78-40.01 Definitions of Statistical Terms
The purpose of including statistical terms in the AACC Approved Methods is to acquaint cereal chemists with the statistical terms that are found in AACC publications. Without proper foundation and background, these definitions should not be used as a basis for undertaking statistical analyses. An elementary textbook, cautiously used, may be helpful. Tables of the various distributions are essential for statistical analysis, and these will be found in most texts.
Many statistical analyses are based on the assumption that a set of measurements, denoted by X_{1}, X_{2}, ... X, represents a random sample of size n from one or more normal probability distributions. Under this assumption it is possible to derive many results regarding the probability distribution of statistics that can be computed from the sample (sums, means, sums of squares, etc.).
A univariate normal distribution is completely identified when specific values are given to two parameters, namely, the mean, µ, and the standard deviation, sigma. One univariate normal distribution may be distinguished from another by noting that they have either different means or different standard deviations, or both.
In some investigations µ and sigma are estimated from a sample, and such terms as confidence limits, standard error of a sample mean, t-distribution, and random sample are encountered. In other investigations different treatments are applied to members of a sample to determine whether the new distributions (populations) thereby created have different means (µ's) or standard deviations (sigmas) or both. Discussion of statistical methods for this purpose involves new terms such as analysis of variance, sampling distributions (t, F, chi^{2}), least significant differences, tests of hypotheses, and level of significance.
Measurements are not always confined to just one characteristic of the units of a population. If two characteristics, X and Y, are involved, it is often assumed that sampling is from a bivariate normal distribution with means µx and µy, standard deviations sigmax and sigmay, and a new constant, rho, called the correlation between X and Y. Attention may be directed toward estimating rho or testing hypotheses that it has some value when a sample of observations consisting of pairs of values -- (X_{1}, Y_{1}), (X_{2}, Y_{2}), ..., (X_{n}, Y_{n}) -- is available. If interest centers on estimating the mean of the Y-values for a fixed value of X (µ_{y|x}), new terms such as simple linear regression and regression coefficients are encountered. Should a problem involve measuring more than two characteristics, it is often assumed that samples are drawn from a multivariate normal distribution; and a discussion of statistical analyses involves terms such as multiple correlation coefficients, partial correlation coefficients, and multiple linear regression.
If one is dealing with two or more characteristics in which the numerical measures cannot be assigned to the units, it may be that the units can be classified into different categories. In this case analyses involve contingency tables and the chi-square criterion for testing hypotheses about suspected relationships among the characteristics.
The need for definition of terms used in discussions of statistical analyses is apparent. In addition, certain symbols have gained wide acceptance and are often used in publications without being explicitly defined. Definitions for these symbols are numbered, in boldface, and numbers only are used for cross-reference between Glossary and Symbols
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78-60.01 Statistical Evaluation of Test Methods—Estimation of Variance in Analytical Tests
Variation in analytical results is expressed in terms of "within-laboratory" and "between-laboratory" variations. Within-laboratory variation includes variation of replications, analysts, and days. Between-laboratory variation is within-laboratory variation plus the variation among laboratories. This design is applicable to analyzing variability in any analytical method. Modification of design may be necessary in some cases, but this should be done under proper guidance.
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