Value central

One way to characterize the data in Table 4.1 is to assume that the masses of individual pennies are scattered around a central value that provides the best estimate of a penny s true mass. Two common ways to report this estimate of central tendency are the mean and the median. 

If the mean or median provides an estimate of a penny s true mass, then the spread of the individual measurements must provide an estimate of the variability in the masses of individual pennies. Although spread is often defined relative to a specific measure of central tendency, its magnitude is independent of the central value. Changing all... 

Realizing that our data for the mass of a penny can be characterized by a measure of central tendency and a measure of spread suggests two questions. Eirst, does our measure of central tendency agree with the true, or expected value Second, why are our data scattered around the central value Errors associated with central tendency reflect the accuracy of the analysis, but the precision of the analysis is determined by those errors associated with the spread. 

Precision is a measure of the spread of data about a central value and may be expressed as the range, the standard deviation, or the variance. Precision is commonly divided into two categories repeatability and reproducibility. Repeatability is the precision obtained when all measurements are made by the same analyst during a single period of laboratory work, using the same solutions and equipment. Reproducibility, on the other hand, is the precision obtained under any other set of conditions, including that between analysts, or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the reproducibility of an analysis can be no better than its repeatability. 

Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magnitude and direction. Indeterminate errors need not affect the accuracy of an analysis. Since indeterminate errors are randomly scattered around a central value, positive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. 

An analysis, particularly a quantitative analysis, is usually performed on several replicate samples. How do we report the result for such an experiment when results for the replicates are scattered around a central value To complicate matters further, the analysis of each replicate usually requires multiple measurements that, themselves, are scattered around a central value. 

Developing a meaningful method for reporting an experiment s result requires the ability to predict the true central value and true spread of the population under investigation from a limited sampling of that population. In this section we will take a quantitative look at how individual measurements and results are distributed around a central value. 

In the previous section we introduced the terms population and sample in the context of reporting the result of an experiment. Before continuing, we need to understand the difference between a population and a sample. A population is the set of all objects in the system being investigated. These objects, which also are members of the population, possess qualitative or quantitative characteristics, or values, that can be measured. If we analyze every member of a population, we can determine the population s true central value, p, and spread, O. 

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. 

The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is reported by stating the mean or median. The range, standard deviation, or variance may be used to report the data s spread. Data also are characterized by their errors, which include determinate errors... 

The distribution of the results of an analysis around a central value is often described by a probability distribution, two examples... 

For example, the lowest octave band corresponds to a frequency range between 22 and 45 Hz. Its central value is... 

Median The central value of a series of observations ranked in order of magnitude. 

The normal distribution describes the way measurement results are commonly distributed. This type of distribution of data is also known as a Gaussian distribution. Most measurement results, when repeated a number of times, will follow a normal distribution. In a normal distribution, most of the results are clustered around a central value with fewer results at a greater distance from the centre. The distribution has an infinite range, so values may turn up at great distances from the centre of the distribution although the probability of this occurring is very small. 

This is the Menzel-Minnaert-Unsold interpolation formula (often used assuming Roo = 1). It gives a better approximation to stellar absorption-line profiles (which are definitely not flat-bottomed) than does the exponential formula the shape of the corresponding curve of growth is much the same, but its use leads to a b-parameter that is about 25 per cent higher for the same observational data. Denoting the central value of p by po, the Doppler part of the curve is given by... 

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). 

The collection of data often results in a somewhat randomly organized list of observations distributed in some way around a central value. 

The values are spread around this central value, extending over a range. 

The spread which may or may not be uniform, reflecting the shape of the curve. Usually, most values lie close to the central value. 

The distribution can then often be defined sufficiently in terms of the central value, a quantity expressing the degree of spread, and the general form of the distribution (the shape of the curve). The equation of the curve may be then written as (Davis and Goldsmith, 1972)... 

The arithmetic mean ( mean or average ) is the commonest measure of location or central value and is given by the sum of all observations divided by their number. That is... 

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 ... 

The classical and most used estimator for a central value is the arithmetic mean x (x mean in R-notation). Throughout this book the term mean will be used for the arithmetic mean. For a normal or approximately normal distribution the mean is the best (most precise) central value. 

A robust measure for the central value—much less influenced by outliers than the mean—is the median xm (x median). The median divides the data distribution into two equal halves the number of data higher than the median is equal to the number of data lower than the median. In the case that n is an even number, there exist two central values and the arithmetic mean of them is taken as the median. Because the median is solely based on the ordering of the data values, it is not affected by extreme values. 

There are several other possibilities for robustly estimating the central value. Well known are M-estimators for location (Huber 1981). The basic idea is to use a function iji that defines a weighting scheme for the objects. The M-estimator is then the solution of the implicit equation... 

Depending on the choice of the i/i-function, the influence of outlying objects is bounded, resulting in a robust estimator of the central value. Moreover, theoretical properties of the estimator can be computed (Maronna et al. 2006). 

MAD is based on the median xM as central value the absolute differences x — xM are calculated and MAD is defined as the median of these differences. In the case of a normal distribution MAD can be used for a robust estimation of the theoretical standard deviation cr by... 

The above measures of spread are expressed in the same unit as the data. If data with different units should be compared or the spread should be given in percent of the central value it is better to use a measure which is dimension free. Such a measure is... 

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. 

Tests for Two Central Values of Dependent Pairs of Samples... 

Tests for Several Central Values of Independent Samples... 

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