Statistical treatment of random errors

When N exceeds 10 it is snggested that an observation be rejected if its deviation from the mean of the others exceeds 2.65, where S is the estimated standard deviation of the others from their mean this corresponds to abont a 1 percent probabihty that the suspect observation is valid. 

Whether the criteria for rejection are those recommended above or alternative criteria, it is important that the chosen criteria be used consistently. The criteria must be decided upon ahead of time selection of criteria after the measnrements have been made runs the grave risk of introducing bias. If a series of measnrements presents a problem that cannot be treated objectively with previonsly selected criteria, then if at all possible the entire series should be rejected and a new series of measnrements made after careful review of the experimental procedure to minimize the possibility of large errors, mistakes, or bias. 

The next three subsections describe the background and principles of random error treatment, and they introduce two important quantities standard deviation a- and 95 percent confidence limits. The four subsections following these— Uncertainty in Mean Valne, Small Samples, Estimation of Limits of Error, and Presentation of Numerical Results—are essential for the kind of random error analysis most frequently required in the experiments given in this book. The Student t distribution is particularly important and useful. 

Let it be supposed that a very large number of measurements x, (/ = 1, 2. V) are made of a physical quantity x and that these are snbject to random errors ,. For simplicity we shall assume that the trae value Xq of this quantity is known and therefore that the errors are known. We are here concerned with the freqnency n( ) of occurrence of errors of size . This can be shown by means of a bar graph, like that of Fig. 3, in which the error scale is divided into ranges of equal width and the height of each bar represents the nnmber of 

A typical distribution of errors. The bar graph represents the actual error frequency distribution 73(e) for 376 measurements the estimated normal error probability function P(e) is given by the dashed curve. Estimated values of the standard deviation cr and the 95 percent confidence limit A are indicated in relation to the normal error curve. 

Statistics is the study of a large set of people, objects, or numbers, called a population. The population is not studied directly, because of its large size or inaccessibility. A subset from the population, called a sample, is studied and the likely properties of the population are inferred from the properties of the sample. If several repetitions of a measurement can be made, this set of measurements can be considered to be a sample from a population. The population is an imaginary set of infinitely many repetitions of the measurement. Statistical analysis can be used to study the properties of the sample and to infer likely properties of the population. 

The infinitely many members in a population of numerical quantities will be distributed among all values in some range according to some probability distribution. We have discussed probability distributions in Chapter 5. A probability density or 

If the probability distribution is normalized the total probability of all occurrences equals imity  

The most important property of a population of numerical quantities is its mean value. If there are no systematic errors, the mean of the population of measurements will equal the correct value of the measined quantity, since random errors are equally likely in either direction and will cancel in taking the mean. If the probability distribution /(x) is normalized the population mean fi is given by Eq. (5.69),  

The population standard deviation is a measure of the spread of the distribution, and is given by Eq. (5. 74) 

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Thus far the discussion in this chapter has concentrated on the statistical treatment of random errors in calibration data and calibration equations, with only passing mention of the implications of the practicalities involved in the acquisition of these data. For example, no mention has been made of what kind of calibration data are (or can be) acquired in common analytical practice, under which circumstances one approach is used rather than another and (importantly) what are the theoretical equations to which the experimental calibration data should be fitted by least-squares regression for different circumstances. Moreover, it is important to address the question of analytical accuracy to complement the discussion of precision that we have been mainly concerned with thus far the meanings of accuracy and precision in the present context are discussed in Section 8.1. The present section represents an attempt to express in algebra the calibration functions that apply in different circumstances, while exposing the potential sources of systematic uncertainty in each case. 

Statistical treatment of random errors, 971-983 Stator, motor, 53 Stern-Volnier equation, 408,409 Stokes... 

APPENDIX 1 Evaluation of Analytical Data 9(>7 alA Precision and Accuracy 967 alB Statistical Treatment of Random Errors 971 alC Hypothesis Testing 983 alD Method of Least Squares 985 Questions and Problems 988... 

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