Statistical statements

At this point, it is appropriate to introduce a number of statistical relationships with which I can describe the distribution of a number of measurements. This section mnst necessarily be somewhat mathematical. However, textbooks on statistics will cover the theoretical basis of these parameters in much detail, and here I will content myself with a number of simple definitive statements. Later, these will become relevant to an understanding of counting statistics. 

Let us assume we have m measurements, x, X2, x, x , each of which is an estimate of some parameter. The nature of the parameter is not important it might be a voltage, a length or, more relevantly, a number of events within a particular count period. The actual form, that is 

Practical Gamma-ray Spectrometry — 2nd Edition Gordon R. Gilmore 2008 John Wiley Sons, Ltd. ISBN 978-0-470-86196-7 

The difference between any particular value, Xj, and the expected value gives some idea of how good an estimate that particular measurement was. Taking the differences for all of the measurements into account would give an idea of the overall uncertainty of the measurements. However, some measurements will be below the expected value and others above taking a simple sum of the differences is hkely to give a result of precisely zero. To get around this, the sum of the square of the differences is used. The resulting factor is called the variance, so that  

Covariance is a measure of the interrelation, or correlation, between x and y. When there is no correlation, as is likely to be in all the cases discussed here, then cov(x, ) = 0. 

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A statistical statement about the smallest amount of analyte that can be determined with confidence. 

Finally, we have seen that the detection limit is a statistical statement about the smallest amount of analyte that can be detected with confidence. A detection limit is not exact because its value depends on how willing we are to falsely report the analyte s presence or absence in a sample. When reporting a detection limit, you should clearly indicate how you arrived at its value. 

Basically, as you read through the books and articles you have chosen, you should be looking for ideas, facts, statistics, statements, speeches, or other information—whether it be a sentence or a complete paragraph—that you feel will be important support material when you assemble your notes into a research paper. 

When n becomes large the t value tends toward the standardized normal value of 1.96 (z = 1.96), which was approximated to 2 above. The 95% confidence interval, calculated by equation 2.13, is sometimes explained much like the expanded uncertainty, as a range in which the true value lies with 95% confidence. In fact, the situation is more complicated. The correct statistical statement is if the experiment of n measurements were repeated under identical conditions a large number of times, 95% of the 95% confidence intervals would contain the population mean. ... 

The infinitely many (7-points represent infinitely many identical copies of our gas model, which started at time t from all the possible phases and which then move independently of each other, under similar conditions (i.e., the function E q, p) is the same for all of them). The fiction of such a host of infinitely many identical and independent gas models allows us to replace certain probability assumptions" by statistical statements. They were explicitly formulated and used first by Maxwell 13) (1878), and on this occasion he used the word statistico-mechanical" to describe the study of such ensembles of gas models (cf. note 3). However, seven years earlier Boltzmann [4] (1871) had already worked with essentially the same kind of ensembles, (cf. note 107). 

After the basic protocol has been worked out, the statistician should prepare a statistical statement. However, before summarizing what is expected by the FDA in this statement, a quick review of a few of the basic tools of statistics might be worthwhile. 

Above all, in seeing the statistical statement through to its full usefulness, be certain before the study begins that any questions raised by the FDA statisticians (or other experts) have been worked out in satisfactory revisions. 

The results of analyses planned in the statistical statement presented in the clinical protocol for the study should be provided, including an investigation of the time course of the responses. Explanations must be given if any analyses in the protocol are not carried out or are modified. 

Boltzmann equation An equation used in me smdy of a collection of particles in non-equilibrium statistical mechanics, particularly meir transport properties. The Boltzmatm equation describes a quantity called me distribution function, which gives a mamematical description of me state and how it is changing. The distribution function depends on a position vector r, a velocity vector v, and the time fi it mus provides a statistical statement about me positions and velocities of the particles at any time. In me case of one species of particle being present, Boltzmann s equation can be written... 

Now we have everything at hand we need to do statistical statements answering practical questions around reliability growth processes. 

Our low resolution RNA modeling protocol (7) generates a series of models for the structure of interest. The method permits us to identify conflicts in the experimental data, by a covariation analysis of the extent to which each experimental constraint is or is not satisfied by each model. By examining the variation in position of a given nucleotide in the different models, the protocol provides a statistical statement about the uncertainty in position of each nucleotide, in analogy with the temperature factors obtained from x-ray crystallography. 

Although quantum mechanics does not allow us to describe the electron in the hydrogen atom as moving in an orbit, it does allow us to make statistical statements about where we would find the electron if we were to look for it. For example, we can obtain the probability of finding an electron at a certain point in a hydrogen atom. Although we cannot say that an electron will definitely be at a particular position at a given time, we can say that the electron is hkely (or not likely) to be at this position. 

If the process X(t) is a stationary, zero mean Gaussian process. Thus, the condition for statlonarlty is converted Into a statistical statement in eq. (30). This allows the derivation of a measure C for nonstatlonarityi, ... 

The model should produce statistical statements on the lifetime in terms of the overall applied stress field, the overall volume of material, and boundary effects. Important input parameters are fiber packing geometry, fiber strength, matrix and interface creep exponents, rate factors in the stress-corrosion chemistry, and the applied stress level. 

XI.2.2 B/flj—There is insufficient interlaboratory test data to establish a statistical statement of bias for the procedure in Appendix XI of Test Method D 2007. 

Bias—There are no interlaboratory test data to establish a statistical statement on bias. 

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