Population, of data

The control chart is set up to answer the question of whether the data are in statistical control, that is, whether the data may be retarded as random samples from a single population of data. Because of this feature of testing for randomness, the control chart may be useful in searching out systematic sources of error in laboratory research data as well as in evaluating plant-production or control-analysis data. ... 

Models are constructed from samples of data and can be used to predict the behavior of the system for all conditions (the population of data). 

These ten results represent a sample from a much larger population of data as, in theory, the analyst could have made measurements on many more samples taken from the tub of low-fat spread. Owing to the presence of random errors (see Section 6.3.3), there will always be differences between the results from replicate measurements. To get a clearer picture of how the results from replicate measurements are distributed, it is useful to plot the data. Figure 6.1 shows a frequency plot or histogram of the data. The horizontal axis is divided into bins , each representing a range of results, while the vertical axis shows the frequency with which results occur in each of the ranges (bins). 

A normally distributed population of data can be characterized by two parameters. The centre, or location, of the population is described by the parameter i... 

As mentioned in Section 6.1.1, analysts generally have only a sample of data from a much larger population of data. The sample is used to estimate the properties, such as the mean and standard deviation, of the underlying population. 

The standard deviation is used to describe the dispersion of individual measurement results. If we make a number of repeated measurements on the same sample, the standard deviation provides an estimate of the expected spread of the results. The standard deviation of the mean describes the dispersion of mean values estimated from a number of samples drawn at random from the same population of data. The standard deviation of the mean will always be smaller than the standard deviation by a factor of +Jn, where n is the number of values that have been averaged to obtain the estimate of the mean. 

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 . 

Statistics should follow the technical scrutiny, not the other way round. A statistical analysis of data of an interlaboratory study cannot explain deviating results nor can alone give information on the accuracy of the results. Statistics only treat a population of data and provide information on the statistical characteristics of this population. The results of the statistical treatment may give rise to discussions on particular data not belonging to the rest of the population, but outlying data can sometimes be closer to the true value than the bulk of the population (Griepink et al., 1993). If no systematic errors affect the population of data, various statistical tests may be applied to the results, which can be treated either as individual data or as means of laboratory means. When different methods are applied, the statistical treatment is usually based on the mean values of replicate determinations. Examples of statistical tests used for certification purposes are described elsewhere (Horwitz, 1991). Together with the technical evaluation of the results, the statistical evaluation forms the basis for the conclusions to be drawn and the possible actions to be taken. 

Standard deviation — (of a population of data) (a) A measure of the -> precision of a population of data. It is the positive square root of the sum of the squares of the deviations between the observations and the - population mean (p), divided by the total number of replicate... 

Squaring the true standard deviation gives a term called the true vuriunce, a2. It can be shown that the standard deviation of means a, calculated for samples taken from the total population of data, will have a true standard deviation equal to a/ n, where n is the sample size. In other words, the spread of these means is less than the spread of the overall data around the group mean. 

In terms of the previously mentioned normal distribution, the probability that a randomly selected observation x from a total population of data will be within so many units of the true mean p can be calculated. However, this leads to an integral which is difficult to evaluate. To overcome this difficulty, tables have been developed in terms of p Ztrue standard deviation a of a particular normal distribution under study is known and assuming that the difference between the sample x and the true mean p is only the result of chance and that the individual observations are normally distributed, then a confidence interval in estimating p can be determined. This measure was referred to previously as the confidence level. 

The objective of statistical samphng is to establish likely values for the true error rate in the population of data being considered. If the tme error rate was known, the probabihties of given numbers of errors in samples could be obtained mathematically using standard statistical distributions. Statistical inference allows the reverse process — from an observed error rate in a sample likely and possible true error rates can be inferred. Likely data population error rates are defined by the 99% single upper confidence limit, and possible data population error rates by the 99.9% single upper confidence limit on the sample error rate. 

Large populations of data (in excess of 5000 items) can be regarded as infinite and thus a binomial approximation to the hypergeometric distribution can be applied. It is assumed that errors occur randomly throughout the data population. If data within the population has been obtained from different sources in different ways, there may be an expectation that error rates for these subpopulations may differ. If this is the case, the data population should be split into strata and analyzed separately. Note that for populations less than 5000 items it is recommended that all items be checked rather than a sample taken. 

The distribution of errors for a particular population of data is given by the two population parameters p and a. The population mean p expresses the magnitude of the quantity being measured the standard deviation o expresses the scatter and is therefore an index of precision. 

Statisticians find it useful to differentiate between the sample mean and the population mean. The sample mean x is the arithmetic average of a limited sample drawn from a population of data. The sample mean is defined as the sum of the measurement values divided by the number of measurements, as given by Equation 5-1, page 92. In that equation, N represents the number of measurements in the sample set. The... 

The population standard deviation a, which is a measure of the precision of a population of data, is given by the equation... 

The two curves in Figure 6-4a are for two populations of data that differ only in their standard deviations. The standard deviation for the data set yielding the broader but lower curve B is twice that for the measurements yielding curve A. The breadth of these curves is a measure of the precision of the two sets of data. Thus, the precision of the data set leading to curve A is twice as good as that of the data set represented by curve B. 

Note that z is the deviation of a data point from the mean relative to one standard deviation. That is, when x — p = cr, zis equal to one when x — p = 2(t, z is equal to two and so forth. Since z is the deviation from the mean relative to the standard deviation, a plot of relative frequency versus z yields a single Gaussian curve that describes all populations of data regardless of standard deviation. Thus, Figure 6-4b is the normal error curve for both sets of data used to plot curves A and B in Figure 6-4a. 

Because of area relationships such as these, the standard deviation of a population of data is a useful predictive tool. For example, we can say that the chances are 68.3 in 100 that the random uncertainty of any single measurement is no more than 1(T. Similarly, the chances are 95.4 in 100 that the error is less than 2cr, and so forth. The calculation of areas under the Gaussian curve is described in Feature 6-2. 

Population mean, The mean value for a population of data the true value for a quantity that is free of systematic error. Population of data The total number of values (sometimes infinite) that a measurement could take also referred to as a universe... 

Population standard deviation, cr A precision parameter based on a population of data. 

Statistical control The condition in which performance of a product or a service is deemed within bounds that have been set for quality assurance defined by upper and lower control limits. Statistical sample A finite set of measurements, drawn from a population of data, often from an infinite number of possible measurements. 

Very often a test population of data is not available or would be prohibitively expensive to obtain. When a test population of data is not possible to obtain, internal validation must be considered. The methods of internal PM model validation include data splitting, resampling techniques (cross-validation and bootstrapping) (9,26-30), and the posterior predictive check (PPC) (31-33). Of note, the jackknife is not considered a model validation technique. The jackknife technique may only be used to correct for bias in parameter estimates, and for the computation of the uncertainty associated with parameter estimation. Cross-validation, bootstrapping, and the posterior predictive check are addressed in detail in Chapter 15. 

The user could also use another way of checking his method. He could decide not to follow ISO Guide 33 and verify if its own values are covered by the population of data delivered in the certification exercise e.g. within the range of lab 02 and 08 taking their own standard deviation into account ... 

Additional characterisation tests are performed to examine the population of data. They do not lead to decisions on whether or not a parameters should be certified or a set of data should be excluded e.g. normality of distribution of means and individual data (Kolmogorov-Smirnov-Lilliefors), consistency between laboratories of variances (Bartlett), etc. Many other tests could be performed before calculating the certified value. No definitive rules are given in the various guides of ISO [1,7]. The basic principle should remain as follows ... 

To determine adhesive failure, it was necessary to apply appropriate algorithms to the data For quantitative analysis data were imported to a spreadsheet, smoothed to remove noise from the LVDTs, and then sorted to remove edge effects. Because there was considerable warp in all specimens due to the durability test, a parabolic function was fit to this distortion and subtracted from the raw data to produce a flat bondline. The data were again sorted (in ascending order) to produce a cumulative frequency distribution of surface irregularities (wood failure). Conceptually, a thickness tolerance could then be specified to define the bondline region as well as a depth tolerance for shallow wood failure. The relative population of data within these regions represented the percent e of adhesive, shallow, and deep wood failure. 

If no systematic errors affect the population of data, various statistical tests may be applied to the results, which can be treated either as individual data or as means of laboratory means. When different methods are applied, the statistical treatment is usually based on the mean values of replicate determinations. 

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