Statistical notions standard deviation

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. 

The superscript used in the coefficient matrices in (6.192) is a reminder that the statistics must be evaluated at the notional-particle location. For example, e = e(X r), and the scalar standard-deviation matrix and scalar correlation matrix p are computed from the location-conditioned scalar second moments X )(X, t). 

Since 1976 it has been recommended by IUPAC that the notion coefficient of variation should no longer be used instead of relative standard deviation. In statistics it is still known as a feature of a distribution derived from the moments. 

Example 46 There seems to be a clear reduction in the residual standard deviation, and the F-test supports this notion F (6.6/4.5) = 2.15, with F(26, 26, 0.05) = 1.93. Point No. 8 (see Fig. 4.21) is now only 0.011 AU above the parabola, which means it is barely outside the 2 band all other residuals are smaller. From the practical point of view there is little incentive for further improvement the residual standard deviation +4.5 mAU is now only about twice the experimental standard deviation (repeatability), which is not all that bad when one considers that two dilutions and a derivatization step are involved. The scatter appears to increase towards higher concentrations Indeed, this may be so, but to underpin the case statistically one would have to run at least eight repeats at a low and another eight at a high concentration if Shigh = 2 low, because Fcrit(7, 7,0.05) = 3.8. Should 5high only be 1.5 - low, then the experimental plan would call for nhigh = niow 18. 

Knowledge of statistics (Chapter 26) is basic to effective handling of the last unit operation, evaluation of data. Many chemists rarely go beyond a calculation of the standard deviation for a set of determinations and have the erroneous notion that the use of more advanced statistical methods is restricted to enormous bodies of data. A point often overlooked is that a relatively small number of systematically planned observations may yield more information than a larger number of repeated identical observations. For example, in the simple matter of running triplicate analyses of a sample, it may be best to weigh three samples of substantially different sizes. The results obtained may reveal determinate errors that would be unsuspected with samples of equal size. 

A more rigorous definition of uncertainty (Type A) relies on the statistical notion of confidence intervals and the Central Limit Theorem. The confidence interval is based on the calculation of the standard error of the mean, Sx, which is derived from a random sample of the population. The entire population has a mean /x and a variance a. A sample with a random distribution has a sample mean and a sample standard deviation of x and s, respectively. The Central Limit Theorem holds that the standard error of the mean equals the sample standard deviation divided by the square root of the number of samples ... 

Population and sample are discussed in Sect. 20.2. The properties of a population are studied in a representative random sample taken from that population. In pharmacy preparation practice populations are for instance batches of dosage units. Their properties are measured by analytical or biological assays and summarised as means, standard deviations and many other sample statistics. Some basic notions of probability distributions are briefly discussed. 

Quite a different notion is the distribution of a sample statistic such as the sample mean or the sample standard deviation. For instance samples taken from a population with parameters p and have themselves a distribution with mean p and variance = o /n, where n is the sample size. The square root of is often called the standard error of the mean or SEM ... 

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