Variance comparison

The extremly wide range is the consequence of the large sample variance. Comparison of two variances... 

The measured results are tested for variance homogeneity by means of the multiple variance comparison according to COCHRAN [KRAUSE and METZLER, 1983] ... 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

So D may be estimated either from the measured delay time, or from the variance of the response pulse. In the former case, comparison of equations... 

If improvement in precision is claimed for a set of measurements, the variance for the set against which comparison is being made should be placed in the numerator, regardless of magnitude. An experimental F smaller than unity indicates that the claim for improved precision cannot be supported. The technique just given for examining whether the precision varies with the two different analytical procedures, also serves to compare the precision with different materials, or with different operators, laboratories, or sets of equipment. 

In order to compare populations based on their respective samples, it is necessaiy to have some basis of comparison. This basis is predicated on the distribution of the t statistic. In effecd, the t statistic characterizes the way in which two sample means from two separate populations will tend to vaiy by chance alone when the population means and variances are equal. Consider the following ... 

Standard Costs for Budgetary Control For convenience and simplicity, we shall consider the total cost of a manufac tured product to be the sum of the material, labor, and overhead costs. Standard costs are those that have been predetermined and budgeted for the manufacture of a given amount of product in a given time. The deviation of the actual cost from the standard cost is called the variance. It is far easier to make comparisons between periods by using variances than by using actual production data. The different variances for materi, labor, and overhead costs are listed in Table 9-38. 

More terms of the series are usually not justifiable because the higher moments cannot be evaluated with sufficient accuracy from e)meri-mental data. A comparison of the fourth-order GC with other distributions is shown in Fig. 23-12, along with calculated segregated conversions of a first-order reaction. In this case, the GC is the best fit to the original. At large variances the finite value of the ordinate at... 

Based on the survey, the actual costs normally associated with the maintenance operation were reduced by more than 50 per cent. The comparison of maintenance costs included the actual labor and overhead of the maintenance department. It also included the actual materials cost of repair parts, tools and other equipment required to maintain plant equipment. The analysis did not include lost production time, variances in direct labor or other costs that should be directly attributed to inefficient maintenance practices. 

A comparison of two or more means can be made with a one-way analysis of variance. This tool compares... 

COMPARISON OF MORE THAN TWO MEANS (ANALYSIS OF VARIANCE)... 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

The distance from the origin or dj is a measure of the information contained in the corresponding row or column. If the distance is relatively large, in comparison to others, then the corresponding row or column can be seen to contribute more information (inertia, variance) to the result of the analysis. In the case when the distance is zero, then the row or column carries no information (inertia, variance) since all its elements are zero. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

When there are many samples and many attributes the comparison of profiles becomes cumbersome, whether graphically or by means of analysis of variance on all the attributes. In that case, PCA in combination with a biplot (see Sections 17.4 and 31.2) can be a most effective tool for the exploration of the data. However, it does not allow for hypothesis testing. Figure 38.8 shows a biplot of the panel-average QDA results of 16 olive oils and 7 appearance attributes. The biplot of the... 

There is a plethora of model adequacy tests that the user can employ to decide whether the assumed mathematical model is indeed adequate. Generally speaking these tests are based on the comparison of the experimental error variance estimated by the model to that obtained experimentally or through other means. 

Since the germination data obtained for the controls, regardless of solvent or pretreatment, fit normal distributions with different means and variances according to seed species, statistical analyses were made using the standard normal deviates for each species. To allow comparisons between species the germination data are shown as percent of the germination observed in the appropriate control. 

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