Normal probability paper

What is clear without the further aid of statistics is that the methanol concentration is the most important factor. Equally, it is clear that the citric acid concentration is not significant nor are three of the four interactions. Are the methanol concentration main effect and/or the interaction between the methanol and citric acid concentrations significant One way forward is to plot the data from Table 6 on normal probability paper. If all these data are insignificant then they will lie on a straight line. If values are observed that are a long way off the line it is likely that the effects or interactions are significant. 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

Normal probability paper is obtained by adjusting the vertical in such a way that the plot of P versus X is a straight line. Thus, data that follow a normal probability distribution will produce a straight line when plotted on normal probability paper, as shown in Figure 6. 

Figure 6. Plot of cumulative percent probability versus X plotted on normal probability paper.

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

Fig. 2.19 Graphical presentation of size analysis data on normal probability paper...

By another technique, plotting on normal probability paper, the significant parameters can be detected without prior knowledge of the experimental error variance. This very useful technique is described in Chapter 6. 

A plot of P(y) against y on normal probability paper will give a result as shown in Fig. 6.7b. 

The analysis utilized the approach of normal probability plots, as no reliable historical estimate of variation was available. For each primary response the observed factor effects were plotted on normal probability paper in rank order (Fig. 2). Here, factor effects are estimated as the difference of the average response at the high level (-I-) from the average response of the low level (-). On these plots, values that deviate markedly from the general trend line indicate significant effects. The largest factor effects (in absolute value) are labeled. 

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] ... 

The levels of Mn-SOD in sera from 194 male and 207 female healthy adult individuals were examined by ELISA. As shown in Fig. 14, the frequency distribution of serum Mn-SOD levels for the normal adult male was found to follow a normal distribution pattern. The distribution for the normal female adult was found to be slightly skewed, but the plotting of the cumulative frequency using normal probability paper gave a near-straight line. 

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. 

As has been emphasized in this chapter, many statistical tests assume that the data used are drawn from a normal population. One method of testing this assumption, using the chi-squared test, was mentioned in the previous section. Unfortunately, this method can only be used if there are 50 or more data points. It is common in experimental work to have only a small set of data. A simple visual way of seeing whether a set of data is consistent with the assumption of normality is to plot a cumulative frequency curve on special graph paper known as normal probability paper. This method is most easily explained by means of an example. 

Use normal probability paper to investigate whether the data below could have been drawn from a normal population ... 

Tabie 3.5 Data for normal probability paper example... 

Normal probability paper has a non-linear scale for the percentage cumulative frequency axis, which will convert this S-shaped curve into a straight line. A graph plotted on such paper is shown in Figure 3.4 the points lie approximately on a straight line, supporting the hypothesis that the data come from a normal distribution. 

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. 

Most elements are properly plotted on lognormal, rather than normal, probability paper. Although many geochemical distributions are not truly lognormal, a few are more closely approximated by a normal distribution. A signal exception to the approximation of lognormality is the distribution of pH... 

Cumulative drop size distributions can be plotted conveniently on linear or log probability paper. A straight line on linear or normal probability paper means that the drop sizes follow a normal or Gaussian distribution. If data form a straight line on log probability paper, the distribution is referred to as lognormal. 

More convincing than that is the plot of the logarithms of these 45 data on the normal probability paper. Effectively the data are almost aligned along a straight line as shown in Fig. 4.20. The designer re-run all the statistics on these 45 tests and gets... 

This equation has several properties that render the estimation of v and Dl easier The plot of the outlet concentration curve as a function of J = (U - l)/ /t7 corresponds to a normal probability distribution with a mean ft/ = 0 and a standard deviation a/ = The plot C/Co versus J on normal probability paper... 

In practical terms, the type and token cumulative frequency distributions may be tested for log-normality by plotting these functions on normal probability paper with the logarithm of frequency as the abscissa. When this test was applied to the medical use type and token frequency distributions, the log-normal model was found to describe both distributions very well over two orders of magnitude. Figure 1 shows this relationship for the use type distribution and also for the cluster type distribution. The cluster types involve a different counting than use types, so that two clusters are the same if all their components are Identical. The total number of cluster types is simply the sum of the last column in Table II. 

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