Standard deviation sample size

Sample size Standard deviation Standard error Minimum maximum Standardized ... 

Reasonable and justifiable acceptance criteria for the particle size distribution of samples from a future study, either scale-up or validation, would be the following each sample median should fall within the range 5.28-6.22 urn, and the particle size standard deviation should fall with the range 0.40-0.56 jim. 

The tantalum boat atomization technique provides very low detection limits coupled with small sample size. The reproducibility of the sample size is apparently the limiting factor on precision of results. Figure 10-25 shows results obtained on replications of zinc samples. Relative standard deviations of between 2.8 and 3.4% were obtained for zinc concentrations between 1 X 10 and 5 x 10 g. Some typical absolute detection limits with the tantalum boat systems are given in Table 10-4. They range from 10 to... 

In 10 repeat measurements using laser diffraction of a natural lacustrine sample with grains ranging from 1 to 100 [tm, the standard deviation of the mean size (8.59 pim) was only 0.03 urn. A single measurement made on each of 10 subsamples split from the same bulk sample, the standard deviation of the mean size was only 0.09 [im. Other statistical values from these two tests all had small deviations [63], However, particle size results of non-spherical particles from different instruments are often less consistent with different degrees of deviation when compared to other methods and is often a function of sample polydispersity and particle shape. 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

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... 

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 quantity of sample required comprises two parts the volume and the statistical sample size. The sample volume is selected to permit completion of all required analytical procedures. The sample size is the necessary number of samples taken from a stream to characterize the lot. Sound statistical practices are not always feasible either physically or economically in industry because of cost or accessibiUty. In most sampling procedures, samples are taken at different levels and locations to form a composite sample. If some prior estimate of the population mean, and population standard deviation. O, are known or may be estimated, then the difference between that mean and the mean, x, in a sample of n items is given by the following ... 

If the standard deviation of the lot caimot be estimated, a sampling program of greater sample size is required to generate an estimate of the standard deviation for future sampling operations. In some cases, sample size can be increased and sampling costs reduced by the use of automatic samplers. These offer a substantial reduction in labor costs but an increase in capital costs. 

Since the t distribution relies on the sample standard deviation. s, the resultant distribution will differ according to the sample size n. To designate this difference, the respec tive distributions are classified according to what are called the degrees of freedom and abbreviated as df. In simple problems, the df are just the sample size minus I. In more complicated applications the df can be different. In general, degrees of freedom are the number of quantities minus the number of constraints. For example, four numbers in a square which must have row and column sums equal to zero have only one df, i.e., four numbers minus three constraints (the fourth constraint is redundant). 

Chi-Square Distribution For some industrial applications, produrt uniformity is of primary importance. The sample standard deviation. s is most often used to characterize uniformity. In dealing with this problem, the chi-square distribution can be used where = (.s /G ) (df). The chi-square distribution is a family of distributions which are defined by the degrees of freedom associated with the sample variance. For most applications, df is equal to the sample size minus 1. 

If X is not normally distributed, tlien X, tlie mean of a sample of n observations on X, is approximately normally distributed witlt mean p and standard deviation, provided tlte sample size n is large (>30). Tliis result is based on an... 

FIGURE 11.23 Power analysis.The desired difference is >2 standard deviation units (X, - / = 8). The sample distribution in panel a is wide and only 67% of the distribution values are > 8. Therefore, with an experimental design that yields the sample distribution shown in panel a will have a power of 67% to attain the desired endpoint. In contrast, the sample distribution shown in panel b is much less broad and 97% of the area under the distribution curve is >8. Therefore, an experimental design yielding the sample distribution shown in panel B will gave a much higher power (97%) to attain the desired end point. One way to decrease the broadness of sample distributions is to increase the sample size. 

The size of beads was uniform and consistent, the mean size of beads with 3% alginate and based on measurement of 20 samples the mean value for the beads diameter was 4.85 mm, with a standard deviation of 0.3 mm and calculated variance of 0.1 mm. The standard deviation was less than 5%. The data for the batch fermentation experiment with 50gl 1... 

Figure 1.2. The range R(n) for size of sample n, with n - 2. .. 40 (left). The line gives the tabulated values.range R is given as y = R/sx in units of the experimental standard deviation. A total of 8190 normally distributed values with mean 0 and standard deviation 1 was simulated. (See Section 3.5.5.) The righthand figure gives the distribution of ranges found after simulating 100 sets of n = 10 normally distributed values.

Figure 2.2. Examples of correlations with high and low coefficients of determination. Data were simulated for combinations of various levels of noise (a = 1,5, 25, top to bottom) and sample size (n - 10, 20, 40, left to right). The residual standard deviation follows the noise level (for example, 0.9, 5.7, 24.7, from top to bottom). Note that the coefficient 0.9990 in the top left panel is on the low side for many analytical calibrations where the points so exactly fit the theoretical line that > 0.999 even for low n and small calibration ranges.

The point that needs to be made is that with sample size as small as it is here (n = 10), the distribution can strongly vary in appearance from one sample to the next, much more so than with n - 100 as in Fig. 1.10 for example, vectors C 46 (column 1) and C 55 (column 20) of file TABLET C.dat are the extremes, with standard deviations of 2.88 and 1.21. The corresponding Huber s k-values for the largest residual in each vector are 6.56 (this looks very much like an outlier, kc = 3.5) and 2.11 (far from being an outlier). The biggest k-value is found for vector C 49 (column 8) at 7.49 Fig. 4.25 shows the results for this vector as they are presented by program HUBER. 

Note. If the N dimensions yield very different numerical values, such as 105 3 mmol/L, 0.0034 0.02 meter, and 13200 600 pg/ml, the Euclidian distances are dominated by the contributions due to those dimensions for which the differences A-B, AS, or BS are numerically large. In such cases it is recommended that the individual results are first normalized, i.e., x = (x - Xn,ean)/ 5 t, where Xmean and Sx are the mean and standard deviation over all objects for that particular dimension X, by using option (Transform)/(Normalize) in program DATA. Use option (Transpose) to exchange columns and rows beforehand and afterwards The case presented in sample file SIEVEl.dat is different the individual results are wt-% material in a given size class, so that the physical dimension is the same for all rows. Since the question asked is are there differences in size distribution , normalization as suggested above would distort tbe information and statistics-of-small-numbers artifacts in the poorly populated size classes would become overemphasized. 

In this study the reader is introduced to the procedures to be followed in entering parameters into the CA program. For this study we will keep Pm = 1.0. We will first carry out 10 runs of 60 iterations each. The exercise described above will be translated into an actual example using the directions in Chapter 10. After the 10-run simulation is completed, determine (x)6o, y)60, and d )6o, along with their respective standard deviations. Do the results of this small sample bear out the expectations presented above Next, plot d ) versus y/n for = 0, 10,20, 30,40, 50, and 60 iterations. What kind of a plot do you get Determine the trendline equation (showing the slope and y-intercept) and the coefficient of determination (the fraction of the variance accounted for by the model) for this study. Repeat this process using 100 runs. Note that the slope of the trendline should correspond approximately to the step size, 5=1, and the y-intercept should be approximately zero. 

In Figure 14(a) the bright-held TEM cross-sectional image of the reference AuAg sample is shown. The size distribution of the clusters has an average diameter = 11.7nm and a standard deviation of the experimental bimodal distribution o- = 6.4nm. The effect of... 

In reality, the queue size n and waiting time (w) do not behave as a zero-infinity step function at p = 1. Also at lower utilization factors (p < 1) queues are formed. This queuing is caused by the fact that when analysis times and arrival times are distributed around a mean value, incidently a new sample may arrive before the previous analysis is finished. Moreover, the queue length behaves as a time series which fluctuates about a mean value with a certain standard deviation. For instance, the average lengths of the queues formed in a particular laboratory for spectroscopic analysis by IR, H NMR, MS and C NMR are respectively 12, 39, 14 and 17 samples and the sample queues are Gaussian distributed (see Fig. 42.3). This is caused by the fluctuations in both the arrivals of the samples and the analysis times. 

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