Pooled standard deviation

It is unclear, however, how many degrees of freedom are associated with f(a, v) since there are two sets of independent measurements. If the variances sa and sb estimate the same O, then the two standard deviations can be factored out of equation 4.19 and replaced by a pooled standard deviation. Spool, which provides a better estimate for the precision of the analysis. Thus, equation 4.19 becomes... 

To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an T-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated... 

Since the standard deviations can be pooled, the test statistic is calculated using equation 4.20... 

Since Fgxp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated. 

Since a pooled standard deviation could not be calculated, the test statistic, fgxp, is calculated using equation 4.19... 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

The goal of a collaborative test is to determine the expected magnitude of ah three sources of error when a method is placed into general practice. When several analysts each analyze the same sample one time, the variation in their collective results (Figure 14.16b) includes contributions from random errors and those systematic errors (biases) unique to the analysts. Without additional information, the standard deviation for the pooled data cannot be used to separate the precision of the analysis from the systematic errors of the analysts. The position of the distribution, however, can be used to detect the presence of a systematic error in the method. 

From equation (5) the pooled standard deviation sp is given by ... 

Residual standard deviation within-day effects and for pooled data. 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

The standard deviation can be used as a measure of risk. For example, if we pool the risk of random demands by several customers in one warehouse, the above formula indicates that the risk increases slower than the mean demand. It is therefore generally a good idea to pool risks. 

The three sets of replicate results below were accumulated for the analysis of the same sample. Pool these data to obtain the most efficient estimate of the mean analyte content and the standard deviation. 

The precisions were found to be similar and the pooled standard deviation (absolute) computed to be 0.071%. Comment upon the efficiency of the mixing. 

A better estimate of the standard deviation may often be obtained by the pooling of results from more than one set. Thus, 5 may be calculated from K sets of data. 

Before compound collections are screened, the collection is calibrated to determine the concentration at which it should be tested. Five hundred samples that are representative of the entire collection (single compounds, pools of combinatorial compounds, or extracts) are serially diluted to identify the concentration at which 2 to 3% of the compounds show at least 80% inhibition of tumor cells within two to three standard deviations from the mean. This concentration is then used to screen the whole collection. 

Because of some peak tailing, the number of theoretical plates was based on peak width at one-half peak height Ns5.S4 he pooled standard deviation (all temperatures) of retention time measurements (dfs34) was t 0.007 minutes. 

Coefficient of variation (replicates at 50%) Pooled standard deviation (25 at 5 levels) Correlation coefficient (linear fit)... 

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