Small Sample Distributions

The population mean and standard deviation cannot be determined for an infinite population hence, they must be estimated from a sample of size n. When fi and a are estimated from small samples, fi x and o s, the uncertainty in their estimates may be large, depending on the size of n, thus the confidence interval described in Equation 3.8 must be inflated accordingly by use of the /-distribution. When n is small, say 3 to 5, the uncertainty is large, whereas when n is large, say 30 to 50, the uncertainty is much smaller. 

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The molecular dipole moment is perhaps the simplest experimental measure of charge density in a molecule. The accuracy of the overall distribution of electrons in a molecule is hard to quantify, since it involves all of the multipole moments. Experimental measures of accuracy are necessary to evaluate results. The values for the magnitudes of dipole moments from AMI calculations for a small sample of molecules (Table 4) indicate the accuracy you may... 

Applieation for approximately 40 years Design parameters treated as random variables Small samples used to obtain statistieal distributions... 

In reality, it is impossible to know the exaet eumulative failure distribution of the random variable, beeause we are taking only relatively small samples of the... 

So how does one infer that two samples come from different populations when only small samples are available The key is the discovery of the t-distribution by Gosset in 1908 (publishing under the pseudonym of Student) and development of the concept by Fisher in 1926. This revolutionary concept enables the estimation of ct ( standard deviation of the population) from values of standard errors of the mean and thus to estimate... 

In a qualitative way, the non uniformity of the beam was demonstrated in the experiment with a glass slide (1.25) the darkening of which varied considerably across the beam. Such an experiment is a quick way to check intensity distribution in an x-ray beam or to locate a region of uniform density for the positioning of a small sample. 

Let a volume (Vi) be injected onto a column resulting in a rectangular distribution of sample at the front of the column. According to the principle of the Summation of Variances, the variance of the final peak will be the sum of the variances of the sample volume plus the normal variance of a peak for a small sample. 

Thus, one can be far from the ideal world often assumed by statisticians tidy models, theoretical distribution functions, and independent, essentially uncorrupted measured values with just a bit of measurement noise superimposed. Furthermore, because of the costs associated with obtaining and analyzing samples, small sample numbers are the rule. On the other hand, linear ranges upwards of 1 100 and relative standard deviations of usually 2% and less compensate for the lack of data points. 

In its simplest form an aliquot of the aqueous solution is sh2dcen with an equal volume of an immiscible organic solvent. Llmited to small sample volumes and solutes-with large distribution constants. 

Huan et al. [41] measured the behavior of a small fluidized bed consisting of 45-80 mustard seeds in a small-bore vertical magnet. The small sample size allowed short pulses, and spatial distribution of collision correlation times and granular temperature were measured directly and compared with the hydrodynamic theory of Garzo and Dufty [42], This paper [41] contains an excellent survey of previous experiments on fluidized beds. 

A small sample of a coal slurry containing particles with equivalent spherical diameters from 1 to 500 pm is introduced into the top of a water column 30 cm high. The particles that fall to the bottom are continuously collected and weighed to determine the particle size distribution in the slurry. If the solid SG is 1.4 and the water viscosity is 1 cP, over what time range must the data be obtained in order to collect and weigh all the particles in the sample ... 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

Two of the major points to be made throughout this chapter are (1) the use of the appropriate statistical tests, and (2) the effects of small sample sizes (as is often the case in toxicology) on our selection of statistical techniques. Frequently, simple examination of the nature and distribution of data collected from a study can also suggest patterns and results which were unanticipated and for which the use of additional or alternative statistical methodology is warranted. It was these three points which caused the author to consider a section on scattergrams and their use essential for toxicologists. 

Sample distribution frequency in any one series should not be more than every two weeks and not less than every four months. A frequency greater than once every two weeks could lead to problems in turn-round of samples and results. If the period between distributions extends much beyond four months, there will be unacceptable delays in identifying analytical problems and the impact of the scheme on participants will be small. The frequency also relates to the field of application and amount of internal quality control that is required for that field. Thus, although the frequency range stated above should be adhered to, there may be circumstances where it is acceptable for a longer time scale between sample distribution, e.g. if sample throughput per annum is very low. Advice on this respect would be a function of the Advisory Panel. 

The median of an even number of results is nothing but the average of the two middle values provided the results are listed in order whereas for an odd number of results the median is the middle value itself. However, the mean and the median are exactly identical in the case of a truly symmetrical distribution. In short, median is an useful measure specifically when dealing with very small samples. 

Treatment of wood with multi-component systems is likely to result in separation of the components when large wood samples are treated. This has been likened to the action of a chromatography column (Schneider, 1995). This is a significant problem that is often only encountered during scale-up of laboratory-based studies, where satisfactory results were previously obtained on small wood samples. Similarly, treatment of large wood samples can often lead to considerable variability in results due to inhomogeneous distribution, which again may not be evident with small samples treated under laboratory conditions. 

Many methods have been used to quantify steroidal compounds. These include RIA, gas chromatogra-phy-mass spectrometry (GC/MS), high-performance liquid chromatography (HPLC), and liquid chroma-tography-mass spectrometry (LC/MS). Although these techniques are successful in the analysis of steroids, it has been difficult to achieve quantitative analysis of small samples of neurosteroids because of their low concentrations in nervous tissues. Highly specific analytical methods are required to analyze small quantities of neurosteroids and their sulfates. Only with extremely sensitive methods of analysis is it possible to discover whether neurosteroids are synthesized in nervous tissues in quantities sufficient to affect neuronal activity, and whether these neurosteroids are distributed uniformly in brain. 

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