Parent population

A set of chromosomes is called a population, and creation of a population from a parent population is called generation. 

Confidence Interval for a Variance The chi-square distribution can be used to derive a confidence interval for a population variance <7 when the parent population is normally distributed. For a 100(1 — Ot) percent confidence intei val... 

The standard deviation, Sj, is the most commonly used measure of dispersion. Theoretically, the parent population from which the n observations are drawn must meet the criteria set down for the normal distribution (see Section 1.2.1) in practice, the requirements are not as stringent, because the standard deviation is a relatively robust statistic. The almost universal implementation of the standard deviation algorithm in calculators and program packages certainly increases the danger of its misapplication, but this is counterbalanced by the observation that the consistent use of a somewhat inappropriate statistic can also lead to the right conclusions. 

All the possible measures of a given set of variables in all the possible subjects that exist is termed the population for those variables. Such a population of variables cannot be truly measured for example, one would have to obtain, treat and measure the weights of all the Fischer-344 rats that were, are, or ever will be. Instead, we deal with a representative group, a sample. If our sample of data is appropriately collected and of sufficient size, it serves to provide good estimates of the characteristics of the parent population from which it was drawn. 

In devising a model for an analytical operation, we identify a target population to which we want our conclusions to apply This will differ from the parent population from which the samples are actually taken The difference may be reduced by random selection of individual portions (increments) for analysis so that each part of the population has an equal chance of selection Genuinely random sampling is difficult because bias, unconscious or deliberate, is readily introduced Untrained individuals often have difficulty in accepting that an apparently unsystematic sampling pattern must be followed to be valid ... 

Estimator a rule or method for estimating a parameter of the parent population, usually from an incomplete sample. 

In general statistics there is a difference between the parent population of a random variable, e.g. x (sometimes also characterized by capital letters) and a single realization of the parent population expressed, e.g., as single measurements, xh of the variable x. The parent population means an infinity of values which follow a certain distribution function. In the reality of experimental sciences one always has single realizations, x , of the random variable x. 

Normally the true parameters (so-called parent population parameters) of distributions are not known. For empirical distributions they have to be estimated (symbol A) on the basis of a limited number, n, of observations (so-called sample parameters). Estimates of the most important parameters are ... 

Both the sample and the whole sampling program are the basis for answering the specific analytical question. This means the samples must represent the quality of the whole object under investigation, the so-called parent population defining the specific task or problem. 

Each sample contains an uncertainty - the sampling error - arising from the heterogeneity of the parent material. The potential hazard resulting from extrapolation from the smaller portion (laboratory sample) to the larger portion (parent population) becomes obvious with the aid of the following theoretical example ... 

The influence of emission of some airborne pollutants on an agricultural area has to be investigated. From an area of 10000 square meters, ten soil samples were taken from the surface soil horizon. 1 g of each sample was recently analyzed in the laboratory. This means that the results, for instance a pollutant concentration, obtained from ten 1 g samples are used to estimate the level of pollution in the whole area with a medium depth of the surface horizon of 30 cm and a medium soil density of 1.5 g cm-3. A parent population of more than 109 g has to be assessed from a total sample mass of 10 g The necessity of extrapolation over a range of several orders of magnitude indicates the problems connected with sampling process. 

Fig. 4-1 shows the different steps of sample preparation, starting with the parent population, according to the IUPAC, Analytical Chemistry Division, Commission on Analytical Nomenclature [HORWITZ, 1990]. 

The term specimen is of particular importance in environmental investigations. It is a specifically selected portion of a material taken from a dynamic system and assumed (on the basis of a priori knowledge or experience of the environment) to be representative of the parent population at the time it is taken and at the site from which it is taken. Even when the sampling error cannot be determined, taking a specimen is sometimes the only way of obtaining information on the state of pollution of the environment. The strongly time-dependent discharge of waste water from small factories is a typical example. 

A further difficulty of sampling in the environment must be taken into consideration in many queries, the parent population, e.g. the volume of polluted air over a town, can be defined only very subjectively. 

Random samples have to be selected in such a manner that any portion of the population has an equal (or known) chance of being chosen. But random sampling is, in reality, quite difficult. A sample selected haphazardly is not a random sample. Thus, random samples have to be obtained by using a random sampling process (for instance with random number generation for specimen selection). The samples must reflect the parent population on the basis of an equal probability distribution. 

Stratified samples consist of portions obtained from identified subparts (strata) of the parent population. Within each stratum, the samples are taken randomly. 

A normally distributed parent population X is characterized by its expected mean value /< and standard deviation a. If the samples taken from this population are representative, the sample average x is an unbiased estimate of /c... 

If the purpose of sampling is the detailed description of the composition of an object, the character of the internal correlation has to be investigated. The methods of autocorrelation and/or semivariogram analysis, as described in Sections 6.6 and 4.4.2, may be useful for clarification of the internal spatial and/or temporal relationships existing within the parent population to be sampled. Geostatistical methods, e.g. kriging, enable undistorted estimation of the composition of unsampled locations in the area of investigation. 

Simple random sampling. Simple random sampling is performed directly on the whole population (area or section) under investigation. Any increment taken from the parent population has an equal chance of being selected. In practice, the problem is that the sample has to be taken in space or time after random number generation, not haphazardly. 

Stratified random sampling. The parent population has first to be subdivided or... 

As we saw in Section 3.1.1, the familiar bell-shaped curve describes the sampling distributions of many experiments. Many distributions encountered in chemistry are approximately normal [3], Regardless of the form of the parent population, the central limit theorem tells us that sums and means of samples of random measurements drawn from a population tend to possess approximately bell-shaped distributions in repeated sampling. The functional form of the curve is described by Equation 3.19. 

In Example 4.9, the results from Example 4.8 are used to compute the residual variance and / -ratios for the data set described in Figure 4.16. The / -values for the 10 unknown spectra are shown in Table 4.2. The unknown spectrum contaminated with a minor level of an impurity is shown in the first row. All samples in the training set have small residual variances and / -ratios less than the critical value of F = 4.105. The unacceptable unknown spectrum has a very large F-value, indicating with a high degree of confidence that it is not a member of the parent population represented by the training set. 

The reason why there is a factor of I — 1 when using measurements in a number of samples to estimate statistics is because one degree of freedom is lost when determining variance experimentally. For example, if we record one sample, the sum of squares l =i (xi —x)2 must be equal to 0, but this does not imply that the variance of the parent population is 0. As the number of samples increases, this small correction is not very important, and sometimes ignored. 

A second facility that is sometimes useful is the random number generator function. There are several possible distributions, but the most usual is the normal distribution. It is necessary to specify a mean and standard deviation. If one wants to be able to return to the distribution later, also specify a seed, which must be an integer number. Figure A. 15 illustrates the generation of 10 random numbers coming from a distribution of mean 0 and standard deviation 2.5 placed in cells A1 -A10 (note that the standard deviation is of the parent population and will not be exactly the same for a sample). This facility is very helpful in simulations and can be employed to study die effect of noise on a dataset. 

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