Area under curve normal distribution

Table 2.26b Areas Under the Normal Distribution Curve from 0 to z 2.122...

Relationship between confidence intervals and results of a significance test, (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on fexp. The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a,v). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. 

For example, the proportion of the area under a normal distribution curve that lies to the right of a deviation of 0.04 is 0.4840, or 48.40%. The area to the left of the deviation is given as 1 - P. Thus, 51.60% of the area under the normal distribution curve lies to the left of a deviation of 0.04. When the deviation is negative, the values in the table give the proportion of the area under the normal distribution curve that lies to the left of z therefore, 48.40% of the area lies to the left, and 51.60% of the area lies to the right of a deviation of -0.04. 

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... 

The statistics of the normal distribution can be applied to give more information about random-walk diffusion. 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 = + v/(2/V) 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 2 J, that is, 2V(2Dr t) is equal to about 5%. 

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

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). 

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. 

Instead of calculating f z), most people look at the area under die normal distribution curve. This is proportional to the probability that a measurement is between certain limits. For example die probability that a measurement is between one and two... 

The 5.15 value used to estimate sigma for each of the variation components originates from the AIAG calculations for KUK2, Ky, which uses 99.0% of the area under the normal distribution curve (5.15a). Note also that the R R and estimated PV are not additive to 100% of total variation. Using X and s for this product, estimated total variation is equal to 0.48(6 x 0.081a), which compares well with the gage R R value of 0.50. 

These areas under the normal distribution curve can be given probability interpretations. For example, if an experiment yields a nearly normal distribution... 

The sample standard deviation(s) provides an empirical measure of uncertainty (Le., expected spread) and is frequently used for that purpose. If a distribution is Gaussian 68.3% of the values will fall between 1 standard deviation ( ls), 95.4% within 2s, and 99.7% within 3s from the mean. This concept is shown in Kgure 2.7. This spread provides a range of measurements as well as a probability of occurrence. Most often, the uncertainty is cited as 2 standard deviations, since approximately 95% of the area under the normal distribution curve is contained within these bormdaries. Sometimes 3 standard deviations are used, to account for more than 99% of the lrea under the curve. Thus, if the... 

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. 

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). 

Note that the normal distribution curve has a mathematical equation and integrating the equation of this curve, for example between p — 2a and p + 2a, irrespective of the values of p and a, will always give the answer 0.954. So 95.4 per cent of the area under the normal curve is contained between p — 2a and p + 2a and it is this area calculation that also tells us that 95.4 per cent of the individuals within the population will have data values in that range. 

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

The normal distribution is commonly encountered in the cumulative form, that is, as the fraction of particles larger (oversized) or smaller (undersized) than a particular tt value. Since the total area under the normal curve equals unity, the area under one tail of the curve from t, to oo gives the fraction of the population having t values greater than the integration limit t . 

The area under the Normal curve is of considerable interest in Statistics. That is, it is of considerable interest to define and quantify the area bounded by the Normal curve at the top and the x-axis at the bottom. This area will be defined as 1.0, or as 100%. Given this interest, the final point in Section 6.6 raised an issue that appears problematic. That is, it appears that, if the two lower slopes of the Normal curve never quite reach the x-axis, the area under the curve is never actually fully defined and can therefore never be calculated precisely. Fortunately, this apparent paradox can be solved mathematically. In the Preface of this book I noted that, in several cases, I had resisted the temptation to provide an explanation of subtle points. This case, I believe, is a worthwhile exception. An understanding of the qualities of the Normal distribution and the Normal curve is extremely helpful in setting the scene for topics covered in Chapters 7 and 8, namely statistical significance and clinical significance. 

Of particular interest in Statistics is that the means of many large samples taken from a particular population are approximately distributed in this Normal fashion, i.e., they are said to be Normally distributed. This is true even when the population data themselves are not Normally distributed. The mathematical properties of a true Normal distribution allow quantitative statements of the area under the curve between any two points on the x-axis. In Section 6.6.1 it was shown that the total area under the Normal curve is 1, or 100%. It is also of interest to know the proportion of the total area under the curve that lies between two points that are equidistant from the mean. These points are typically represented by multiples of the SD. From the properties of the mathematical equation that governs the shape of the Normal curve, it can be shown that ... 

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. 

Having introduced the normal distribution and discussed its basic properties, we can move on to the common statistical tests for comparing sets of data. These methods and the calculations performed are referred to as significance tests. An important feature and use of the normal distribution function is that it enables areas under the curve, within any specified range, to be accurately calculated. The function in Equation (1) is integrated numerically and the results presented in statistical tables as areas under the normal curve. From these tables, approximately 68% of observations can be expected to lie in the region bounded by one standard deviation from the mean, 95% within jjl 2o, and more than 99% within x 3a. 

This function corresponds to the area under the normal error distribution curve from its maximum value to a value z. The error function has a value of 0 when z is zero, and a value of one when z is infinity. The error function complement is simply defined as... 

Determine gt, which is the specification expressed in multiples of the standard deviation a, by using standard tables of the normal distribution. The area under the normal curve between a — 5 and a + 5 must equal the fraction of spot samples specified in step 1. This value, which equals e, is just another way of looking at the probability of a spot sample composition x lying within the range between a — 8 and a + 8. Thus, Z is the normalized standard deviation corresponding to this area. 

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