Sample mean value

Figure 16-3. Bergum acceptance curve for evaluating content uniformity (CU) data. Note The curve shows that for a relative standard deviation (%RSD) of 4.5%, a sample mean value (SMV) of 100% must be achieved for the manufacturing process to be judged as validated.

SMV, sample mean value RSD, relative standard deviation w, number of replicates USP, United State Pharmacopeia IP, Japanese Pharmacopeia EP, European Pharmacopeia. 

For the purposes of comparing assay, content uniformity, and dissolution data, simple statistics such as sample mean value (SMV) and relative standard deviation (%RSD) derived from experience of performing the tests over long periods of time can be used as acceptance criteria. Alternatively, more sophisticated statistics such as the z-test, F-test or t-test as shown in Table 16-4 can be applied [17-19,25]. In the case of evaluating CU data, it can be concluded that results from two labs are equivalent based on applying the simple statistics of the difference between the SMV from Lab A and Lab B (Table 16-4) not to be more than 2.0%. In the other examples where the more sophisticated statistics such as z-test, F-test, or t-test are applied (Table 16-4), results from two labs are considered to be equivalent because the calculated statistics in each case (z-calculated values of 0.32/0.64, F-calculated value of 0.14, or T-calculated values of 0.30/0.60) are less than the predicted statistics (z-critical value of 1.64, F-critical values of 3.18, or T-critical value of 1.73) [19,25]. 

Sample mean value (SMV) = 95-105% Difference between the SMV (i.e., mean difference) is <2% RSD (n = 3) <2.0%. In this example, the transfer was successfully completed because all predetermined acceptance criteria were met. 

Sigurdsson and Franzson (1988), based on iodine in 24-h urine samples, mean values. 

Negataki, 1993), results from different studies, based on casual urine samples, mean values SD. 

The temperature limit of the thermal stability of bitumen and its components was found to be relatively uniform independent of the origin of the samples. Mean values of this temperature with very small coefficients of variation were calculated (Tables 4-67 and 4-68). The sequence of the cracking temperatures ... 

Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known O. The standard deviation for the mean value. Ox, which also is known as the standard error of the mean, is... 

Defining the sample s variance with a denominator of n, as in the case of the population s variance leads to a biased estimation of O. The denominators of the variance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample s variance, however, substituting X for p, removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the member can always be deduced from the remaining - 1 members andX For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be... 

Significance testing for comparing two mean values is divided into two categories depending on the source of the data. Data are said to be unpaired when each mean is derived from the analysis of several samples drawn from the same source. Paired data are encountered when analyzing a series of samples drawn from different sources. 

Unpaired Data Consider two samples, A and B, for which mean values, Xa and Ab, and standard deviations, sa and sb, have been measured. Confidence intervals for Pa and Pb can be written for both samples... 

A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analysis, or for other samples. A f-test is used to compare mean values, and an F-test to compare precisions. Comparisons between two sets of data require an initial evaluation of whether the data... 

In this problem you will collect and analyze data in a simulation of the sampling process. Obtain a pack of M M s or other similar candy. Obtain a sample of five candies, and count the number that are red. Report the result of your analysis as % red. Return the candies to the bag, mix thoroughly, and repeat the analysis for a total of 20 determinations. Calculate the mean and standard deviation for your data. Remove all candies, and determine the true % red for the population. Sampling in this exercise should follow binomial statistics. Calculate the expected mean value and expected standard deviation, and compare to your experimental results. 

Determine the %CO for each sample, and report the mean value and the 95% confidence interval. 

A two-sample chart is divided into four quadrants, identified as (-P, -p), (-, -p), (-, -), and (-P, -), depending on whether the points in the quadrant have values for the two samples that are larger or smaller than the mean values for samples X and Y. Thus, the quadrant (-P, -) contains all points for which the result for sample X is larger than the mean for sample X, and for which the result for sample Y is less than the mean for sample Y. If the variation in results is dominated by random errors. 

A visual inspection of a two-sample chart provides an effective means for qualitatively evaluating the results obtained by each analyst and of the capabilities of a proposed standard method. If no random errors are present, then all points will be found on the 45° line. The length of a perpendicular line from any point to the 45° line, therefore, is proportional to the effect of random error on that analyst s results (Figure 14.18). The distance from the intersection of the lines for the mean values of samples X and Y, to the perpendicular projection of a point on the 45° line, is proportional to the analyst s systematic error (Figure 14.18). An ideal standard method is characterized by small random errors and small systematic errors due to the analysts and should show a compact clustering of points that is more circular than elliptical. 

In the two-sample collaborative test, each analyst performs a single determination on two separate samples. The resulting data are reduced to a set of differences, D, and a set of totals, T, each characterized by a mean value and a standard deviation. Extracting values for random errors affecting precision and systematic differences between analysts is relatively straightforward for this experimental design. 

Finally, the data for each analyst can be reduced to separate mean values, A . The variance of the individual means about the global mean is called the between-sample variance, s, and is calculated as... 

Once a significant difference has been demonstrated by an analysis of variance, a modified version of the f-test, known as Fisher s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values Xj and X2 is the f-test described in Chapter 4, except that Spool is replaced by the square root of the within-sample variance obtained from an analysis of variance. 

The principal tool for performance-based quality assessment is the control chart. In a control chart the results from the analysis of quality assessment samples are plotted in the order in which they are collected, providing a continuous record of the statistical state of the analytical system. Quality assessment data collected over time can be summarized by a mean value and a standard deviation. The fundamental assumption behind the use of a control chart is that quality assessment data will show only random variations around the mean value when the analytical system is in statistical control. When an analytical system moves out of statistical control, the quality assessment data is influenced by additional sources of error, increasing the standard deviation or changing the mean value. 

Construction of Property Control Charts The simplest form for a property control chart is a sequence of points, each of which represents a single determination of the property being monitored. To construct the control chart, it is first necessary to determine the mean value of the property and the standard deviation for its measurement. These statistical values are determined using a minimum of 7 to 15 samples (although 30 or more samples are desirable), obtained while the system is known to be under statistical control. The center line (CL) of the control chart is determined by the average of these n points... 

Property control charts can also be constructed using points that are the mean value, Xj, for a set of r replicate determinations on a single sample. The mean for the ith sample is given by... 

Suppose for one sample that n measurements are made of an isotope (R) the measured ratios are Ri, R2.Rp- A simple mean value (R) is given by Ri R2 R3 etc. 

Mean values from duplicate analyses of each of three samples by atomic absorption spectrophotometry. 

Confidence Interval for a Mean For the daily sample tensile-strength data with 4 df it is known that P[—2.132 samples exactly 16 do fall witmn the specified hmits (note that the binomial with n = 20 and p =. 90 would describe the likelihood of exactly none through 20 falling within the prescribed hmits—the sample of 20 is only a sample). 

The mean value x of a property x is a statistic based on a sample of n items defined by... 

The micropore volume varied from -0.15 to -0.35 cmVg. No clear trend was observed with respect to the spatial variation. Data for the BET surface area are shown in Fig. 14. The surface area varied from -300 to -900 mVg, again with no clear dependence upon spatial location withm the monolith. The surface area and pore volume varied by a factor -3 withm the monolith, which had a volume of -1900 cm. In contrast, the steam activated monolith exhibited similar imcropore structure variability, but in a sample with less than one fiftieth of the volume. Pore size, pore volume and surface area data are given in Table 2 for four large monoliths activated via Oj chemisorption. The data in Table 2 are mean values from samples cored from each end of the monolith. A comparison of the data m Table 1 and 2 indicates that at bum-offs -10% comparable pore volumes and surface areas are developed for both steam activation and Oj chemisorption activation, although the process time is substantially longer in the latter case. 

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