Grand Mean

Since a series of t-tests is cumbersome to carry out, and does not answer all questions, all measurements will be simultaneously evaluated to find differences between means. The total variance (relative to the grand mean xqm) is broken down into a component Vi variance within groups, which corresponds to the residual variance, and a component V2 variance between groups. If Hq is true, Vi and V2 should be similar, and all values can be pooled because they belong to the same population. When one or more means deviate from the rest, Vj must be significantly larger than Vi. 

If Hq were to be retained, the individual means Xmeanj could not be distinguished from the grand mean xgm would then be taken as the average variance associated with Xgm and - 1 degrees of freedom. 

Interpretation of Table 1.16 Since V2 is significantly larger than V, the groups cannot all belong to the same population. Therefore, both the grand mean Xgm> which is equal to 219.93 -h 35 = 6.28, and the associated standard deviation /(49.28 -h 34) = 1.2 are irrelevant. The question of which... 

The difference between the mean of the data group (experimental or control) and the grand mean of the data... 

X represents the grand mean of all the data (from both groups). 

The mean of each row, and the difference of each row mean from the grand mean (this estimates the influence of the values of the factor corresponding to the rows)... 

Any difference between the actual data and the corresponding values calculated from the grand mean and the influences of the row and columns factors (this estimates the error variability). 

Solvent number Row diff. from grand mean Solvent number Row diff from grand mean... 

COMPARISON OF PRECISION AND ACCURACY FOR METHODS AND LABORATORIES USING THE GRAND MEAN FOR SAMPLES No. 1-3 (Collabor GM Worksheet), OR BY USING A SPIKED RECOVERY STUDY FOR SAMPLES No. 4-7 (Collabor TV Worksheet)... 

Table 34-5 Individual sample analysis estimated accuracy using grand mean calculation...

To compute the results shown in Table 34-5 for production samples, the accuracy of each set of replicates for each sample, method, and location was individually calculated using the root mean square deviation equation as shown in equations 34-5 and 34-6 in standard symbolic and MathCad notation, respectively. The standard deviation of each set of sample replicates yields an estimate of the accuracy for each sample, for each method, and for each location. The accuracy is calculated where each y is an individual replicate measurement GM is the Grand Mean of the replicate measurements for each sample, both methods, both locations and N is the number of replicates for each sample, method, and location. The results found in Table 34-5 represent samples 1-3. Note Each sample had a Grand Mean computed by taking the mean for all measurements made for each of the samples 1-3. 

The analytical results for each sample can again be pooled into a table of precision and accuracy estimates for all values reported for any individual sample. The pooled results for Tables 34-7 and 34-8 are calculated using equations 34-1 and 34-2 where precision is the root mean square deviation of all replicate analyses for any particular sample, and where accuracy is determined as the root mean square deviation between individual results and the Grand Mean of all the individual sample results (Table 34-7) or as the root mean square deviation between individual results and the True (Spiked) value for all the individual sample results (Table 34-8). The use of spiked samples allows a better comparison of precision to accuracy, as the spiked samples include the effects of systematic errors, whereas use of the Grand Mean averages the systematic errors across methods and shifts the apparent true value to include the systematic error. Table 34-8 yields a better estimate of the true precision and accuracy for the methods tested. 

GRAND MEANS FOR EACH ROW (USE IF NO TRUE VALUE IS AVAILABLE) ... 

In terms of confidence limits the two Grand Means can be written as 38.5 + 6.4 mg/m2 for the EC plot and 49.9 + 32.7 mg/m for the GF plot at the 90 level. This statement emphasizes the extent to which sampling variability can affect the confidence with which an analytical result is known. Unless the sampling program is designed to measure and identify the source of the variability much effort towards improvement of the quality of the chemical analyses can be wasted (4). The difficulty of improving the sampling procedures to reduce the variability is illustrated by calculation of the number of samples that would have to be analyzed to obtain estimates known to have an uncertainty less than 10 at the 90 confidence level (4). This would require 106 analyses from the EC plot and 2140 from the GF. Both sample sizes... 

Figure 10.2 Statistical process control charts for clearings. Top panel runs chart showing clearings as a function of measurement number. Middle panel x-bar chart with dashed upper control limit (UCL) and lower control limit (LCL) solid horizontal line is the grand mean, X. Bottom panel range chart with dashed upper control limit (UCL) solid horizontal line is the average range, r.

The analysis of chromatographic data is usually performed on normalized chromatograms, which is an attempt to account for the mass injected. However, the closure of analytical data is a problem with normalized data which has been described elsewhere (11). We examined our data for this problem by plotting the grand mean variation over all 368 peaks versus the standard deviations of these peaks. Closure did not occur in the unnormalized data. 

Analysis of Variance (ANOVA). Keeping in mind that the total variance is the sum of squares of deviations from the grand mean, this mathematical operation allows one to partition variance. ANOVA is therefore a statistical procedure that helps one to learn whether sample means of various factors vary significantly from one another and whether they interact significantly with each other. One-way analysis of variance is used to test the null hypothesis that multiple population means are aU equal. 

Progress Report No 9, July-Sept 1955. Although the method of temp, measurement in deton of solid expls was fairly accurate, the reproducibility needed improvement. A method for detn of temp of deton of NG is described and the values are tabulated. The grand mean is ca 4215°K... 

The certified value is usually taken as the grand mean of the valid results. The organizer uses standard deviation as the basis for calculating the measurement uncertainty. Results from the laboratories will include their own estimates of measurement uncertainty and statements of the metrological traceability of the results. There is still discussion about the best way to incorporate different measurement uncertainties because there is not an obvious statistical model for the results. One approach is to combine the estimates of measurement uncertainty as a direct geometric average and then use this to calculate an uncertainty of the grand mean. Type A estimates will be divided by /n n is the number of laboratories), but other contributions to the uncertainty are unlikely to be so treated. 

Nevertheless, the pattern was unresponsive to our standard tests for process instability, and individual batch results were well within established control limits for this product (180 to 220 mg). The grand mean of 99.0% is 2.0 mg below the theoretical tablet potency, probably because of below-target purity of... 

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