Within-groups mean square

The n will always fall between the minimum and maximum numbers of observations. For the analyst example, the between-groups mean square = 0.004383, the within-groups mean square = 0.000127, and n = 4.93. Therefore, the measurement standard deviation (cr) = /(0.000127) = 0.011 mM, and the between-analysts standard deviation (of) = /[(0.004383 -0.000127)/ 4.93] = 0.029 mM. 

When duplicate or split samples are sent for analysis, the repeatability and reproducibility can be calculated from an ANOVA of the data with the laboratories as the grouping factor. If the between-groups mean square is significantly greater than the within-groups mean square, as determined by an F test, then the variance due to laboratory bias can be computed as described in chapter 2. 

The within groups mean square allows estimation of the... 

The organizing laboratory will collect the results, test for normality of variances and outliers, and then determine repeatability precision and reproducibility precision, either directly or from an analysis of variance (ANOVA) on the data. A one way ANOVA, with the laboratories being the factor studied, will give the repeatability variance as the within groups mean square, and the between groups mean square is the repeatability plus the number of replicates times the laboratory variance. For duplicate determinations these quantities may be obtained directly from the means of the two results and their differences, which will subtract out the laboratory bias. 

In an ANOVA involving more than two groups, we estimate the underlying variability from more than two samples, and yet we are interested in the extent to which (only) two of the means differ from each other. Therefore, when comparing the means of two samples, the pooled standard deviation from the two-sample case, is replaced by an estimate that captures the variability across all groups in the analysis - the mean square error or the within-samples mean square. Recall from Section 11.4 that this quantity has the same interpretation as the pooled standard deviation, the typical spread of data across all observations. 

In our example of three treatment groups there are three pairwise comparisons of interest. Therefore, each pairwise comparison will be tested at an a level of 0.05/3 = 0.01667. This a level will require defining a critical value from the t distribution with 12 (that is, 15 - 3) df that cuts off an area of 0.00833 (half of 0.01667) in the right-hand tail. Use of statistical software reveals that the critical value is 2.77947. From inspection of the ANOVA table presented as Table 11.4 the within-samples mean square (mean square error) can be seen to be 1. The final component needed for the MSD is ... 

The quantity represented by the letter "q" is determined from a table of values used just for this test. Two characteristics are needed to determine the appropriate value of q each time that it is used. These characteristics are represented by the letters "a" and "v." The letter a represents the number of groups, which in this example is 3. The letter v represents the df, which in this test is the df associated with the within-samples mean square. In this case, the value of v is 12, as calculated for and shown in the ANOVA summary table in Table 11.4. From the table of q values for Tukey s test (provided in Appendix 5) the value of q associated with an... 

To obtain the distribution extrapolated to zero concentration, the distribution at each concentration is divided into a number of zones within the weight fraction zone 0 to 1. Then for each zone a plot of s or 1/s versus the sample concentration is made and extrapolated to obtained the sedimentation coefficient at zero concentration, sq A plot of weight fraction versus so is the corrected integral distribution at zero concentration. The differential distribution, dc/ds, can be obtained by fitting groups of points with a sliding least mean squares cubic fit. 

In this equation, the term SSW is refereed to as the the sum of squares within groups or error sum of squares. The quantity SSW when divided by the appropriate degrees of freedom J(I-l) is referred to as the mean square or error mean square and is denoted by MSW- As Eq. (1.114) is not particularly convenient for calculation purposes, it can be presented in the more usable form ... 

To properly understand CMP film thickness control, the CMP engineer should understand the sources of thickness variation and how they impact the total film thickness uniformity. Nonuniformity can be grouped in two categories—random variation and systematic variation. Examples of random variation include wafer-to-wafer (WTW), run-to-run (RTR), and some elements of within-wafer (WIW) variations. Elements of random variation add to the total thickness variation by their root mean square [19]... 

The next step is to calculate the 3 x 3 matrix of pooled within groups sums of (mean corrected) squares and cross products (SSCP) by the formula ... 

By dividing the sums of squares by their corresponding degrees of freedom, we can obtain quantities that are estimates of the between-groups and within-groups variations. These quantities are called mean square values and are defined as... 

An estimate of the average variance over all k groups represents the "typical" spread of data over the entire study or experiment. This variability is often referred to as random variation or noise. In the ANOVA strategy this number is called the within-group variance (or mean square error), and is calculated as a weighted average of the sample variances ... 

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