Central value, statistical

One basic way of summarizing data is by the computation of a central value. The most commonly used central value statistic is the arithmetic average, or the mean. This statistie is partieularly useful when applied to a set of data having a fairly symmetrieal distribution. The mean is an efficient statistie in that it summarizes all the data in the set and because each piece of data is taken into account in its computation. The formula for eomputing the mean is... 

FIG. 23-44 Schematic representation of time-averaged distribution and spread for a continuous plume. and o2 are the statistical measures of crosswind and vertical dimensions 4.3oy is the width corresponding to a concentration 0.1 of the central value when the distribution is of gaussian form (a corresponding cloud height is 2.15o2). (Redrawn from Pasquill and Smith, Atmospheric Diffusion, 3d ed., Ellis Norwood Limited, Chichester, U.K, 1983). 

Toxicology has long recognized that no population, animal or human, is completely uniform in its response to any particular toxicant. Rather, a population is composed of a (presumably normal) distribution of individuals some resistant to intoxication (hyporesponders), the bulk that respond close to a central value (such as an LD50), and some that are very sensitive to intoxication (hyperresponders). This population distribution can, in fact, result in additional statistical techniques. The sensitivity of techniques such as ANOVA is reduced markedly by the occurrence of outliers (extreme high or low values, including hyper- and hyporesponders), which,... 

Section 1.6.2 discussed some theoretical distributions which are defined by more or less complicated mathematical formulae they aim at modeling real empirical data distributions or are used in statistical tests. There are some reasons to believe that phenomena observed in nature indeed follow such distributions. The normal distribution is the most widely used distribution in statistics, and it is fully determined by the mean value p. and the standard deviation a. For practical data these two parameters have to be estimated using the data at hand. This section discusses some possibilities to estimate the mean or central value, and the next section mentions different estimators for the standard deviation or spread the described criteria are fisted in Table 1.2. The choice of the estimator depends mainly on the data quality. Do the data really follow the underlying hypothetical distribution Or are there outliers or extreme values that could influence classical estimators and call for robust counterparts ... 

In statistics it is useful to work with the concept of squared deviations from a central value the average of the squared deviations is denoted variance. Consequently, the unit of the variance is the squared data unit. The classical estimator of the variance is the sample variance (v, x var), defined as the squared standard deviation. 

The normal distribution has some properties that are very important in understanding statistical results. The curve is symmetrical about the central value p. 68,27% of the values he widiin p + la, 95,45% within p + 2a and 99,73% within p + 3a. 

Another common category of descriptive statistics is the measure of dispersion of a set of data about a central value. The range is the arithmetic difference between the greatest (maximum) and the least (minimum) value in a data set. While this characteristic is easily calculated and is useful in initial inspections of data sets,... 

Condensation of information is obtained through formal calculations of central values and dispersions without prejudice as to the type of distribution. Interpretation of the meaning of the information obtained is another matter. Here knowledge of statistical decision theory is helpful.Statistical tools should be employed as aids to common sense. 

We have discussed, in previous sections, ways of estimating, for a normally distributed population, the central value (mean, x), the spread of results (standard deviation, s), and the confidence limits (t test). These statistical values hold strictly for a large population. In analytical chemistry, we typically deal with fewer th 10 results, and for a given analysis, perhaps 2 or 3. For such small sets of data, other estimates may be more appropriate. 

Si>mc elementary concepts of the distribution and its characterization will be di.scu.ssed here. A more comprchen.sive computer analysis is listed in the Statistical Analysis section. A database that is presumed to have a normal distribution may be characterized by two types of statistical parameters one that establishes its central value and one that charac-tcri/cs the spread or tlLspersion of values around the central value. 

The majority of statistical calculations and resulting decisions arc based on sample estimates for. y, S , and S. In certain circumstances the true values are known and slightly different procedures are used for statistical decisions. The standard deviation gives an estimate of the dispersion about the central value or mean in measurement units. For a normal distribution the interval of a about the mean /a will contain 68..1 percent of all values in the population the 2a interval contains 95.5 percent, and the .V interval contains 99.7 percent. A relative or unit-free indicator of the dispersion is the coefficient of variation ... 

Most industrial mixtures fail to conform to the statistically ideal pattern of equi-sized particles distinguishable only by colour. It is for this reason that equations (2.5) and (2.6) are so important in establishing the best attainable limits of mixture quality. The statistically precise work of Stange was applied to real powder mixtures by Poole, Taylor and Wall . This application involved assumptions and estimations which in view of the central value of the equation are worthy of investigation. 

As was explained in Section 2.9.10, the reduced and oxidized ions of a redox couple interact with the solvent dipoles by ion-dipole interaction. This influences the energy of the electronic states. The fluctuation of the solvent molecules around the ion with only a statistical equilibrium solvation leads to a distribution of the electron energies around a central value of Gaussian form. Two energy distribution functions describe the energy distribution, one for the reduced ions (the occupied states) and the other for the oxidized ions (the unoccupied states). This was shown in Figure 2.33. The development of two different distribution functions is based on stable oxidation states. In each state the ion-dipole interaction can achieve a quasi equilibrium distribution. 

A set of results has a statistical distribution, the shape of which is affected by these errors. The most commonly occurring distribution is the Normal Distribution. It is characterized by a bell-shaped (Gaussian) curve with a central value (a) and a width which is expressed in terms of the standard deviation (5). The central value, /, is a combination of the true value and the systematic error or bias of the method. The standard deviation of a method is a measure of the magnitude of the random errors. 

W median. Positional statistical parameter central value of those observed. 

When a given measurement (length, temperature, velocity, etc.) is made several times, a number of different values may be obtained due to random errors of measurement. In general, the observed magnitudes will tend to cluster about a central value and be less numerous with displacement from this central value. When the frequency of occurrence (y) of a given observed value is plotted vs the value of the variable of interest (x), a bell shaped curve will often be obtained particularly when the number of determinations (N) is large. In statistical parlance, such curves are called distributions. 

Interpreting Control Charts The purpose of a control chart is to determine if a system is in statistical control. This determination is made by examining the location of individual points in relation to the warning limits and the control limits, and the distribution of the points around the central line. If we assume that the data are normally distributed, then the probability of finding a point at any distance from the mean value can be determined from the normal distribution curve. The upper and lower control limits for a property control chart, for example, are set to +3S, which, if S is a good approximation for O, includes 99.74% of the data. The probability that a point will fall outside the UCL or LCL, therefore, is only 0.26%. The... 

Many distribution functions can be apphed to strength data of ceramics but the function that has been most widely apphed is the WeibuU function, which is based on the concept of failure at the weakest link in a body under simple tension. A normal distribution is inappropriate for ceramic strengths because extreme values of the flaw distribution, not the central tendency of the flaw distribution, determine the strength. One implication of WeibuU statistics is that large bodies are weaker than small bodies because the number of flaws a body contains is proportional to its volume. 

Monte Carlo simulation is a numerical experimentation technique to obtain the statistics of the output variables of a function, given the statistics of the input variables. In each experiment or trial, the values of the input random variables are sampled based on their distributions, and the output variables are calculated using the computational model. The generation of a set of random numbers is central to the technique, which can then be used to generate a random variable from a given distribution. The simulation can only be performed using computers due to the large number of trials required. 

Populations are very large collections of values. In practice, experimental pharmacology deals with samples (much smaller collections) from a population. The statistical tools used to deal with samples differ somewhat from those used to deal with populations. When an experimental sample is obtained, the investigator often wants to know about two features of the sample central tendency and variability. The central tendency refers to the most representative estimate of the value, while the variability defines the confidence that the estimate is a true reflection of that value. Central tendency estimates can be the median (value that divides the sample into two equal halves) or the... 

Descriptive statistics quantify central tendency and variance of data sets. The probability of occurrence of a value in a given population can be described in terms of the Gaussian distribution. 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

---