Error Calculation

11 Key excerpts on "Error Calculation"

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  Theory and Design for Mechanical Measurements, International Adaptation

  • Richard S. Figliola, Donald E. Beasley(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  C H A P T E R 5 Uncertainty Analysis 5.1 INTRODUCTION Whenever we plan a test or later report a test result, we want to know something about the quality of the result. Uncertainty analysis provides a methodical approach to estimating the quality of the result from a test. This chapter focuses on how to estimate the “± what?” in a test result. Suppose the competent dart thrower of Chapter 1 tossed several practice rounds of darts at a bull’s-eye. This would give us a good idea of the thrower’s tendencies. Then, let the thrower toss another round. Without looking, can you guess where the darts will hit? Test measurements that include systematic and random error components are much like this. We can calibrate a measurement system to get a good idea of its behavior and accuracy. However, from the calibration we can only estimate how well any measured value might estimate the true value in a subsequent measurement. Errors are a property of the measurement. Measurement is the process of assigning a value to a physical variable based on a sampling from the population of that variable. Error causes a difference between the value assigned by measurement and the true value of the population of the variable measured. Measurement errors are introduced from various elements, for example, the individual instrument calibrations, the data set finite statistics, and the approach used. If we knew the exact value of an error, we would just correct for it. But most often we do not know the true value of the measured variable but rather only know its measured value instead. So we will not know the exact values of the errors affecting the measurement. Instead, we draw from what we do know to estimate a probable range to define the limits of error in the measurement. This estimate of the limits of the error is an assigned value called the uncertainty.

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  Essential Mathematics and Statistics for Forensic Science

  • Craig Adam(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  10 Statistics in the evaluation of experimental data: computation and calibration Introduction: What more can we do with statistics and uncertainty?

  Having a sound understanding of the fundamentals of statistics associated with random variations in measurement allows us to develop further a range of calculations that may be applied in the analysis of experimental data. This chapter will deal with two important areas. First, when the outcomes of experimental work are used in further calculations, how can we determine the uncertainty in our results at the end of such a process? This requires computational skills in the propagation of errors through a variety of functions and formulae. Second, how may experimental uncertainties be accommodated in graphical analysis, particularly graphs related to the calibration or interpretation of analytical measurements? To deal with this, a more detailed understanding of topics such as the identification of outliers and the use of linear regression techniques is required. However, let us begin by reviewing our current understanding of the statistical interpretation of uncertainty.

  10.1 The propagation of uncertainty in calculations 10.1.1 Review of uncertainty in experimental measurement

  Throughout much of this book attention has been given to understanding and quantifying the uncertainty associated with experimental measurements. It is useful now to review how this topic has developed up to this point. In Section 1.3, discussion focused on a qualitative appreciation of uncertainty, its origins and how its magnitude might be estimated in particular instances, for example using the range method. Later, in Section 9.1.2, it was found that, since a series of repeated measurements effectively samples a normal distribution, the concept of confidence limits could be invoked to interpret meaningfully numerical estimates of uncertainty. The crucial parameter in such calculations is the standard deviation for a set of measurements. However, in dealing with a sample taken from a much larger population, the calculation of the standard error, followed by the use of the t

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  Physics Lab Experiments

  • Matthew French(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  DATA ANALYSIS AND ERRORS Perhaps at school, once a measurement was completed and a result calculated it was thought the experiment was complete. Some- times the result may have even been compared with an accepted value. For instance, an experiment might have been performed to measure the acceleration due to gravity, g. Perhaps the oscillations of a simple pendulum were timed and the result substituted into the familiar equation: 2 l T g p  (4.1) At university level, experiments must be taken further to include an estimate of the error in the result. Scientists are usually interested in making measurements, not for their own sake, but to test a theory. Theories can be accepted or rejected on the basis of experimental measurements, therefore it is vital to have some indication of the accuracy of the results. Every time a measurement is made there will be a degree of uncertainty in the result: an instrument may be difficult to read, it may fluctuate, it may be badly calibrated, a genuine mistake may be made reading the instrument or the result could be limited by the precision of the instrument. It is vital that a competent scientist CHAPTER 4 114 • Physics Lab Experiments has an understanding of the terminology of errors as well as how to calculate, evaluate and develop techniques to minimize the sig- nificance of measurement error in their work. It is beneficial to adopt an inquisitive mind-set where questions are always asked and improvements to the apparatus and techniques which may reduce measurement error are always considered: this can be summarized as “be awkward”! Suppose a large underground cavern is to be found. The hypoth- esis might be that due to the void in the ground the measured value of g near the cavern will be less than 9.81 m/s 2 . Of course, the famil- iar value of g=9.81 m/s 2 is not exact and so also has an error which is neglected in this illustrative example.

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  Dealing with Data

  The Commonwealth and International Library: Physics Division

  • Arthur J. Lyon, W. Ashhurst(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  One can be exact and completely error-free only in pure mathematics or formal logic; and one achieves it there at the expense of no longer saying anything factual about the real world. Knowledge of the world, of na-ture, or of human society is always incomplete, approximate, and subject to error; and the results of physical measurements are no exception to this rule. We might say that liability to error and uncertainty is the price we must pay if we want to be able to make statements having real factual content. In mathematical physics, for EXPERIMENTAL ERRORS 3 example, it is possible to make statements which are exact, but they are essentially hypothetical in character, stating that // certain assumptions hold, such and such conclusions must follow. In the experimental sciences we wish to discover what assumptions and laws are actually valid, and within what con-ditions and limits they remain valid ; but even our most con-fident judgements will be subject to elements of qualitative un-certainty and our best numerical values to some degree of inexactness. The levels of precision actually achieved in modern scientific and technological work, and the enormous successes achieved in the practical application of scientific results, show that such limitations do not prevent continual progress and improve-ment. 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.

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  Principles of Nuclear Radiation Detection

  • Geoffrey G. Eichholz(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 3 COUNTING STATISTICS AND ERROR DETERMINATIONS INTRODUCTION

  In reporting the outcome of any laboratory measurements, it is incumbent upon the experimenter to report not only the results of his measurements, but also the accuracy with which they were made or, more appropriately, the confidence with which they are held to be valid. Consequently, it is important that the experimentalist be aware of the various sources of error in his work and able to assess their significance. The mathematical framework for this assessment is the science of statistics.

  It is particularly important that statistics be applied to any laboratory work which includes radiation detection as part of its data collection, since the very process of radioactive decay is probabilistic in nature and the result of any counting experiment is perforce a random quantity. Since the result of any physical measurement can be considered as a random variable, the basic ideas developed in this chapter are not limited to measurements of radiation. They are applicable to all situations in which measurements are subject to some source of random error.

  UNCERTAINTY IN THE MEASUREMENT PROCESS

  The errors that accompany most laboratory measurements can be divided into two classes: systematic errors and random errors. Systematic errors result from faults in the measurement process or its interpretation that lead to a bias in the results. For example, if one measured the length of a table with a tape measure that had been stretched through misuse, the result would consistently be less than the actual length of the table. The measured length would be biased toward low values by the incorrect tape; the results would have a systematic error.

  On the other hand, the proper use of accurately calibrated instruments is no guarantee that the result of a measurement will be errorless. For example, even if perfectly accurate tape measures were used, it is unlikely that forty people would agree to the nth significant figure on the length of a table. Individual variation in holding the tape, reading the scale, etc. would produce a range of results for the measured length, regardless of how carefully the measurements were made. However, it is unlikely that there would be any bias in this set of measurements. A result chosen from this group is likely to contain an error, but the error will be random, not systematic.

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  Introduction to Contextual Maths in Chemistry

  • Fiona Dickinson, Andrew McKinley(Authors)
  • 2021(Publication Date)
  • Royal Society of Chemistry

    (Publisher)

  CHAPTER

  Learning Points: What We'll Cover

  • □ Precision, accuracy and errors in the context of experimental measurement
  • □ The correct use of significant figures in calculating and reporting experimental values
  • □ The standard error and its use in combination with the appropriate number of significant figures
  • □ Propagating experimental uncertainties through mathematical relationships to report the uncertainty correctly in calculated quantities
  • □ Statistical tests to qualify any outliers in a data series and to check whether data sets are statistically different

  Why This Chapter Is Important

  • In chemistry, we validate our theoretical models through experimental measurement; these measurements are subject to laws of probability and we need to know whether the variations in our measurements are significant.
  • Significant figures tell us everything about the measurements made, so it is important that we all use the same rules for reporting these and carrying them through calculations.
  • Experimental uncertainty (‘experimental error’) is unavoidable, so we must be able to quantify ‘the bounds of experimental error’ in order to allow for them effectively in our analyses.
  • We need to make sure that we have a common sense approach to handling errors; if our calculated values have an uncertainty greater than the value itself, we need to be sure that we have handled the errors correctly.

  Passage contains an image Experimental Uncertainty and Significant Figures: What Are the Bounds of Experimental Error?

  It's easy not to think about the number of numbers in a number, but in science the number of significant figures is representative of how well we know a value. Some values we know to a high precision; for example, the speed of light is 299 792 458 m s−1 . Thousands of measurements went in to knowing this value so accurately, but we will often approximate it to a value with just one significant figure (3 × 108 m s−1 )†

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  Applied Metrology for Manufacturing Engineering

  • Ammar Grous(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  We note that the error is totally insignificant to the point where it is sought in the seventh decimal place, which does not count because the results are rounded to three decimal places. We note that the arithmetic mean remained equal to 0.6547740. The quantified uncertainty is then:

  1.9. Principle of uncertainty calculation: types A and B

  When performing measuring experiments on mechanical parts, they should be repeated several times under the same conditions in the hope of obtaining a good trend. Unfortunately, there are always dispersions, and for this reason, we fall back on statistical modeling. Thus, we consider a mathematical expectation of the first order, which is the mean µ, and then, we calculate the variance σ2 and a standard deviation σ on a sample of size n. We then carry out mathematical statistics approaches [NIS 94, PRI 96, TAY 94].

  From a practical measure for true value xi , we have to make calculations of the average of n samples, the values being, obviously, assumed identical. The ensuing systematic errors may however be reduced by applying corrections. This appeals to a sense of analysis that the operator is, unfortunately, not always expected to master. The metrologist should be rigorous to obtain a good correlation between physical measurements and figures that are expected to reflect his or her numerical representation. Knowledge of the measuring process and the fundamental principles from physics is one of the best guarantors of the conduct of the metrology project. In practice, errors are not discussed quickly. Various errors occur during the measurement. We recall a few of them:

  – temperature and pressure; – precision of instruments and position of the feature being measured;

  – deformation of the mechanical part (we discuss this in Chapter 3 );

  – disruption of the quantity measured by the presence of the measuring instrument; – error due to the measurement method itself; – error due to the operator, and so on.

  The errors and their causes being enumerated, we now attempt to deal with the propagation law and the required corrections. Rigorous work of good metrologists means thinking about the errors not yet identified. Those already identified will be subject to potential adjustments to compensate. While these adjustments would be judiciously realized, there is still doubt on the value of the correction, and this is where a rigorous mind is needed. Among the various corrections, three categories should be defined: calibration, environmental, and standardization corrections:

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  The Uncertainty of Measurements

  Physical and Chemical Metrology: Impact and Analysis

  • Shri Krishna Kimothi(Author)
  • 2001(Publication Date)
  • ASQ Quality Press

    (Publisher)

  Since the true value of a characteristic of a product under measurement cannot be known, a measure- ment result always has some error associated with it. No measurement has ever been made that did not contain some error. We do not realize this at times due to our misconceptions about the perfectness of measurement systems. Numerically, measurement error has been defined as the result of a mea- surement minus the true value of the measurand. The true value, as explained above, cannot be known; since measurement error is linked with the true value by definition, measurement error also cannot be known. Measurement error is therefore a qualitative concept only. Metrologists believe in the existence of errors, but they also believe that, like true value, the error cannot be known exactly. The concept of measurement error has been defined by Churchill Eisen- hart (1963) as follows: A conscious characteristic of measurement is disagreement of repeated measure- ments of the same quantity. Experience shows that when high accuracy is sought repeated measurements of the same quantity by a particular measurement process do not yield uniformly the same number. We explain these discordances by saying that the individual measurements are affected by errors which we interpret to be the manifestation of variations in the executions of the process of measurement resulting from the imperfection of instruments and of organs of sense and from the difficulty of achieving (or even specifying with a convenient number of words) the ideal of perfect control of conditions and procedures. There are various sources of errors that affect the measurement process and hence the measurement observations. Some measurement errors may be attributed to human ignorance and negligence.

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  Physics in Laboratory. Experiments for Engineering Physics Courses

  • Giulio Mazzi, Paolo Ronchese, Pierluigi Zotto(Authors)
  • 2022(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Coarse systematic errors, deriving from actual defects or faults of the used instrumenta-tion, which modify in a known way the measure, can be removed, or at least reduced below the device precision, by using well calibrated instruments or by applying appropriate cor-rections to the obtained results. On the contrary it is very dif fi cult or even impossible to remove, or just to reduce, those systematic errors deriving from the impossibility to build perfect instruments, or appearing, in the case of indirect measurements, because of a de-pendence from other physical quantities which are in their turn source of systematic errors. Casual errors are those appearing in an unpredictable way (that is why they are called casual ) during every measurement: if the device precision is suf fi cient to detect them, they cause a random difference among the performed measurements of the same quantity. Some causes originating casual errors are for instance: • fl oating experimental conditions (e.g. an unstable temperature) • external distortions of a measurement (e.g. a vibration) • unclear de fi nition of the quantity to be measured (e.g. the diameter of a non perfectly spherical body) __________________________________________________________________________________ 12 Physics in Laboratory - Experiments for Engineering Physics Courses The presence of casual errors in any measurement is unavoidable; yet, analysing several measures of the same quantity allows the reduction of their effect. 2.3 Arithmetic Mean Principle Suppose to measure n times a physical quantity X using the same device and in the same conditions. The n obtained measures, i. e. the values of X resulting from n measurements, are after removing all systematic error contributions. The simplest graphical representation of the n measures consists in a cartesian diagram, like the one shown in the fi gure, corresponding to a set of 200 values of the same length measured by using a nonius whose precision is 0.1mm.

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  Data Analysis for Physical Scientists

  Featuring Excel®

  • Les Kirkup(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  Chapter 5 Measurement, error and uncertainty 5.1 Introduction Chemists, physicists and other physical scientists are proud of the quantitative nature of their disciplines. By subjecting nature to ever closer examination, new relationships between quantities are discovered, and established relationships are pushed to the limits of their applicability. When ‘numbers’ emerge from an experiment, they can be subjected to quantitative analysis, compared to the ‘numbers’ obtained by other experimenters and be expressed in a clear and concise manner using tables and graphs. If an unfamiliar experiment is planned, an experimenter will often carry out a pilot experiment. The purpose of such an experiment might be to assess the effectiveness of the experimental methods being used, or to offer a preliminary evaluation of a theoretical prediction. It is also possible that the experimenter is acting on instinct or intuition. If the results of the pilot experiment are promising, the experimenter typically moves to the next stage in which a more thorough investigation is undertaken and where there is increased emphasis on the quality of the data gathered. The analysis of these data often provides crucial and defensible evidence sought by the experimenter to support (or refute) a particular theory or idea. The goal of an experiment might be to determine an accurate value for a particular quantity such as the electrical charge carried by an electron. Experimenters are aware that influences exist, some controllable and others less so, that conspire to adversely affect the values they obtain. Despite an experimenter’s best efforts, some uncertainty in an experimentally determined value remains. In the case of the charge on the electron, its value is recognised to be of such importance that considerable effort has gone into establishing an 168 accurate value for it. Currently (2011) the best value 1 for the charge on the electron is (1.602176487 ± 0.000000040) × 10 −19 C.

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  Measurements from Maps

  Principles and Methods of Cartometry

  • D H Maling(Author)
  • 2016(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  We shall see later that the arithmetic mean of these measurements may be considered to be the most probable value of the measurements which have been made. Analysis of the spread of the measurements through the standard deviation (or the standard error of sampling theory) will provide a measure of 84 Errors of Measurement and Their Analysis 85 their precision. The errors which occur in each measurement are subject to certain mathematical laws, and although it is not possible to determine their absolute magnitude, it is possible to evaluate the statistical probability of whether errors of a certain magnitude will occur. THE CAUSES OF ERRORS The causes of errors arising in laboratory measurements have been sum-marised by Lyon (1970) under the following headings: Measurement stage (1) Personal (2) Instrumental (3) Environmental (4) Natural fluctuations of the quantity itself Reduction stage (5) Computational (6) Errors in auxiliary data (7) Errors in theoretical assumptions. It is convenient to split the analysis of these causes into the two stages already noted in Chapter 3. The first is the measurement stage, which involves the actual manipulation of the instruments. This is followed by the reduction stage, in which the results of the measuring process are converted into their ground dimensions and any other kind of numerical reduction and corrections are applied to take into account the geometrical properties of the map or photograph used for measurement. Some of the possible causes of error arising in the simplest case of measuring the length of a line may be listed as follows. Measurement Stage Personal errors include mistakes in reading the scale or in manipulating the dividers. They also include the personal equation of the observer, which is the way in which a reading lying between the closest subdivisions of the scale is estimated visually.

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