Normal probability plots calculations

The variance ratio (F value) is not readily calculated because replicated data are not available to allow the residual error term to be evaluated. However, it is usual practice to use the interaction data in such instances if the normal probability plot has shown them to be on the linear portion of the graph. By grouping the interaction terms from Table 7 as an estimate of the residual error,... 

Excluding the intercept term, diere are seven coefficients. A normal probability plot can be obtained as follows. First, rank die seven coefficients in order. Then, for each coefficient of rank p calculate a probability (p — 0.5)/7. Convert diese probabilities into expected proportions of die normal distribution for a reading of appropriate rank using an appropriate function in Excel. Plot the values of each of die seven effects (horizontal axis) against die expected proportion of normal distribution for a reading of given rank. 

The resulting print-out from MINITAB is given in Figure 2. However, the easiest way to interpret the data is to look at the pareto chart of calculated effects (Fig. 3) and related normal probability plot (Fig. 4). The vertical line on the pareto chart in Figure 3 corresponds to a P-value of 0.10 for each calculated effect. In other words. 

Calculate the normal probability plot for the residuals and compute the correlation coefficient. 

Minitab will give a normal probability plot directly. The result is shown in Figure 3.5. The program uses a slightly different method for calculating the percentage cumulative frequency but the difference is not important. 

There is no necessity in practice for the manual calculation of all these results, which would clearly be too tedious for routine use. The application of a spreadsheet program to some regression data is demonstrated in Section 5.9. Every advantage should also be taken of the extra facilities provided by programs such as Minitab, for example plots of residuals against x or y values, normal probability plots for the residuals, etc. (see also Section 5.15). 

The quality of the regression model is assessed in view of numerical and graphical information, which includes the model variance, confidence intervals on the parameter estimates, the linear correlation coefficient, residual and normal probability plots. The model variance is defined 5 = [(y-y) (y-y)]/v, where yand y are the measured and calculated vectors of the dependent variable respectively, v is the number of degrees of freedom (v= N- k +1)) and k is the number of independent variables included in the model. The linear correlation coefficient is defined by = [(y - (y - p)] /[(y -yf y- y)], where y is the mean of y. The variance and... 

Calculating the values, using appropriate formulae compute the parameter estimates, the normal probability plot of the parameters, and if appropriate, the SSRi and F-value for each of the parameters. 

Without going into the details of the calculation of all of the values, as an Excel spreadsheet was used to obtain the values, the summary results are presented here. The normal probability plot of the parameters is shown in Fig. 4.5. From this figure, it can be seen that there are 2 really significant values (which could well not be white noise), /5q and fis, which imply that the only significant factor is E. This suggests that the model can be written as... 

It is not possible to assess the effect of the substrate surface independently however, the powder can be characterized with respect to effective size by means of a simple sieve analysis that can be used to get a cumulative size distribution. If the total fraction that passes each size sieve is plotted against the sieve opening on normal probability paper, the mean weight-diameter from the 50% point is obtained and the standard deviation from the sizes corresponding to the 13% and/or 83% diameter. The effective (surface mean) particle diameter can then be calculated by means of the equation given by Orr and Dallavalle [66] ... 

Before statistical parameters were developed, the mean of the results reported by each participant in the water metals and water trace elements studies were plotted on normal probability paper to determine the distribution. Values showing a gross deviation from the normal distribution were then rejected as nonrepresentative because of errors in calculation, dilution, or other indeterminate factors and were not used in subsequent calculations. For the water nutrients study, a somewhat more sophisticated, and more objective, computer-programmed technique was used for rejection of outliers. As verified by plotting of the data on probability paper, however, the results were about the same. 

In order to eheek the model adequacy it is necessary to calculate the residuals, make the plots of normal probability and residuals versus fitted values. Furthermore, response surface plots can be realized using the regression model expression. 

From the 11 measurements carried out over a 2-year period including all seasons, Papastefanou and loannidou (1995) reported that the activity median aerodynamic diameter (AMAD) varied from 0.76 to 1.18 pm (average 0.90 pm) and the geometric standard deviation (a ) varied from 1.86 to 2.77 (average 2.24). The AMAD and (ag) calculations were made by plotting the cumulative distributions on log-normal probability paper. They also showed that 60% of the Be activity was associated with particles with diameter smaller than 1.1 pm. 

Fig. 2. Golden Mean process language Word 00 has zero probability aU others have nonzero probability. The logarithm base 2 of the word probabilities is plotted versus the binary string, represented as base-2 real number O.s . To allow word probabihties to be compared at different lengths, the distribution is normalized on [0,1]—that is, the probabilities are calculated as densities.

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

Figure 20.5.3 plots tlie pdf of the log-noniial distribution for a = 0 and (3=1. Probabilities concerning random variables liaving a log-normal distribution can be calculated using tables of the normal distribution. If X lias a log-normal distribution witli parameters a and p, then In X lias a normal distribution with p = a and o = p. Probabilities concerning X can tlierefore be converted into equivalent probabilities concerning In X. Suppose, for example, tliat X lias a log-nonnal distribution with a = 2 and p = 0.1. Then... 

Figure 12. Potential energy contour plots for He + I Cl(B,v = 3) and the corresponding probability densities for the n = 0, 2, and 4 intermolecular vibrational levels, (a), (c), and (e) plotted as a function of intermolecular angle, 0 and distance, R. Modified with permission from Ref. 40. The I Cl(B,v = 2/) rotational product state distributions measured following excitation to n = 0, 2, and 4 within the He + I Cl(B,v = 3) potential are plotted as black squares in (b), (d), and (f), respectively. The populations are normalized so that their sum is unity. The l Cl(B,v = 2/) rotational product state distributions calculated by Gray and Wozny [101] for the vibrational predissociation of He I Cl(B,v = 3,n = 0,/ = 0) complexes are shown as open circles in panel (b). Modified with permission from Ref. [51].

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