Error, normally distributed

Box and Draper (1965) derived a density function for estimating the parameter vector 6 of a multiresponse model from a full data matrix Y, subject to errors normally distributed in the manner of Eq. (4.4-3) with a full unknown covariance matrix E. With this type of data, every event u has a full set of m responses, as illustrated in Table 7.1. The predictive density function for prospective data arrays Y from n independent events, consistent with Eqs. (7.1-1) and (7.1-3), is... 

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. 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

Understanding the distribution allows us to calculate the expected values of random variables that are normally and independently distributed. In least squares multiple regression, or in calibration work in general, there is a basic assumption that the error in the response variable is random and normally distributed, with a variance that follows a ) distribution. 

The distribution of measurements subject to indeterminate errors is often a normal distribution. 

A second example is also informative. When samples are obtained from a normally distributed population, their values must be random. If results for several samples show a regular pattern or trend, then the samples cannot be normally distributed. This may reflect the fact that the underlying population is not normally distributed, or it may indicate the presence of a time-dependent determinate error. For example, if we randomly select 20 pennies and find that the mass of each penny exceeds that of the preceding penny, we might suspect that the balance on which the pennies are being weighed is drifting out of calibration. 

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

Normal distribution curves showing the definition of detection limit and limit of identification (LOI). The probability of a type 1 error is indicated by the dark shading, and the probability of a type 2 error is indicated by light shading. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

The most commonly used form of linear regression is based on three assumptions (1) that any difference between the experimental data and the calculated regression line is due to indeterminate errors affecting the values of y, (2) that these indeterminate errors are normally distributed, and (3) that the indeterminate errors in y do not depend on the value of x. Because we assume that indeterminate errors are the same for all standards, each standard contributes equally in estimating the slope and y-intercept. For this reason the result is considered an unweighted linear regression. 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

The basic underlying assumption for the mathematical derivation of chi square is that a random sample was selected from a normal distribution with variance G. When the population is not normal but skewed, square probabihties could be substantially in error. 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

Both func tions are tabulated in mathematical handbooks (Ref. I). The function P gives the goodness of fit. Call %q the value of at the minimum. Then P > O.I represents a believable fit if ( > 0.001, it might be an acceptable fit smaller values of Q indicate the model may be in error (or the <7 are really larger.) A typical value of for a moderately good fit is X" - V. Asymptotic Iy for large V, the statistic X becomes normally distributed with a mean V ana a standard deviation V( (Ref. 231). 

Experience gained in the ZAF analysis of major and minor constituents in multielement standards analyzed against pure element standards has produced detailed error distribution histograms for quantitative EPMA. The error distribution is a normal distribution centered about 0%, with a standard deviation of approximately 2% relative. Errors as high as 10% relative are rarely encountered. There are several important caveats that must be observed to achieve errors that can be expected to lie within this distribution ... 

Measurement noise covariance matrix R The main problem with the instrumentation system was the randomness of the infrared absorption moisture eontent analyser. A number of measurements were taken from the analyser and eompared with samples taken simultaneously by work laboratory staff. The errors eould be approximated to a normal distribution with a standard deviation of 2.73%, or a varianee of 7.46. 

The numerator is a random normally distributed variable whose precision may be estimated as V(N) the percent of its error is f (N)/N = f (N). For example, if a certain type of component has had 100 failures, there is a 10% error in the estimated failure rate if there is no uncertainty in the denominator. Estimating the error bounds by this method has two weaknesses 1) the approximate mathematics, and the case of no failures, for which the estimated probability is zero which is absurd. A better way is to use the chi-squared estimator (equation 2,5.3.1) for failure per time or the F-number estimator (equation 2.5.3.2) for failure per demand. (See Lambda Chapter 12 ),... 

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 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 precondition for the use of the normal distribution in estimating the random error is that adequate reliable estimates are available for the parame-rcrs ju. and cr. In case of a repeated measurement, the estimates are calculated using Eqs. (12.1) and (12,3). When the sample size iiicrease.s, the estimates m and s approach the parameters /c and cr. A rule of rhumb is that when s 30. the normal distribution can be osecl,... 

There are some restrictions that we do not consider here. Our primary requirement is that the y, are normally distributed (for a given set of Xjj) about their mean true values with constant variance. We also, for the present, assume that the errors in the Xjj are negligible relative to those in y,. 

Secondly, knowledge of the estimation variance E [P(2c)-P (2c)] falls short of providing the confidence Interval attached to the estimate p (3c). Assuming a normal distribution of error In the presence of an Initially heavily skewed distribution of data with strong spatial correlation Is not a viable answer. In the absence of a distribution of error, the estimation or "krlglng variance o (3c) provides but a relative assessment of error the error at location x Is likely to be greater than that at location 2 " if o (2c)>o (2c ). Iso-varlance maps such as that of Figure 1 tend to only mimic data-posltlon maps with bull s-eyes around data locations. 

For complex reactions more than one dependent variable is measured. The fitting procedure should take all the observed variables into account. When each of the variables has a normally distributed error, all data are equally precise, and there is no correlation between the variables measured, parameters can be estimated by minimizing the following function ... 

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