Area under curve standard normal distribution

The random number generated is the cumulative probability, and the cumulative probability is the area under the standard normal distribution curve. Since the standard normal distribution curve is symmetrical, the negative values of Z and the corresponding area are found by symmetry. For example, as described in the two previous problems,... 

Fig. 2.4. Symmetric interval about the mean, containing 95% of the total area under the standard normal distribution curve.

If T is normally distributed witli mean p and standard deviation a, then tlie random variable (T - p)/a is normally distributed with mean 0 and standard deviation 1. The term (T - p)/a is called a standard normal variable, and tlie graph of its pdf is called a "standard normal curve. Table 20.5.2 is a tabulation of areas under a standard normal cur e to tlie right of Zo of r normegative values of Zo. Probabilities about a standard normal variable Z can be detennined from tlie table. For example,... 

P(u < V < h) = the probability that v will fall between a and h (, (((/) = the cumulative area under the standard normal r-distribution curve... 

The detailed shape of a normal-distribution curve is determined by its mean and standard deviation values. For example, as shown in F jure 19.6, an e >oiment widh a small standard deviation will produce a tall, narrow curve whereas a large standard deviation will result in a short, wide curve. However, it is important to note that since die normal probability distribution represents all possible outcomes of an experiment (with the total of probabilities equal to 1), the area under any given normal distribution should alwsg be equal to 1. Also, note normal distribution is symmetrical about the mean. 

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

In Table 19.H> z = 1 represents one standard deviation above the mean and 34.13% of the total area under a standard normal curve. On the other hand, z = — 1 represents one stan dard deviation below the mean and 34.13% of the total area, as shown in Figure 19.7. Therefore, for a standard normal distribution, 68% of the data foil in the interval of z = — 1 to z = 1 (—s to s). Similarly, z = —2 and z = 2 (two standatd deviations helow and above the... 

FIGURE 9.12 Meaning of standard deviation for a normal distribution. The hatched area represents 68% of total area under curve. 

Fig. 3.8. Random sample of 10 elements taken from a standardized normal distribution. Each element represents a region whose area is equal to 1/10 of the total area under the curve.

The standard normal distribution results from the special case wherein pi = 0 and area under the curve from —< to + > is exactly 1.0. If one can develop a table of random numbers for a uniform distribution over the interval 0-1, it is possible to map a set of equivalent values for the standard normal distribution, as in Figure 10. The value along the ordinate represents the probability that the random variable X lies in the interval —< to x. For any random number we can compute the equivalent value x. This latter value is called the random normal deviate. 

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. 

The normal probability distribution may be obtained from the standard normal frequeney distribution (Fig. 14.2). The probability that a reading will fall between (t<7)i and ta)2 in Fig. 14.4 will be the area under the frequency curve between these two points. Table 14.3 gives the probability for a value to fall between x = 0 and x = ta on the standard normal distribution eurve. This corresponds to the area under the curve between these two points. 

A graphical interpretation of these expressions is shown in Fig. 21.3 where each probability corresponds to an area under the /(x) curve. Equation 21-6 and Fig. 21.3 demonstrate that if a random variable x is normally distributed, there is a very high probability (0.9973) that a measurement lies within 3a of the mean x. This important result provides the theoretical basis for widely used SPC techniques. Similar probability statements can be formulated based on statistical tables for the normal distribution. For the sake of generahty, the tables are expressed in terms of the standard normal distribution, A(0,1), and the standard normal variable, z = x — jL)/a. 

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. 

The shape of the Normal distribution is shown in Figure 3 for an arbitrary mean, /i= 150 and varying standard deviation, ct. Notice it is symmetrical about the mean and that the area under each curve is equal representing a probability of one. The equation which describes the shape of a Normal distribution is called the Probability Density Function (PDF) and is usually represented by the term f x), or the function of A , where A is the variable of interest or variate. 

If a result is quoted as having an uncertainty of 1 standard deviation, an equivalent statement would be the 68.3% confidence limits are given by Xmean 1 Sjc, the reason being that the area under a normal distribution curve between z = -1.0 to z = 1.0 is 0.683. Now, confidence limits on the 68% level are not very useful for decision making because in one-third of all... 

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... 

Determining the area under the normal curve is a very tedious procedure. However, by standardizing a random variable that is normally distributed, it is possible to relate all normally distributed random variables to one table. The standardization is defined by the identity z = (x- p)/C7, where z is called the unit normal. Further, it is possible to standardize the sampling distribution of averages x by the identity z = (x - p)/(c/Vn). 

The normal distribution is generally written as P(x) = (l/ /2TTstandard deviation of the distribution, the whole being normalized so that the area under the curve is equal to 1.0. 

The statistics of the normal distribution can now be applied to give more information about the statistics of random-walk diffusion. It is then found that the mean of the distribution is zero and the variance (the square of the standard deviation) is na2), equal to the mean-square displacement, . The standard deviation of the distribution is then the square root of the mean-square displacement, the root-mean-square displacement, + f . The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of J (the root-mean-square displacement) on either side of it, is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2f is equal to the total area under the curve minus the area under the curve up to 2f. This is found to be equal to about 5%. Some atoms will have gone further than this distance, but the probability that any one particular atom will have done so is very small. 

Figure 2.1. The standardized normal or Gaussian distribution. The shaded area as a fraction as the entire area under the curve is the probability of a result between Xj and X2.

Since the area under the curve is always unity, a narrow distribution will show larger values of f x) at the maximum, whereas a broader distribution will have a smaller value for the function at the maximum. This is why the standard deviation appears in the denominator of the preexponential normalization factor. 

The origin of this concept is that the fraction of the total area under a normal distribution curve between the 16 and 84% points is twice the standard deviation. The smaller CV, the more nearly uniform the crystal sizes. Products of DTB crystallizers, for instance, often have CVs of 30-50%. The number is useful as a measure of consistency of operation of a crystallizer. Some details are given by Mullin (1972, pp. 349, 389). 

In pharmaceutical research and drug development, noncompartmental analysis is normally the first and standard approach used to analyze pharmacokinetic data. The aim is to characterize the disposition of the drug in each individual, based on available concentration-time data. The assessment of pharmacokinetic parameters relies on a minimum set of assumptions, namely that drug elimination occurs exclusively from the sampling compartment, and that the drug follows linear pharmacokinetics that is, drug disposition is characterized by first-order processes (see Chapter 7). Calculations of pharmacokinetic parameters with this approach are usually based on statistical moments, namely the area under the concentration-time profile (area under the zero moment curve, AUC) and the area under the first moment curve (AUMC), as well as the terminal elimination rate constant (Xz) for extrapolation of AUC and AUMC beyond the measured data. Other pharmacokinetic parameters such as half-life (t1/2), clearance (CL), and volume of distribution (V) can then be derived. 

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. 

This rather complicated equation can be interpreted as follows. The function f (x) is proportional to the probability that a measurement has a value v for a normally distributed population of mean /< and standard deviation a. The function is scaled so that the area under the normal distribution curve is 1. 

Standard score, z Ordinate of normal distribution curve, p z) Area under normal curve, cumulative probability, t/= p z)dz o ... 

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