Statistics random errors

To improve the statistical precision, replicate samples are processed for each set of conditions. Our error analysis methods have been described previously (28, 54). The cited measurement uncertainties represent single standard deviations at the 68% confidence level. In the case of yield branching ratios these uncertainties follow directly from statistical random error analysis. Speculative estimates of the contributions from possible systematic mechanistic errors have not been included. 

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. 

If systematic errors due to the analysts are significantly larger than random errors, then St should be larger than sd. This can be tested statistically using a one-tailed F-test... 

In the previous section we described several internal methods of quality assessment that provide quantitative estimates of the systematic and random errors present in an analytical system. Now we turn our attention to how this numerical information is incorporated into the written directives of a complete quality assurance program. Two approaches to developing quality assurance programs have been described a prescriptive approach, in which an exact method of quality assessment is prescribed and a performance-based approach, in which any form of quality assessment is acceptable, provided that an acceptable level of statistical control can be demonstrated. 

Different tests for estimation the accuracy of fit and prediction capability of the retention models were investigated in this work. Distribution of the residuals with taking into account their statistical weights chai acterizes the goodness of fit. For the application of statistical weights the scedastic functions of retention factor were constmcted. Was established that random errors of the retention factor k ai e distributed normally that permits to use the statistical criteria for prediction capability and goodness of fit correctly. 

Assumption 4 There is no systematic association of the random error for any one data point with the random error for any other data point. Statistically this is expressed as Correlation ( , 8 ) = 0. For u, V = 1, 2,. . . n, u V,... 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

The flowsheet shown in the introduction and that used in connection with a simulation (Section 1.4) provide insights into the pervasiveness of errors at the source, random errors are experienced as an inherent feature of every measurement process. The standard deviation is commonly substituted for a more detailed description of the error distribution (see also Section 1.2), as this suffices in most cases. Systematic errors due to interference or faulty interpretation cannot be detected by statistical methods alone control experiments are necessary. One or more such primary results must usually be inserted into a more or less complex system of equations to obtain the final result (for examples, see Refs. 23, 91-94, 104, 105, 142. The question that imposes itself at this point is how reliable is the final result Two different mechanisms of action must be discussed ... 

If systematic errors can be traced, and perhaps eliminated, and personal errors can be minimized, the remaining random errors can be analyzed by statistical methods. This procedure will be summarized in the following sections. 

By careful proceeding of measurements random variations can be minimized, but fundamentally not eliminated. The appearance of random errors follow a natural law (often called the Gauss law ). Therefore, random variations may be characterized by mathematical statistics, namely, by the laws of probability and error propagation. 

Allow estimates to be made of the magnitude of the noise and/or other random error, if for no other reason than to compare our results to so as to tell if they are statistically significant. 

In both experiments, Conditions 1 and 2 together mean that all results from the experiment will be the same in the first scenario, and all results except the ones corresponding to the effective catalyst will be the same while that one will differ. Condition 3 means that we do not need to use any statistical or chemometric considerations to help explain the results. However, for pedagogical purposes we will examine this experiment as though random error were present, in order to be able to compare the analyses we obtain in the presence and in the absence of random effects. The data from these two scenarios might look like that shown in Table 10-4. 

Continuing from our previous discussion in Chapter 18 from reference [1], analogous to making what we have called (and is the standard statistical terminology) the a error when the data is above the critical value but is really from P0, this new error is called the [3 error, and the corresponding probability is called the (3 probability. As a caveat, we must note that the correct value of [3 can be obtained only subject to the usual considerations of all statistical calculations errors are random and independent, and so on. In addition, since we do not really know the characteristics of the alternate population, we must make additional assumptions. One of these assumptions is that the standard deviation of the alternate population (Pa) is the same as that of the hypothesized population (P0), regardless of the value of its mean. 

In both the linear and the nonlinear cases the total variation of the residuals is the sum of the random error, plus the departure from linearity. When the data is linear, the variance due to the departure from nonlinearity is effectively zero. For a nonlinear set of data, since the X-difference between adjacent data points is small, the nonlinearity of the function makes minimal contribution to the total difference between adjacent residuals and most of that difference contributing to the successive differences in the numerator of the DW calculation is due to the random noise of the data. The denominator term, on the other hand, is dependent almost entirely on the systematic variation due to the curvature, and for nonlinear data this is much larger than the random noise contribution. Therefore the denominator variance of the residuals is much larger than the numerator variance when nonlinearity is present, and the Durbin-Watson statistic reflects this by assuming a value less than 2. 

One common characteristic of many advanced scientific techniques, as indicated in Table 2, is that they are applied at the measurement frontier, where the net signal (S) is comparable to the residual background or blank (B) effect. The problem is compounded because (a) one or a few measurements are generally relied upon to estimate the blank—especially when samples are costly or difficult to obtain, and (b) the uncertainty associated with the observed blank is assumed normal and random and calculated either from counting statistics or replication with just a few degrees of freedom. (The disastrous consequences which may follow such naive faith in the stability of the blank are nowhere better illustrated than in trace chemical analysis, where S B is often the rule [10].) For radioactivity (or mass spectrometric) counting techniques it can be shown that the smallest detectable non-Poisson random error component is approximately 6, where ... 

If a large number of readings of the same quantity are taken, then the mean (average) value is likely to be close to the true value if there is no systematic bias (i.e., no systematic errors). Clearly, if we repeat a particular measurement several times, the random error associated with each measurement will mean that the value is sometimes above and sometimes below the true result, in a random way. Thus, these errors will cancel out, and the average or mean value should be a better estimate of the true value than is any single result. However, we still need to know how good an estimate our mean value is of the true result. Statistical methods lead to the concept of standard error (or standard deviation) around the mean value. 

Indeterminate errors, also called random errors, on the other hand, are errors that are not specifically identified and are therefore impossible to avoid. Since the errors cannot be specifically identified, results arising from such errors cannot be immediately rejected or compensated for as in the case of determinate errors. Rather, a statistical analysis must be performed to determine whether the results are far enough off-track to merit rejection. 

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... 

Analytical quality control (QC) efforts usually are at level I or II. Statistical evaluation of multivariate laboratory data is often complicated because the number of dependent variables is greater than the number of samples. In evaluating quality control, the analyst seeks to establish that replicate analyses made on reference material of known composition do not contain excessive systematic or random errors of measurement. In addition, when such problems are detected, it is helpful if remedial measures can be Inferred from the QC data. 

Significance testing is another topic of statistics. What does this mean A typical situation for analytical chemists is where a reference material together with its certified value and uncertainty is given on the one hand and on the other hand there is the result of the analysis. The analyst wants to know whether the bias of his result is significant or just due to random errors. 

A 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, y0, X2,yi), x, ys),..., ( n,yn) where the values jCi 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. 

Deviations that arise probabilistically and have two characteristics (a) the magnitude of these errors is more typically small, and (b) positive and negative deviations of the same magnitude tend to occur with the same frequency. Random error is normally distributed, and the bell-shaped curve for frequency of occurrence versus magnitude of error is centered at the true value of the measured parameter. See Statistics (A Primer)... 

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