Statistics sampling

It must be stressed, however, that the whole object may be the analytical sample, e.g. a specimen of moon-rock. Ideally this sample would be analysed by non-destructive methods. Occasionally the bulk material may be homogeneous (some water samples) and then only one increment may be needed to determine the properties of the bulk. This increment should be of suitable size to provide samples for replicate analyses. 

The errors arising in sampling, particularly in the case of heterogeneous solids, may be the most important source of uncertainty in the subsequent analysis of the material. If we represent the standard deviation of the sampling operation (the sampling error) by ss and the standard deviation of the analytical procedures (the analytical error) by sA, then the overall standard deviation sT (the total error) is given by 

Example 1. If the sampling error is +3 per cent and the analytical error is 1 per cent, from equation (1) we can see that the total error sT is given by 

in the above example, the analytical error was 0.2 per cent then the total error sT would be equal to 3.006 per cent. Hence the contribution of the analytical error to the total error is virtually insignificant. Youden7 has stated that once the analytical uncertainty is reduced to one-third of the sampling uncertainty, further reduction of the former is not necessary. It is most important to realise that if the sampling error is large, then a rapid analytical method with relatively low precision may suffice. 

In designing a sampling plan the following points should be considered 8 

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These equations apply when an entire population is available for measurement. The most common situation in practical problems is one in which the number of measurements is smaller than the entire population. A group of selected measurements smaller than the population is called a sample. Sample statistics are slightly different from population statistics but, for large samples, the equations of sample statistics approach those of population statistics. 

Mitschele, J. Small Sample Statistics, /. Chem. Educ. 1991, 68, 470M73. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as X, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like [L and O are not random variables they are fixed constants. 

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of average of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

If the value of a sample statistic 6 is used to estimate a parameter 0 of the population, this statistic is called an estimator and its value for the sample the estimate. Sample mean x and variance s2 are the usual estimators of the population mean g and... 

The distribution of a sample statistic is a sampling distribution. A particularly important result concerns the variance of the sample mean given by... 

The useful equations (4.3.6) and (4.3.7), which are valid only for smooth monotonous functions, can be translated into relationships between the corresponding sample statistics... 

B.2.6 Recommendations for criteria of rejecting an observation 3B.2.7 Sampling statistics... 

Describe the various means and ways usually adopted for Sampling Statistics . Give suitable examples. 

By comparing the kinetic parameters of a donor-only sample to a donor-plus acceptor sample ( 50 cells/sample), statistically significant shifts can be determined for donor-acceptor pairs that have an average separation of <17 nm. This technique has been used to measure conformational changes in the CD4 antigen of human peripheral blood T-cells (7), as well as the relationships between various CD3 antigens and nearby accessory molecules (8). 

The general strategy of equating parameters to statistics is of course not restricted to moments. Reliance on sample percentiles (e.g., sample median) can lead to estimators that are not excessively sensitive to outliers. In general, to fit a distribution with k parameters, k parameters must be equated to distinct sample statistics. 

Sampling error In surveys, investigators frequently take measurements (or samples) on the parameters of interest, from which inferences to the true but unknown population are inferred. The inability of the sample statistics to represent the true population statistics is called sample error. There are many reasons why the sample may be inaccurate, from the design of the experiment to the inability of the measuring device. In some cases, the sources of error may be separated (see Variance components). 

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