Standard deviation Statistics

Fig. 13.6 Classical trajectory Monte Carlo (CTMC) ionization and charge transfer cross sections, with statistical standard deviations, for specific initial f plotted against reduced impact speed v/ve. The cross sections are given in units of a 2, where a = rraQ. Circles are for = 2, squares for i = 14. Also included for comparison are points from approximate CTMC calculations with the Na target core held fixed for i = 14 (triangles) and ( = 2...

Counting statistics. The statistical standard deviation of the number of counts in the photopeak is equal to the square root of the integrated photopeak counts. For most useful determinations it is assumed that sufficient activity is produced in order to assure errors due to counting statistics are less than 1%. 

MHz, where the uncertainty is the statistical standard deviation of a single measurement. 

The choice of which parameter to use, and the limits of regions A to E, should be investigated first for each system studied. The figure of merit here is the difference between S (W) and Sbuik ( buik), expressed as a number of statistical standard deviations. 

One estimate for the accuracies of the molecular constants can be obtained from the statistical standard deviations. However, experience shows that these estimates are invariably too low, owing to the neglect of systematic errors. More reliable estimates can be found by two methods. In the first, several different bands with one state in common are analyzed independently and the results compared. Thus values for the ground-state rotational constants of glyoxal-d2 obtained from the independent analyses of five bands are reproduced in Table 3. It is seen that the values forA", B , and C are consistent to 0.00025, 0.00004, and 0.00004 cm"1, respectively, whereas the standard deviations of these constants obtained from the analysis of the 0-0 band are 0.00004, 0.000008, and 0.000009 cm"1, respectively. In this example a realistic estimate for the accuracy is roughly five times the standard deviation. 

In the second method, the ground-state rotational constants obtained from the analysis of an optical spectrum are compared with the constants obtained from a microwave investigation. Some values for cw-glyoxal are given in Table 4. Again it is found that it is necessary to multiply the statistical standard deviations of the constants by factors from 3 to 6 to obtain realistic estimates of the accuracy. The correlation matrix obtained from the analysis of the optical spectrum is given in Table 5. 

Chemical composition standards are certified for given concentrations, with a statistical (standard deviation) range given. If your method falls two standard deviations from the certified value, there is a 95% chance there is a significant (non-random) difference between the results. Depending on the concentration levels being measured, you may establish that the measurement should be within, for example, 2% of the certified value, or perhaps 10% if it is a trace analysis, and so forth. 

Because strength depends critically on porosity, this property should be always measured first. To allow a comparison of individual strength values which were determined on different agglomerates they must be adjusted to fit a representative porosity. Then a larger number of results should be averaged and presented together with the statistical standard deviation or the minimum and maximum deviation of single values. 

EP13-R. 1995. Laboratory statistics—Standard deviation, 1st ed. A report. 

Resulting /c-effective values for the modeled system, including an estimate of the statistical standard deviation of the -effective... 

The method allows variables to be added or multiplied using basic statistical rules, and can be applied to dependent as well as independent variables. If input distributions can be represented by a mean, and standard deviation then the following rules are applicable for independent variables ... 

The model building step deals with the development of mathematical models to relate the optimized set of descriptors with the target property. Two statistical measures indicate the quality of a model, the regression coefficient, r, or its square, r, and the standard deviation, a (see Chapter 9). 

If all sources of systematic error can be eliminated, there will still remain statistical errors. These errors are often reported as stcindard deviations. What we would particularly like to estimate is the error in the average value, (A). The standard deviation of the average value is calculated as follows ... 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

It is much easier to determine the standard deviation using a scientific calculator with built-in statistical functions. ... 

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

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

The mean and standard deviation for the differences are 2.25 and 5.63, respectively. The test statistic 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. 

Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. ... 

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. 

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. 

Control charts were originally developed in the 1920s as a quality assurance tool for the control of manufactured products.Two types of control charts are commonly used in quality assurance a property control chart in which results for single measurements, or the means for several replicate measurements, are plotted sequentially and a precision control chart in which ranges or standard deviations are plotted sequentially. In either case, the control chart consists of a line representing the mean value for the measured property or the precision, and two or more boundary lines whose positions are determined by the precision of the measurement process. The position of the data points about the boundary lines determines whether the system is in statistical control. 

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... 

Usually, 10 to 20 measurements are made of the isotope ratio for one substance. Sometimes, one or more of these measurements appears to be sufficiently different from the mean value that the question arises as to whether or not it should be included in the set at all. Several statistical criteria are available for reaching an objective assessment of the reliability of the apparently rogue result (Figure 48.10). Such odd results are often called outliers, and ignoring them gives a more precise mean value (lower standard deviation). It is not advisable to remove such data more than once in any one set of measurements. 

Statistically, the standard deviation (SD) from the mean Is given by the formula,... 

Statistically, a similar Indication of precision could be achieved by utilising the 95% probability level if the results fell on a "Gaussian" curve, viz., the confidence would lie within two standard deviations of the mean. R 2 x SD = 56.3 24.8... 

In Chaps. 5 and 6 we shall examine the distribution of molecular weights for condensation and addition polymerizations in some detail. For the present, our only concern is how such a distribution of molecular weights is described. The standard parameters used for this purpose are the mean and standard deviation of the distribution. Although these are well-known quantities, many students are familiar with them only as results provided by a calculator. Since statistical considerations play an important role in several aspects of polymer chemistry, it is appropriate to digress into a brief examination of the statistical way of describing a distribution. 

The degree of data spread around the mean value may be quantified using the concept of standard deviation. O. If the distribution of data points for a certain parameter has a Gaussian or normal distribution, the probabiUty of normally distributed data that is within Fa of the mean value becomes 0.6826 or 68.26%. There is a 68.26% probabiUty of getting a certain parameter within X F a, where X is the mean value. In other words, the standard deviation, O, represents a distance from the mean value, in both positive and negative directions, so that the number of data points between X — a and X -H <7 is 68.26% of the total data points. Detailed descriptions on the statistical analysis using the Gaussian distribution can be found in standard statistics reference books (11). 

Statistical quaUty control charts of variables are plots of measurement data, preferably the average result of repHcate analyses, vs time (Fig. 2). Time is often represented by the sequence of batches or analyses. The average of all the data points and the upper and lower control limits are drawn on the chart. The control limits are closely approximated by the sum of the grand average plus for the upper control limit, or minus for the lower control limit, three times the standard deviation. 

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 ... 

Quantitative stmcture—activity relationships have been estabUshed using the Hansch multiparameter approach (14). For rat antigoiter activities (AG), the following (eq. 1) was found, where, as in statistical regression equations, n = number of compounds, r = regression coefficient, and s = standard deviation... 

Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as X, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like [L and O are not random variables they are fixed constants. 

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