Central value, measures

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. 

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. 

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 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... 

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. 

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 Sections 1.6.3 and 1.6.4, different possibilities were mentioned for estimating the central value and the spread, respectively, of the underlying data distribution. Also in the context of covariance and correlation, we assume an underlying distribution, but now this distribution is no longer univariate but multivariate, for instance a multivariate normal distribution. The covariance matrix X mentioned above expresses the covariance structure of the underlying—unknown—distribution. Now, we can measure n observations (objects) on all m variables, and we assume that these are random samples from the underlying population. The observations are represented as rows in the data matrix X(n x m) with n objects and m variables. The task is then to estimate the covariance matrix from the observed data X. Naturally, there exist several possibilities for estimating X (Table 2.2). The choice should depend on the distribution and quality of the data at hand. If the data follow a multivariate normal distribution, the classical covariance measure (which is the basis for the Pearson correlation) is the best choice. If the data distribution is skewed, one could either transform them to more symmetry and apply the classical methods, or alternatively... 

Weekly values of average temperature (AVETEMP) and of heating (HEATDD) and cooling (COOLDD) degree days were calculated from the daily values measured in Central Park (in Manhattan) obtained from the National Weather Service. 

If one wants to test the adequacy of the model and the effects or coefficients, replicates of the entire design or measurements at a central point of the design are necessary. The real position of the central point is easily obtained by the use of Eq. 3-8 which in our simple case immediately leads to x, = central value. 

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,... 

Assuming that m0 is approximately the central value of all frequencies measured, the following transformations can be performed ... 

Table 1.1 Effect of various factors on pesticide exposure. All data are unit exposure values ( xg/kg of active ingredient (a.i.) handled), taken from PHED (1992). Values are central tendency measures based on high confidence data sets...

Two other measures of average value are the mode and the median. The sample mode is the most frequently observed value of the variable, and corresponds to the maximum in the population distribution curve. The median is the value of the observation for which half of the values in the population fall above it, and half below. If the number of observations is odd, the central value when the observations are ranked is taken as the median. If there is an even number of observations, the median is the arithmetic mean of the two central ranked observations. 

One of the objectives of data evaluation is to obtain values for the central value and dispersion that are as efficient as possible for a given expenditure of time and money. For a gaussian distribution the mean is the most efficient estimator for measuring the... 

The measures of dispersion partly reflect the estimator used, and an estimate of dispersion can be obtained either from the same series of observations used to obtain the central value or from a separate series. For a gdussian distribution the standard deviation and the mean are independent. 

Each individual measurement of any physical quantity yields a value A. But, independently of any possible observation errors associated with imperfect experimental measurements, the outcomes of identical measurements in identically prepared microsystems are not necessarily the same. The results fluctuate around a central value. It is this collection or Spectrum of values that characterizes the observable A for the ensemble. The fraction of the total number of microsystems leading to a given A value yields the probability of another identical measurement producing that result. Two parameters can be defined the mean value (later to be called the expected value ) and the indeterminacy (also called uncertainty by some authors). The mean value A) is the weighted average of the different results considering the frequency of their occurrence. The indeterminacy AA is the standard deviation of the observable, which is defined as the square root of the dispersion. In turn, the dispersion of the results is the mean value of the squared deviations with respect to the mean (A). Thus,... 

The arithmetic mean is not a perfect measure of the true central value of a given data set. Arithmetie means can overemphasize the importance of one or two extreme data points. Many measurements of a normally distributed data set will have an arithmetie mean that elosely approximates the true central value. 

Because one analysis gives no information about the variability of results, chemists usually carry two to five portions (replicates) of a sample through an entire analytical procedure. Individual results from a set of measurements are seldom the same (see Figure 5-1), so we usually consider the best estimate to be the central value for the set. We justify the extra effort required to analyze several samples in two ways. First, the central value of a set should be more reliable than any of the individual results. Usually, the mean or the median is used as the central value for a set of replicate measurements. Second, an analysis of the variation in data allows us to estimate the uncertainty associated with the central result. 

The most widely used measure of central value is the mean, x. The mean, also called the arithmetic mean, or the average, is obtained by dividing the sum of replicate measurements by the number of measurements in the set ... 

Median The central value in a set of replicate measurements. For an odd number of data points, there are an equal number of points above and below the median for an even number of data points, the median is the average of the central pair. 

---