Random errors statistical treatment

The precision of a result is its reproducibility the accuracy is its nearness to the truth. A systematic error causes a loss of accuracy, and it may or may not impair the precision depending upon whether the error is constant or variable. Random errors cause a lowering of reproducibility, but by making sufficient observations it is possible to overcome the scatter within limits so that the accuracy may not necessarily be affected. Statistical treatment can properly be applied only to random errors. 

Randomization means that the sequence of preparing experimental units, assigning treatments, miming tests, taking measurements, and so forth, is randomly deterrnined, based, for example, on numbers selected from a random number table. The total effect of the uncontrolled variables is thus lumped together into experimental error as unaccounted variabiUty. The more influential the effect of such uncontrolled variables, the larger the resulting experimental error, and the more imprecise the evaluations of the effects of the primary variables. Sometimes, when the uncontrolled variables can be measured, their effect can be removed from experimental error statistically. 

B.2.1 Statistical treatment of finite samples 3B.2.2 Distribution of random errors 3B.2.3 Significant figures 3B.2.4 Comparison of results 3B.2.5 Method of least squares... 

As the name suggests, indeterminate errors cannot be pin-pointed to any specific well-defined reasons. They are usually manifested due to the minute variations which take place inadvertently in several successive measurements performed by the same analyst, using utmost care, under almost identical experimental parameters. These errors are mostly random in nature and ultimately give rise to high as well as low results with equal probability. They can neither be corrected nor eliminated, and therefore, form the ultimate limitation on the specific measurements. It has been observed that by performing repeated measurement of the same variable, the subsequent statistical treatment of the results would have a positive impact of reducing their importance to a considerable extent. 

LEAST SQUARES TREATMENT. Here, we deal briefly with the statistical algorithm, also known as linear regression analysis, for systems where Y (the random variable) is linearly dependent on another quantity X (the ordinary or controlled variable). The procedure allows one to fit a straight line through points (xi,yi), (x2,y2), ( 3,ys), , (Xn,yn) where the values X are defined before the experiment and y, values are obtained experimentally and are subject to random error. The best fit line through such a series of points is called a least squares fit, and the protocol provides measures of the reliability of the data and quality of the fit. 

There is, however, one obvious difference between a mathematical model and a physical model (or the real system itself). The response of the former to the same set of conditions is always identical. In physical experiments, where results are measured rather than calculated, there are inevitably random errors which may be appreciable. As already pointed out, mathematical models are usually to some extent imperfect in other words, they do contain systematic errors. The important point is that these imperfections are always reproduced in the same way, even though their ultimate source may have been random errors in data on which the model was based. This point has been stressed because it is important to recognize that only partial use of methods from statistical treatments of design of experiments is involved in what follows. The use of these methods here is only for the purpose of studying the geometry of response with respect to the controllable variables. No consideration of probability or of error enters into the discussion. 

Statistical overview of randomized, controlled trials that reduce random errors and may detect small treatment effects that are not apparent in a single study. 

In the analysis of the effect on the calculated quantity of random errors in measured quantities it is unfortunate that the only model susceptible to an exact statistical treatment is the linear one (II). Here we have attempted to characterize the frequency distribution of the error in the calculated vapor composition by the standard methods and have not included a co-variance term for each pair of dependent variables (12). Our approach has given a satisfactory result for the methanol-water-sodium chloride system but it has not been tested on other systems and perhaps of more importance, it has not been possible, so far, to confirm the essential correctness of the method by an independent procedure. Work is currently being undertaken on this project. 

Chapter 3 Using Spreadsheets in Analytical Chemistry 54 Chapter 4 Calculations Used in Analytical Chemistry 71 Chapter 5 Errors in Chemical Analyses 90 Chapter 6 Random Errors in Chemical Analysis 105 Chapter 7 Statistical Data Treatment and Evaluation 142 Chapter 8 Sampling, Standardization, and Calibration 175... 

The consideration of systematic experimental errors involves semantic problems as well as questions of proper statistical treatment. Although random errors are well-defined in the mathematical sense, the same cannot be said for systematic errors. 

Why have we gone to the trouble of classifying different types of error Because once we can identify the systemic errors we can correct for them, and a statistical treatment of the random error will allow us to estimate what the true result is and what uncertainty there may be about that result. Figure 1.5 brings together this discussion and shows the relationships between the true value of the measurand, the errors in a single measurement result, and the distribution of random errors. 

Random-effect model. A term which is used in at least two rather different senses by statisticians in the context of drug development. (1) A model for which more than one term is assumed random but the treatment effect is assumed fixed. (All statistical models, including so-called fixed ones have at least one error term which is random.) (2) A model in which the treatment effect itself is assumed to vary randomly from unit to unit. For balanced designs, random-effect models of the first sort can lead to identical inferences to fixed-effect models. Even for balanced designs, random-effect models of the second sort will not. 

Replication means that the basic experimental measurement is repeated. For example, if one is measuring the CO2 concentration of blood, those measurements would be repeated several times under controlled circumstances. Replication serves several important functions. First, it allows the investigator to estimate the variance of the experimental or random error through the sample standard deviation (s) or sample variance (i ). This estimate becomes a basic unit of measurement for determining whether observed differences in the data are statistically significant. Second, because the sample mean (x) is used to estimate the true population mean (/a), replication enables an investigator to obtain a more precise estimate of the treatment effect s value. If s is the sample variance of the data for n replicates, then the variance of the sample mean is = s /n. 

There are three classes of errors systematic errors are reproducible and the cause is related to some physical law and may be eliminated by appropriate corrective actions random errors are unpredictable and irreproducible and can be characterized by the laws of statistics finally, inadmissible errors occur as a result of mishandling, incorrect instrument operation, or even poor record keeping. Figure 2.6 shows a hierarchical chart that outlines the three types of errors (Chaouki et al., 2007). Most treatments of error consider systematic and random errors. Inadmissible errors might somehow be related to outliers, which are values that appear to be completely unrelated to a rational data set. 

The centerpiece of statistical treatment of data is the Gaussian Error function which gives mathematical expression for the behavior of random... 

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