Normal error probability function

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. 

A probability function derived in this way is approximate the true probability function cannot be inferred from any finite number of measurements. However, it can often be assumed that the probability function is represented by a Gaussian distribution called the normal error probability function,... 

The dashed curve in Fig. 3 represents a normal error probability function, with a value of (T calculated with Eq, (13) from the 376 errors... 

The usual assumptions leading to the normal error probability function are those required for the validity of the central limit theorem The assumptions leading to this theorem are sufficient but not always altogether necessary the normal error probability function may arise at least in part from circumstances different from those associated with the theorem. The factors that in fact determine the distribution are seldom known in detail. Thus it is common practice to assume that the normal error probability function is applicable even in the absence of valid a priori reasons. For example, the normal error probability function appears to describe the 376 measurements of Fig. 3 quite well. However, a much larger number of measurements might make it apparent that the true probability function is slightly skewed or flat topped or double peaked (bimodal), etc. 

The error in D is a strong function of A[H+]/[H+], A[HA]/[HA] and A hpl/ HpL > small changes in [HA],9hpl or pH can produce large deviations of the D value. The principal source of error probably arises from the pH measurement. For example the error in the normalized D and D values for a trivalent two-order complex may be expressed ... 

PERMUTATIONS AND COMBINATIONS PROBABILITY DENSITY FUNCTION PROBABLE ERROR NORMAL ERROR CURVE STATISTICS (A Primer)... 

The normal probability function as expressed by Eq. (14) is useful in theoretical treatments of random errors. For example, the normal probability distribution function is used to establish the probability Hthat an error is less than a certain magnitude 8, or conversely to establish the limiting width of the range, —8 to 8, within which the integrated probability P, given by... 

Maximum likelihood (ML) estimation can be performed if the statistics of the measurement noise Ej are known. This estimate is the value of the parameters for which the observation of the vector, yj, is the most probable. If we assume the probability density function (pdf) of to be normal, with zero mean and uniform variance, ML estimation reduces to ordinary least squares estimation. An estimate, 0, of the true yth individual parameters (pj can be obtained through optimization of some objective function, 0 (0 ). ModeP is assumed to be a natural choice if each measurement is assumed to be equally precise for all values of yj. This is usually the case in concentration-effect modeling. Considering the multiplicative log-normal error model, the observed concentration y is given by ... 

Fortunately, the necessary mathematics for higher numbers of stages are aU worked out for us in the mathematics of probability. MTien the number of stages is large, such a curve becomes the normal curve of error. By substituting Kj K -f- 1) for and 1/(1 -f- K) for q in the probability function, ... 

The Gaussian distribution is also called the normal distribution or normal error distribution. It is associated with a limiting form of binomial distribution. The conditions for the Gaussian distribution are N very large and p = 5, that is, the probability of success is j and the probability of failure is the chances for success and failure are absolutely at random, no bias. The Gaussian distribution is a continuous function ... 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

The probability density of the normal distribution f x) is not very useful in error analysis. It is better to use the integral of the probability density, which is the cumulative distribution function... 

The basic premise of the SLIM technique is that the probability of error associated with a task, subtask, task step, or individual error is a function of the PIFs in the situation. As indicated in Chapter 3, an extremely large number of PIFs could potentially impact on the likelihood of error. Normally the PIFs that are considered in SLIM analyses are the direct influences on error such as levels of training, quality of procedures, distraction level, degree of feedback from the task, level of motivation, etc. However, in principle, there is no reason why higher level influences such as management policies should not also be incorporated in SLIM analyses. 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

One must note that probability alone can only detect alikeness in special cases, thus cause-effect cannot be directly determined - only estimated. If linear regression is to be used for comparison of X and Y, one must assess whether the five assumptions for use of regression apply. As a refresher, recall that the assumptions required for the application of linear regression for comparisons of X and Y include the following (1) the errors (variations) are independent of the magnitudes of X or Y, (2) the error distributions for both X and Y are known to be normally distributed (Gaussian), (3) the mean and variance of Y depend solely upon the absolute value of X, (4) the mean of each Y distribution is a straight-line function of X, and (5) the variance of X is zero, while the variance of Y is exactly the same for all values of X. 

Error function relates to the normal density of probability (pdf)/x(x) of Gauss. fx(x) is given by... 

A graphical display of the residuals tells us a lot about our data. They should be normally distributed (top left). If the variances increase with the concentration, we have inhomogeneous variances, called heteroscedasticity (bottom left). The consequences are discussed in the next slide. If we have a linear trend in the residuals, we probably used the wrong approach or we have a calculation error in our procedure (top right). Non-linearity of data deliver the situation described on bottom right, if we nevertheless use the linear function. 

Fig. 9.4.3 Normal-probability plot (a) and lognormal probability plot (b) for In particles shown in Figure 9.4.2. The ordinate stands for the cumulative percent of particles, with diameters smaller than d on the abscissa. This follows an error function and should give a straight line if the plot obeys the correspondent distribution, as seen in case (b). (From Ref. 4.)...

The joint solution is p = 3.2301 and a = 2.9354. It might not seem obvious, but we can also derive asymptotic standard errors for these estimates by constructing them as method of moments estimators. Observe, first, that the two estimates are based on moment estimators of the probabilities. Let x, denote one of the 500 observations drawn from the normal distribution. Then, the two proportions are obtained as follows Let z,(2.1) = l[x, < 2.1] and z,(3.6) = l[x, < 3.6] be indicator functions. Then, the proportion of 35% has been... 

Exercise. The function 77 defined by (1.6) is not the probability density of , but differs from it by a normalization factor. Find this factor and verify that no error is made by treating 77 as probability density in (1.12). 

On the other hand, random errors do not show any regular dependence on experimental conditions, since they are generated by many small and uncontrolled causes acting at the same time, and can be reduced but not completely eliminated. Thus, random errors are observed when the same measurement is repeatedly performed. In the simplest case, the universe of random errors is described by a continuous random variable e following a normal distribution with zero mean, i.e., for a univariate variable, the probability density function is given by... 

In our paper [133] we have performed calculations of the heats of formation using all three parametrizations (MNDO, AMI, PM3) and both types of the variation wave function (SLG and SCF). Empirical functions of distribution of errors in the heats of formation [141] for the SLG-MNDO and SCF-MNDO methods are remarkably close to the normal one. That means that the errors of these two methods, at least in the considered data set, are random. In the case of the SLG-MNDO method, the systematic error practically disappears for the most probable value of the error... 

This Gaussian distribution function is symmetrical about the true value m (here choosen as the origin for x) and thus implies that positive errors are as probable as negative errors. It is normalized, since ... 

Artefactual increases of as much as 50% in total thyroxine, estimated by a competitive protein-binding assay, and of as much as 30% in triiodothyronine resin uptake are probably due to rapid and continuing lipolytic hydrolysis of triglycerides after blood has been drawn (126). Thyroid function tests should therefore always be performed on blood samples taken before (or a sufficient time after) heparin treatment (127). An increase in serum-free thyroxine concentrations has also been reported after low molecular weight heparin, by up to 171% in specimens taken 2-6 hours after injection. When specimens were obtained 10 hours after injection, the effects were smaller, but with concentrations still up to 40% above normal the results can still cause errors of interpretation (128). 

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