Excel, normal distribution

The F distribution, similar to the chi square, is sensitive to the basic assumption that sample values were selected randomly from a normal distribution. The Microsoft Excel function FDIST(X, dfh df2) gives the upper percent points of Table 3-8, where X is the tabular value. The function FINV(Percent, dfh df2) gives the table value. 

Fig. 1.3 shows the normal distributions of the errors of the cc-pV(X -1,X)Z extrapolations (note that the scale is different from Fig. 1.2). In comparison with the cc-pV6Z results, which are included in Fig. 13 as a broad distribution on its scale, the agreement between the R12 energies and the extrapolated energies is excellent, confirming our earlier conclusions from the discussion of the He atom.

In Figure 8, a plot of dN137/dN95 vs. D is shown together with experimental values of the ratio for indicated particle size fractions. The excellence of the fit shows that the data analysis is consistent, although it cannot be used to establish the uniqueness of the normal distribution function in describing the data. 

The remaining chapters of the book introduce some of the advanced topics of chemometrics. The coverage is fairly comprehensive, in that these chapters cover some of the most important advanced topics. Chapter 6 presents the concept of robust multivariate methods. Robust methods are insensitive to the presence of outliers. Most of the methods described in Chapter 6 can tolerate data sets contaminated with up to 50% outliers without detrimental effects. Descriptions of algorithms and examples are provided for robust estimators of the multivariate normal distribution, robust PC A, and robust multivariate calibration, including robust PLS. As such, Chapter 6 provides an excellent follow-up to Chapters 3, 4, and 5. 

Although the standard deviation gives a measure of the spread of a set of results about the mean value, it does not indicate the way in which the results are distributed. To understand this, a large number of results are needed to characterize the distribution. Rather than think in terms of a few data points (for example, six data points) we need to consider, say 500 data points, so the mean, x, is an excellent estimate of the true mean or population mean, jd. The spread of a large number of collected data points will be affected by the random errors in the measurement (i.e., the sampling error and the measurement error) and this will cause the data to follow the normal distribution. This distribution is shown in Equation 2.8 ... 

Similar well fitting simulation curves for the experimental stress-strain data as those shown in Fig. 46b can also be obtained for higher filler concentrations and silica instead of carbon black. In most cases, the log-normal distribution Eq. (55) gives a better prediction for the first stretching cycle of the virgin samples than the distribution function Eq. (37). Nevertheless, adaptations of stress-strain curves of the pre-strained samples are excellent for both types of cluster size distributions, similar to Fig. 45c and Fig. 46b. The obtained material parameters of four variously filled S-SBR composites used for testing the model are summarized in Table 4, whereby both cluster... 

Then work out how many standard deviations corresponding to the area under the normal curve calculated in step 3, using normal distribution tables or standard functions in most data analysis packages. For example, a probability of 0.9286 (coefficient b2) falls at 1.465 standard deviations. See Table A.l in which a 1.46 standard deviations correspond to a probability of 0.927 85 or use the NORMINV function in Excel. 

Excluding the intercept term, diere are seven coefficients. A normal probability plot can be obtained as follows. First, rank die seven coefficients in order. Then, for each coefficient of rank p calculate a probability (p — 0.5)/7. Convert diese probabilities into expected proportions of die normal distribution for a reading of appropriate rank using an appropriate function in Excel. Plot the values of each of die seven effects (horizontal axis) against die expected proportion of normal distribution for a reading of given rank. 

Lainhart et al. (2006) pooled consistently recorded data on head circumferences from 338 well-documented individual with autism spectrum disorder, 2 9 years of age, from 10 centers in the NIH Collaborative Program of Excellence in Autism. In the entire group 17% were macrocephalic, with 12-20% macrocephalic by 3-5 years, a rate that then remained stable. They found that head circumference of the autistic individuals showed a normally distributed curve, with a shift to the right, suggesting that an increase in head circumference may be present in all autistic individuals. They noted that three other studies had not found a normally distributed head circumference (Lainhart et al., 1997 Fombonne et al., 1999 Miles et al., 2000). These three studies, however, included a smaller number of subjects, had more individuals with mental retardation, and included subjects that did not have idiopathic autism. 

Fig. 6 Example of useful visualizations of an HTS campaign on 365,000 compounds of the NIH molecular libraries small molecule repository in an example enzyme assay. The Z-factor for this HTS campaign over all the normalized controls was Z 0.82 and the robust Z-factor for the screened compounds was degraded only slightly to Z -0.8, and the signal-to-background of -24.2 and signal-to-noise of -143.3 indicated an excellent assay and HTS campaign. The (a) frequency distribution of the activities, the (b) 2-D scattergram of activities as a function of plate order, and (c) a 3-D view of the plates and wells against their activities, all indicate a robust close to normally distributed screen, with no serious plate-based artifacts...

Consider the first sample, which has a log-normal distribution with median = 5.83 pm and standard deviation = 0.49 pm. To visualize this distribution, conduct a simulation by generating 10,000 observations from a log-normal distribution using the parameter estimates given above. The histogram from the simulation, Fig. 6, shows the skewness inherent in the log-normal distribution. There is excellent agreement between the actual data and the simulation see Table 6. The same approach may be used for data that is normally distributed. 

Making this substitution into Equation (3.6) or (3.7) reduces the generic normal distribution to one with mean 0 and standard deviation 1, collapsing all possible normal distributions onto a standard curve. Tabulated values of the cumulative distribution function F are usually presented in terms of the transformation variable z. Sample values of F(z) are presented in Table 3.2. Microsoft Excel contains an intrinsic function, NORMSDIST, that produces the cumulative probability for a standard normal variable z given as its argument. A companion function, NORMSINV, outputs the z value for a given F(z). The Microsoft Excel manual or the electronic help files [5, 6] provide command syntax and usage examples. 

The chi-square distribution is used to perform statistical tests on the sample variance. It is highly asymmetric for small values of n, but becomes more symmetric and similar to a normal distribution as n becomes large, such as 20 or 30. The cumulative distribution function of the chi-square distribution is listed in Table 3.4 as a function of v and a, where v = - 1 is the number of degrees of freedom and a is the percentage of the distribution above the particular Microsoft Excel has built-in functions, CHIDIST and CHIINV, that compute a chi-square distribution [5, 6]. 

The log transformation of dose or concentration is easily done with a pocket calculator. Using the formula for the inverse normal distribution in the data sheet Excel, one can easily do the calculation of the probit values. The mean or median is set to 5 and the standard deviation to 1, i.e., the formula will look like this ... 

An unnecessary complication, which was possibly once introduced to make life easier, is the distinction between one- and two-tailed Student /-values (tails are also used in other statistics). Two-tailed probabilities are spread over the two ends of the distribution with half the given probability in each tail, and are denoted by putting a double prime (") after the probability value. One-tailed probabilities are shown as a single prime ( ) and refer to just one tail of the distribution. For example, for a 95% confidence interval and 10 degrees of freedom, 0.025, 10 is equal to o.o5",io> as can be seen from figure 2.9. Annoyingly, in Excel the z values obtained from the normal distribution are always one tailed (=—NORMSINV(p)1) but the Student /-values... 

In days gone by this was achieved using probability paper, specially ruled graph paper which took care of the normal pdf. Nowadays, spreadsheets have functions to perform this calculation in Excel it is NORMSINV(x), where x is the normalized cumulative frequency. If the data are normally distributed this graph should be linear. Obvious outliers are seen as points at the extremes of the x-axis, that is, at values much greater than would be expected. Example 3.1 shows how to determine whether data are normally distributed using a Rankit plot in Excel. 

In Figure 4 we include the predicted PSD for y=0-5, as well as the best fit corresponding to a log-normal distribution. The correlation is excellent, and shows that our solution is indeed similar to experimental distributions. 

An aptitude test is designed to assess students capability for a particular subject and the top 5% students are defined as having excellent command. From previous experience, it is known that the distribution of the scores, 9, of all the excellent students follows a log-normal distribution ... 

In Equation (19.9), z represents the number of standard deviations from the mean. The mathematical fimction that describes a normal-distribution curve or a standard normal curve is rather complicated and may be beyond the level of your current understanding. Most of you will learn about k later in your statistics or engineering classes. For now, using Excel, we have generated a table that shows the areas under portions of the standard normal-distribution curve, shown in Table 19.11. At this stage of your education, it is important for you to know how to use the table and solve some problems. A more detailed explanation will be provided in your future classes. We will next demonstrate how to use Table 19.11, usii a number of example problems. 

ACLE II Areas Under the Standard Normal Curve-the Values were Generated using the Standard Normal Distribution Function of Excel (coi inued) . t. ... 

Correlation gives a quantitative measure of the relationship between two variables - the amount of variance from the common area between them. For data that are normally distributed, the Pearson product-moment correlation coefficient can be calculated by many commercial analysis packages (e.g. SAS, SPSS, MS Excel). The degree of correlation is indicated by a number between—1 and 1. A correlationofO indicates complete independence between the variables, and a correlation of 1 indicates a perfect increasing linear relationship. 

Figure 2 shows the outcome of a Microsoft Excel simulation of 100 values of the nitrate ion concentration in a single water sample (ppm). It is most unlikely that any material would actually be analyzed 100 times, but the data are used to demonstrate methods by which such results are presented. Each result is given to two places of decimals, all the results lying in the range 0.43-0.57ppm. The Ere-quency column in the spreadsheet shows the number of times that each value 0.43, 0.44, 0.45,. .., 0.57 occurs, and this column along with its neighbor headed Value comprises a frequency table. The data in this table can be plotted as a bar chart, as shown in Figure 2. The shape of this bar chart is approximately the same as the smooth ideal curve for the normal distribution shown in Figure lA. An additional column shows the cumulative frequency values, and when these are plotted as a bar chart the curve obtained looks similar to that in Figure IB. The bar chart, frequency table, and cumulative frequencies are just three of several ways in which replicate experimental data can be presented.

For this purpose, you can simulate symmetrical peaks on the normal distribution function using Excel. Via the setting and adaptation of peak parameters, the simulation of the individual peak areas were adapted to the real chromatogram. 

From this report with overlaid peaks, a simulation is created using Excel (using the normal distribution function). This gives us a simulation, which is identical to the one published by A.W. Westerberg - at the same scale (see Figure 7.3). 

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