systematic error. This would mean that it was not accurate as the measurements would lie a long way from the true value. However, the measurement could be very precise as it could be repeated many times yielding the same result. Thus the results may be closely scattered but around the wrong value. A crude example would be measuring the length of a desk to the nearest centimeter using a meter rule on which the first 5 cm was missing. Alternatively a measurement may be taken with an instrument which has large fluctuations in the readings. This would mean that the measurement may be accurate, but not precise. Figure 4.2: Graphs illustrating the differences between the terms accurate and precise. The vertical line corresponds to the true value. The vertical axis is a probability distribution (or the number of measurements) and horizontal axis is the magnitude of the quantity being measured.
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