Sample unbiased

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of O, not s. Although is an unbiased estimator of O, the value of for any randomly selected sample may differ significantly from O. To account for the uncertainty in estimating O, the term z in equation 4.11 is replaced with the variable f, where f is defined such that f > z at all confidence levels. Thus, equation 4.11 becomes... 

When one takes a sample at the rate of 0.3 liter min from a stack discharging 2000 m min to the atmosphere, the chances for error become quite large. If the sample is truly representative, it is said to be both accurate and unbiased. If it is not representative, it may be biased because of some consistent phenomenon (some of the hydrocarbons condense in the tubing ahead of the trap) or in error because of some uncontrolled variation (only 1.23 gm of sample was collected, and the analytical technique is accurate to 0.5 gm) (1). 

If the sample is unbiased, estimate the source mean, so that... 

The source mean is assumed to be the same if the sample is unbiased, as seen by... 

The variance of the sample and the population (source) may also be assumed equal if the sample is unbiased. The variance is S, defined as... 

In this work, examples are shown of the use of the computerized analytical approach in multicomponent polymer systems. The approach works well for both fractionated and whole polymers. The methodology can (1) permit differentiation to be made as to Whether the given sample conprises one conponent or a mixture of several components (2) allow the NMR spectrum of a polymer mixture to be analyzed in an unbiased fashion (3) give information on mole fractions and reaction probabilities that can be significant variables in understanding catalyst structures or polymerization mechanisms. 

Ciccotti, G. Kapral, R. Vanden-Eijnden, E., Blue moon sampling, vectorial reaction coordinates, and unbiased constrained dynamics, Chem. Phys. Chem. 2005, 6, 1809-1814... 

Since the bias function should enhance the sampling of pathways with important work values it can be made to depend on the work only, ir[z 2 ) = n W( (. Z))]. To minimize the statistical error in the free energy difference the bias function needs to be selected such that both the statistical errors of the numerator and the denominator of (7.44) are small. Ideally, the bias function should have a large overlap with both the unbiased work distribution P(W) and the integrand of (7.36), P (W) exp (—j3W). Just as Sun s work-biased ensemble Pa[z( ), the biased path ensemble )] can... 

Figure 65-1 shows a schematic representation of the F-test for linearity. Note that there are some similarities to the Durbin-Watson test. The key difference between this test and the Durbin-Watson test is that in order to use the F-test as a test for (non) linearity, you must have measured many repeat samples at each value of the analyte. The variabilities of the readings for each sample are pooled, providing an estimate of the within-sample variance. This is indicated by the label Operative difference for denominator . By Analysis of Variance, we know that the total variation of residuals around the calibration line is the sum of the within-sample variance (52within) plus the variance of the means around the calibration line. Now, if the residuals are truly random, unbiased, and in particular the model is linear, then we know that the means for each sample will cluster... 

Wang G, Brennan C, Rook M, et al. Balanced-PCR amplification allows unbiased identification of genomic copy changes in minute cell and tissue samples. Nucleic Acids Res. 2004 32 (e76) 1-10. 

Let 1, x2,..., xn be a random sample of N observations from an unknown distribution with mean fi and variance o2. It can be demonstrated that the sample variance V, given by equation A.8, is an unbiased estimator of the population variance a2. 

This shows that V is an unbiased estimator of a2, regardless of the nature of the sample population. 

Sample mean x and variance s2 are convergent and unbiased estimators (e.g., Hamilton, 1964), which implies that the so-called empirical variance a2 given by... 

We therefore make the assumption that the sample data gathered in vector y are only our best estimates of the real (population) values which justifies the bar on the symbol as representing measured values. This notation contradicts the standard usage, but is consistent with the basic definitions of Chapter 4. Indeed, for an unbiased estimate, we can still write that... 

Note that a scalar behaves as a symmetric matrix.) Because of finite sampling, and P cannot be evaluated exactly. Instead, we will search for unbiased estimates a and P of a and P together with unbiased estimates y( and xtj of yt and xu that satisfy the linear model given by equation (5.4.37) and minimize the maximum-likelihood expression in xt and y,. Introducing m Lagrange multipliers A , one for each linear... 

Therefore, the challenge in sampling solids for environmental analysis is to collect a relatively small portion of the sample that accurately represents the composition of the whole. This requires that sample increments be collected such that no piece, regardless of position (or size) relative to the sampling position and implement, is selectively collected or rejected. Optimization of solids sampling is a function of the many variable constituents of coal and is reflected in the methods by which an unbiased sample can be obtained, as is required by coal sampling (ASTM D197). 

Finally it should be emphasized that to obtain unbiased activity values in samples, all operations affecting the activity coefficients (dilution, changes in the composition) must be avoided. 

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