Other Statistical Measures

Other statistical measurements used in geology for particle size distribution characterization (moment, quartile and others) have been defined [135,136]. 

At this point, it should be noted that the statistics mentioned here are for illustrative purposes only. Such statistics describe the goodness of fit of an equation (e.g., R, s) its significance (e.g., F) and the significance of individual variables. For a more thorough statistical assessment, a wide variety of other statistical measures can be applied these were well described by Eriksson et al. (2003). The other statistical measures are likely to be those required for formal validation of a QSAR, frameworks for which were described in detail by Worth et al. (2004a, 2004b). 

Other statistical measures by which the analyst can judge the appropriate statistical convergence of the result... 

There are many different descriptive statistics that can be chosen as a measurement of the central tendency of a data set. These include arithmetic mean, the median, and the mode. Other statistical measures such as the standard deviation and the range are called measures of spread and describe how spread out the data are. 

The statistical measures can be calculated using most scientific calculators, but confusion can arise if the calculator offers the choice between dividing the sum of squares by N or by W — 1 . If the object is to simply calculate the variance of a set of data, divide by N . If, on the other hand, a sample set of data is being used to estimate the properties of a supposed population, division of the sum of squares by W — r gives a better estimate of the population variance. The reason is that the sample mean is unlikely to coincide exactly with the (unknown) true population mean and so the sum of squares about the sample mean will be less than the true sum of squares about the population mean. This is compensated for by using the divisor W — 1 . Obviously, this becomes important with smaller samples. 

A nonparametric approach can involve the use of synoptic data sets. In a synoptic data set, each unit is represented by a vector of measurements instead of a single measurement. For example, for synoptic data useful for pesticide fate, assessment could take the form of multiple physical-chemical measurements recorded for each of a sample of water bodies. The multivariate empirical distribution assigns equal probability (1/n) to each of n measurement vectors. Bootstrap evaluation of statistical error can involve sampling sets of n measurement vectors (with replacement). Dependencies are accounted for in such an approach because the variable combinations allowed are precisely those observed in the data, and correlations (or other dependency measures) are fixed equal to sample values. 

In chapter 2 I introduced the statistics of repeated measurements. Here I describe how these statistics are incorporated into a quality control program. In a commercial operation it is not always feasible to repeat every analysis enough times to apply t tests and other statistics to the results. However, validation of the method will give an expected repeatability precision (sr), and this can be used to calculate the repeatability limit (r), the difference between duplicate measurements that will only be exceeded 5 times in every 100 measurements. 

A set period of time is established for laboratories under test to carry out the prescribed chemical analyses and any other required measurements. A deadline is also given for return of results to the test provider, who will develop the statistics necessary to confirm the validity of the test round and to provide a figure of merit for each participant. The latter score may be as simple as pass/ fail. 

Of course, the probability is small that at any instant, the enantiomeric mixture at equilibrium is exactly equimolar the absence of observable chirality phenomena, such as optical activity, is the result of rapid cancelations of random statistical fluctuations of activity in the time domain of observation. In other words, although, at any instant, the mixture (with a high degree of probability) has an excess of one enantiomer or the other, under measurement conditions, it effectively contains an equal number of enantiomeric molecules. When 10,000,000 dissymmetric [i.e., chiral] molecules are produced under conditions which favor neither enantiomorph, there is an even chance that the product will contain an excess of more than 0.021 % of one enantiomorph or the other. It is practically impossible for the product to be absolutely optically inactive [12],... 

Once b has been determined, it is then possible to predict y and so calculate the sums of squares and other statistics as outlined in Sections 2.2.2 and 2.2.4. For the data in Table 2.6, the results are provided in Table 2.8, using the pseudo-inverse to obtain b and then predict y. Note that the size of the parameters does not necessarily indicate significance, in this example. It is a common misconception that the larger the parameter the more important it is. For example, it may appear that the b22 parameter is small (0.020) relative to the bn parameter (0.598), but this depends on the physical measurement units ... 

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