Normal distribution standard deviation

The Burchell model s prediction of the tensile failure probability distribution for grade H-451 graphite, from the "SIFTING" code, is shown in Fig. 23. The predicted distribution (elosed cireles in Fig. 23) is a good representation of the experimental distribution (open cireles in Fig. 23)[19], especially at the mean strength (50% failure probability). Moreover, the predicted standard deviation of 1.1 MPa con ares favorably with the experimental distribution standard deviation of 1.6 MPa, indicating the predicted normal distribution has approximately the correct shape. 

In this example, the likelihood function is the distribution on the average of a random sample of log-transformed tissue residue concentrations. One could assume that this likelihood function is normal, with standard deviation equal to the standard deviation of the log-transformed concentrations divided by the square root of the sample size. The likelihood function assumes that a given average log-tissue residue prediction is the true site-specific mean. The mathematical form of this likelihood function is... 

Not-Roundness. The size normalized radial standard deviation is illustrated in the NR template shown in Figure 4. Thus, no matter how the profile deviates from a circle the NR value will only indicate the statistical property of the radial distribution. There is no information in this term concerning the sequence of radial terms. 

Three conditions must be satisfied before an observation can be considered an error. First, it should be known what the expected value distribution is for that observation. Second, if the distribution is normal, the standard deviation should be known. Third, it should be decided what is the maximally allowed chance that an observation which is called an error is actually correct. Just as a reminder, in Table 1 we give the relation between the Z-score (this is the number of standard deviations that an observation varies from the expected value) and the chance that such an observation is not an error, assuming a normal distribution. 

The MAPPER procedure [39] has been suggested to alleviate this problem. This procedure is based upon the average chemical shifts of each amino acid type and its standard deviation. It ranks possible assignments according to their probabilities based on their chemical shift statistics. However, the NMR investigation of protein kinases is made easier if the X-ray structure of the particular kinase is available. For a protein with known structure it is possible to predict chemical shifts quite accurately [40]. The chemical shift matching procedure uses these chemical shift values instead of a statistical input of normally distributed standard resonances. 

Arithmetic mean for normai distributions, geometric mean for iog-normal distributions, and the mode for trianguiar distributions. Standard deviation for normai distributions, geometric standard deviation for Iog-normai distributions, and minimum and maximum for trianguiar and uniform distributions. 

In quantifying the goodness of a mix the standard deviation is often employed. This is generally sufficient for a powder that is normally distributed in composition. Imperfect mixing gives distributions that are far from the normal. The standard deviation can be used to classify the mix, but care must be taken when the proportion of samples is outside a specified range where nonnormal composition distributions exist. 

Assuming that the distribution is roughly normal, the standard deviation (SD) is 573, which is characteristic of the flat, wide spread of data. Figure 1.9 shows the distribution of S-Afor four similar parts from the same supplier butto a different vehicle assembler, EastCo. This time, the SD for the distribution is 95, representing a much narrower spread of differences than for WestCo. 

Precision, as defined by Morrison and Skogerboe (1965), is the measure of the reproducibility of replicate analyses. Because reproducibility determined in this manner is considered to be random in nature and hence a normal distribution, the standard deviation is used as a measure of the magnitude of precision. To normalize the standard deviation to make comparisons between different magnitudes of measurements, the relative standard deviation is used. The relative standard deviation is expressed on a percentage basis and is called the percent relative standard deviation (% RSD). It is computed as follows ... 

Pore radius corresponding to the maximum of the log-normal pore size distribution, standard deviation in the pore radius. 

The first application of the Gaussian distribution is in medical decision making or diagnosis. We wish to determine whether a patient is at risk because of the high cholesterol content of his blood. We need several pieces of input information an expected or normal blood cholesterol, the standard deviation associated with the normal blood cholesterol count, and the blood cholesterol count of the patient. When we apply our analysis, we shall anive at a diagnosis, either yes or no, the patient is at risk or is not at risk. 

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. 

Table 2.26a lists the height of an ordinate (Y) as a distance z from the mean, and Table 2.26b the area under the normal curve at a distance z from the mean, expressed as fractions of the total area, 1.000. Returning to Fig. 2.10, we note that 68.27% of the area of the normal distribution curve lies within 1 standard deviation of the center or mean value. Therefore, 31.73% lies outside those limits and 15.86% on each side. Ninety-five percent (actually 95.43%) of the area lies within 2 standard deviations, and 99.73% lies within 3 standard deviations of the mean. Often the last two areas are stated slightly different viz. 95% of the area lies within 1.96cr (approximately 2cr) and 99% lies within approximately 2.5cr. The mean falls at exactly the 50% point for symmetric normal distributions. 

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. 

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. ... 

From Example A6.2 we know that after 100 steps of the countercurrent extraction, solute A is normally distributed about tube 90 with a standard deviation of 3. To determine the fraction of solute in tubes 85-99, we use the single-sided normal distribution in Appendix lA to determine the fraction of solute in tubes 0-84 and in tube 100. The fraction of solute A in tube 100 is determined by calculating the deviation z (see Chapter 4)... 

P(x, t) dx has the familiar bell shape of a normal distribution function [Eq. (1.39)], the width of which is measured by the standard deviation o. In Eq. (9.83), t takes the place of o. It makes sense that the distribution of matter depends in this way on time, with the width increasing with t. 

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). 

Example 3. A centrifugal pump moving a corrosive Hquid is known to have a time-to-failure that is well approximated by a normal distribution with a mean of 1400 h and a standard deviation of 120 h. A particular pump has been in operation for 1080 h. In order to plan maintenance activities the chances of the pump surviving the next 48 h must be deterrnined. 

The population from which the obsei vations were obtained is normally distributed with an unknown mean [L and standard deviation... 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

Both func tions are tabulated in mathematical handbooks (Ref. I). The function P gives the goodness of fit. Call %q the value of at the minimum. Then P > O.I represents a believable fit if ( > 0.001, it might be an acceptable fit smaller values of Q indicate the model may be in error (or the <7 are really larger.) A typical value of for a moderately good fit is X" - V. Asymptotic Iy for large V, the statistic X becomes normally distributed with a mean V ana a standard deviation V( (Ref. 231). 

The ciimnlative prohahility of a normally distributed variable lying within 4 standard deviations of the mean is 0.49997. Therefore, it is more than 99.99 percent (0.49997/0.50000) certain that a random value will he within 4<3 from the mean. For practical purposes, <3 may he taken as one-eighth of the range of certainty, and the standard deviation can he obtained ... 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

Figure 4.6 Shape of the Cumulative Distribution Function (CDF) for an arbitrary normal distribution with varying standard deviation (adapted from Carter, 1986)...

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