Statistical notions population

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 ... 

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. 

Statistical models. A number of statistical dose-response extrapolation models have been discussed in the literature (Krewski et al., 1989 Moolgavkar et al., 1999). Most of these models are based on the notion that each individual has his or her own tolerance (absorbed dose that produces no response in an individual), while any dose that exceeds the tolerance will result in a positive response. These tolerances are presumed to vary among individuals in the population, and the assumed absence of a threshold in the dose-response relationship is represented by allowing the minimum tolerance to be zero. Specification of a functional form of the distribution of tolerances in a population determines the shape of the dose-response relationship and, thus, defines a particular statistical model. Several mathematical models have been developed to estimate low-dose responses from data observed at high doses (e.g., Weibull, multi-stage, one-hit). The accuracy of the response estimated by extrapolation at the dose of interest is a function of how accurately the mathematical model describes the true, but unmeasurable, relationship between dose and response at low doses. 

Figure 1 shows a hypothetical tolerance frequency distribution, f(D)dD, along with its corresponding cumulative distribution, P(D). Thus, when the response is quantal in nature, the function P(D) can be thought of as representing the dose-response either for the population as a whole, or for a randomly selected subject. The notion that a tolerance distribution, or dose-response function, could be determined solely from consideration of the statistical characteristics of a study population was introduced independently by Gaddum (2) and Bliss (3). 

In much of statistics, the notion of a population is stressed and the subject is sometimes even defined as the science of making statements about populations using samples. However, the notion of a population can be extremely elusive. In survey work, for example, we often have a definite population of units in mind and a sample is taken from this population, sometimes according to some well-specified probabilistic rule. If this rule is used as the basis for calculation of parameter estimates and their standard errors, then this is referred to as design-based inference (Lehtonen and Pahkinen, 2004). Because there is a form of design-based inference which applies to experiments also, we shall refer to it when used for samples as sampling-based inference. 

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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