Gaussian distribution Subject

The central-limit theorem (Section III.B) suggests that when a measurement is subject to many simultaneous error processes, the composite error is often additive and Gaussian distributed with zero mean. In this case, the least-squares criterion is an appropriate measure of goodness of fit. The least-squares criterion is even appropriate in many cases where the error is not Gaussian distributed (Kendall and Stuart, 1961). We may thus construct an objective function that can be minimized to obtain a best estimate. Suppose that our data i(x) represent the measurements of a spectral segment containing spectral-line components that are specified by the N parameters... 

The major pathway of coumarin metabolism in most human subjects is 7-hydroxyl-ation to form 7-hydroxy coumarin, which is excreted in the urine as both glucuronic acid and sulfate conjugates. Coumarin 7-hydroxylation activity exhibits a Gaussian distribution in Caucasian populations (Cholerton et al, 1992 Rautio et al, 1992), but some individuals are deficient in this activity. 

Exercise. A cannon projects a ball with initial velocity v at angle 9 with the horizontal. Both v and 9 are subject to uncertainties that can be described by a Gaussian distribution for each. The distributions are centered at v0 and 60, respectively, and so sharp that non-physical values, such as negative v or 9, may be ignored. What is the probability distribution of the distance covered by the cannon ball ... 

The theory of thermal aspects of laser desorption has been developed for a substrate surface subjected to pulsed laser irradiation, assuming that the laser intensity has a Gaussian distribution [21], The given surface is covered with the organic layer, which does not absorb the laser energy. However, the heat flux in the substrate that absorbs the energy heats the sample to the same temperature as the substrate. For this case, the laser intensity flux /(r,t) is given by... 

Figure 14-19 Outline of the relation between xl and x2 values measured by two methods subject to random errors with constant standard deviations over the analytical measurement range. A linear relationship between the target values (XI Target.. X2Targeti) Is presumed.The xlj and x2,- values are Gaussian distributed around Xi Target and X2Targeti. respectively, as schematically shown. 021 (Oyx) is demarcated.

In the softer sciences, the specific form of the relationship may not be known or, worse, it may not even be known whether a relationship exists at all. In that case, the question to be answered by statistics is not how to extract the best numerical parameters from the data, but how to establish whether or not a relationship exists in the first place, ft is here that concepts such as correlation coefficients become relevant. In quantitative chemical analysis, there are few such ambiguities, since the causal relations are usually well-established and seldom at issue. On the other hand, further statistical measures such as confidence limits, based on a (seldom experimentally supported) presumption of a single Gaussian distribution, are more strongly favoring a particular, mathematically convenient model than seems to be realistic or prudent for the subject matter of this workbook, and thereby tend to provide an overly rosy picture of the data. For this reason, statistical measures beyond standard deviations will not be considered here. 

The robustness to sensor drift of the method under study was evaluated using a simple synthetic drift model. A gain for each of the 60 sensors was initiated to 1 after which the gain factor was subject for over 100 random-walk steps taken from a Gaussian distribution with = 0.01. In the on-line learning condition while testing drift robustness, the last unsupervised vector quantization step was run continuously. 

The odor threshold is very specific to substance. It is determined in several measuring series and the results form a Gaussian distribution curve. Since this is ultimately a subjective evaluation, one should not be surprised to find more as well as, less, reliable data in literature. Examples are shown in Table 18.1.8. 

---