Normal distribution cumulative function

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

The probability density of the normal distribution f x) is not very useful in error analysis. It is better to use the integral of the probability density, which is the cumulative distribution function... 

A table of cumulative probabilities (CP) lists an area of 0.975002 for z -1.96, that is 0.025 (2.5%) of the total area under the curve is found between +1.96 standard deviations and +°°. Because of the symmetry of the normal distribution function, the same applies for negative z-values. Together p = 2 0.025 = 0.05 of the area, read probability of observation, is outside the 95% confidence limits (outside the 95% confidence interval of -1.96 Sx. .. + 1.96 Sx). The answer to the preceding questions is thus... 

Figure 3.8. The transformation of a rectangular into a normal distribution. The rectangle at the lower left shows the probability density (idealized observed frequency of events) for a random generator versus x in the range 0 < jc < 1. The curve at the upper left is the cumulative probability CP versus deviation z function introduced in Section 1.2.1. At right, a normal distribution probability density PD is shown. The dotted line marked with an open square indicates the transformation for a random number smaller or equal to 0.5, the dot-dashed line starting from the filled square is for a random number larger than 0.5.

The normal distribution, A Y/l, o 2), has a mean (expectation) fi and a standard deviation cr (variance tr2). Figure 1.8 (left) shows the probability density function of the normal distribution N(pb, tr2), and Figure 1.8 (right) the cumulative distribution function with the typical S-shape. A special case is the standard normal distribution, N(0, 1), with p =0 and standard deviation tr = 1. The normal distribution plays an important role in statistical testing. 

FIGURE 1.8 Probability density function (PDF) (left) and cumulative distribution function (right) of the normal distribution cr2) with mean /a and standard deviation cr. The quantile q defines a probability p. 

Parameter Two distinct definitions for parameter are used. In the first usage (preferred), parameter refers to the constants characterizing the probability density function or cumulative distribution function of a random variable. For example, if the random variable W is known to be normally distributed with mean p and standard deviation o, the constants p and o are called parameters. In the second usage, parameter can be a constant or an independent variable in a mathematical equation or model. For example, in the equation Z = X + 2Y, the independent variables (X, Y) and the constant (2) are all parameters. 

The calculation for the Rankit plot is shown in spreadsheet 3.3. The effects are ordered from most negative to most positive. A column of the rank of each effect is then created, with ties (none here) taking the higher rank (e.g., 1, 2, 4, 4, 5). The column headed z is the point on the cumulative normal distribution of the rank/(A + 1), where N is the number of experiments. The z score is calculated by the function =N0RMSINV (z). When this is plotted against the effect (see figure 3.16), it is clear that copper and lead do, indeed, appear to be off the line, and all the other effects are concluded to be insignificant. 

Suppose a polydisperse system is investigated experimentally by measuring the number of particles in a set of different classes of diameter or molecular weight. Suppose further that these data are believed to follow a normal distribution function. To test this hypothesis rigorously, the chi-squared test from statistics should be applied. A simple graphical examination of the hypothesis can be conducted by plotting the cumulative distribution data on probability paper as a rapid, preliminary way to evaluate whether the data conform to the requirements of the normal distribution. 

FIG. C.1 A normal, or Gaussian, distribution (a) represented as a frequency function (b) represented as a cumulative function and (c) represented as a cumulative function linearized by plotting on probability paper. 

To compute the integral in (9), we use the fact that, for fixed a%2, a x, the quantity Pidki - 9k2 > (o 2 aki)/ )is ttie cumulative distribution function of a standard normal distribution evaluated at... 

These numbers can be obtained using simple functions, e.g. in a spreadsheet, but are often conventionally presented in tabular form. There are a surprisingly large number of types of tables, but Table A.l allows the reader to calculate relevant information. This table is of the cumulative normal distribution, and represents the area to the left of the curve for a specified number of standard deviations from the mean. The number of standard deviations equals the sum of the left-hand column and the top row, so, for example, the area for 1.17 standard deviations equals 0.87900. 

Cumulative distributions can be fitted by a linear function if the data fit a suitable mathematical fimction. This curve fitting gives no insight into the fundamental physics by which the particle size distribution was produced. Three common functions are used to linearize the cumulative distribution the normal distribution fimction, the log-normal distribution function, and the Rosin—Rammler distribution function. By far athe most commonly used is the log-normal distribution function. 

Tables of values of the normal distribution function, f(Z), and the cumulative distribution function, F(Z are given in the appendix of...

The normal probability function table given in the appendix d this book can also be used for values of the log-normal distribution function, f, and the log-normal cumulative distribution function, F. In these tables Z = [ln(d/cy/(In o- )] is used. A plot of the cumulative log-normal distribution is linear on log-normal probability paper, like that shown in Figure 2.11. A size distribution that fits the log-normal distribution equation can be represented by two numbers, the geometric mean size, dg, and the geometric standard deviation,. The geometric mean size is the size at 50% of the distribution, d. The geometric standard deviation is easily obtained finm the following ratios ... 

Fig. 34. Cumulative probabilities for Ae breakdown voltage as a function of D for normal distributions in the diflfusiv-ity D ctd = 0.75. , Data for Fel7Cr in 3.5% NaCl solution at 30 °C. Vc = -0.046 V (SCE). After Shibata (see [65]). Reproduced from J. Electro-chem. Soc. 139, 3434 (1992) by permission of the Electrochemical Society.

Fig. 35. Differential cumulative probabilities for the induction time as a function of D for normal distributions in...

Fig. 6.3 Standardized cumulative volume distribution g3(dp/d5o) as a function of in the form of logarithmic normal distribution for the test conditions given in Fig. 6.2 from [166]... 

The mean and standard deviation of the normal distribution are T and a, respectively. Since the normal distribution is designed for continuous data, the cumulative distribution function is more practical than the probability density function. For a particular data population, the cumulative distribution [2] is as follows ... 

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. 

Cumulative Probability, a, of tbe Standard Normal Distribution as a Function of tbe Standard Variable z. The left column indicates z to the tenths digit and the top row indicates z to the hundredths. For example, the a value for z = 1.38 is 0.91621. 

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]. 

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. 

Figure A.3 Representative probability density functions (top) and cumulative distribution functions (bottom) for the normal distribution.

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