Properties of a Population

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

Significance testing can be divided into a small number of steps. It starts with the formulation of the Null hypothesis. This is the assumption, which is made about the properties of a population of data expressed mathematically, e.g. there is no bias in our measurements . The second step is the formulation of the alternative hypothesis, the opposite of the Null hypothesis, in the above example there is a bias . 

The unknown quantities of interest described in the previous section are examples of parameters. A parameter is a numerical property of a population. One may be interested in measures of central tendency or dispersion in populations. Two parameters of interest for our purposes are the mean and standard deviation. The population mean and standard deviation are represented by p and cr, respectively. The population mean, p, could represent the average treatment effect in the population of individuals with a particular condition. The standard deviation, cr, could represent the typical variability of treatment responses about the population mean. The corresponding properties of a sample, the sample mean and the sample standard deviation, are typically represented by x and s, which were introduced in Chapter 5. Recall that the term "parameter" was encountered in Section 6.5 when describing the two quantities that define the normal distribution. In statistical applications, the values of the parameters of the normal distribution cannot be known, but are estimated by sample statistics. In this sense, the use of the word "parameter" is consistent between the earlier context and the present one. We have adhered to convention by using the term "parameter" in these two slightly different contexts. 

The most important property of a population of numerical quantities is its mean value. If there are no systematic errors, the mean of the population of measurements will equal the correct value of the measured quantity, since random errors are equally likely in either direction and will cancel in taking the mean. If the probability distribution f x) is normalized the population mean p is given by Eq. (5.69),... 

Since the processes are random in nature, we find that the use of statistics to describe the properties of a population of particles, and the particle size distribution is well suited for this purpose since it was originally designed to handle large numbers in a population. 

Periodic box The properties of a population of molecules is represented by the time-average behavior of a single molecule and its associated solvent isolated in a... 

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. 

Parameter n A parameter is a characteristic or property of a population. Often described in a more specific sense as a numerical property of a population. Usually a characteristic of a sample of a population is referred to as a statistic of the sample. 

Mathematical statistics is used to infer the properties of a population from a sample. 

The most important property of a population of numerical values is its mean value. The mean is one of three commonly used averages. The other averages are the median and the mode. The median is defined such that half of the members of the population have a value greater than the median and half have a value smaller then the median. The mode is the most commonly occurring value in the population. The mean value of x for the distribution is denoted by (x) and is defined by... 

The statistical measures can be calculated using most scientific calculators, but confusion can arise if the calculator offers the choice between dividing the sum of squares by N or by W — 1 . If the object is to simply calculate the variance of a set of data, divide by N . If, on the other hand, a sample set of data is being used to estimate the properties of a supposed population, division of the sum of squares by W — r gives a better estimate of the population variance. The reason is that the sample mean is unlikely to coincide exactly with the (unknown) true population mean and so the sum of squares about the sample mean will be less than the true sum of squares about the population mean. This is compensated for by using the divisor W — 1 . Obviously, this becomes important with smaller samples. 

Population Analyses Population analyses are used to gain a detailed understanding of the electronic properties of a molecule. A common feature of most of these analytic tools is the definition of atomic charges. Because there is no... 

The simple trend in the formulas shown by the third-row elements demonstrates the importance of the inert gas electron populations. The usefulness of the regularities is evident. Merely from the positions of two atoms in the periodic table, it is possible to predict the most likely empirical and molecular formulas. In Chapters 16 and 17 we shall see that the properties of a substance can often be predicted from its molecular formula. Thus, we shall use the periodic table continuously throughout the course as an aid in correlating and in predicting the properties of substances. 

Subset of a population that is collected Frank and Todeschini [1994] in order to estimate the properties of the underlying population , e.g., the sample parameters mean x and standard deviation s. In the ideal case of representative sampling, the sample parameter fit the parameter of the population ji and a, respectively. 

Thermodynamic processes play an important, or even dominant, role in all branches of science, from cosmology to biology and from the vastness of space to the microcosmos of living cells. Energy and entropy determine and direct all the processes which occur in the observable world. Thermodynamics only describes the properties of large populations of particles it cannot make any statements about the behaviour of single atoms or molecules. The most important properties of a system are determined by ... 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

In the final analysis we are interested not in toxicity, but rather in risk. By risk we mean the likelihood, or probability, that the toxic properties of a chemical will be produced in populations of individuals under their actual conditions of exposure. To evaluate the risk of toxicity occurring for a specific chemical at least three types of information are required ... 

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