Normal distribution standard

For convenience, the normal distribution can be transformed to a standard normal distribution where the mean is zero and the standard deviation equals 1. The transformation is achieved using Equation 3.5  

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Table 1 Area under the cumulative Standard Normal Distribution (SND)...

A value of Cp = 1.33 would indicate that the distribution of the product characteristics covers 75% of the tolerance. This would be sufficient to assume that the process is capable of producing an adequate proportion to specification. The numbers of failures falling out of specification for various values of Cp and Cp can be determined from Standard Normal Distribution (SND) theory (see an example later for how to determine the failure in parts-per-million or ppm). For example, at Cp = 1.33, the expected number of failures is 64 ppm in total. 

It is evident that an approximate — 1.5cr shift ean be determined from the data and so the Cpi value is more suitable as a model. Using the graph on Figure 6, whieh shows the relationship Cp, (at 1.5cr shift) and parts-per-million (ppm) failure at the nearest limit, the likely annual failure rate of the produet ean be ealeulated. The figure has been eonstrueted using the Standard Normal Distribution (SND) for various limits. The number of eomponents that would fall out of toleranee at the nearest limit, is potentially 30 000 ppm at = 0.62, that is, 750 eomponents of the 25 000 manufaetured per annum. Of eourse, aetion in the form of a proeess eap-ability study would prevent further out of toleranee eomponents from being produeed and avoid this failure rate in the future and a target Cp = 1.33 would be aimed for. 

The standard normal distribution is determined by calculating a random variable 7. where... 

The probability function for the standard normal distribution is then... 

The t (Student s t) distribution is an unbounded distribution where the mean is zero and the variance is v/(v - 2), v being the scale parameter (also called degrees of freedom ). As v -> < , the variance —> 1 (standard normal distribution). A t table such as Table 1-19 is used to find values of the t statistic where... 

For np > 5 and n( 1 - p) > 5, an approximation of binomial probabilities is given by the standard normal distribution where z is a standard normal deviate and... 

A conceptual definition of the follows from consideration of a set of numbers drawn at random from the standard normal distribution, the one with mean zero and standard deviation one. Ordering this set of numbers gives a sequence called order statistics. The are the... 

Fig. 4.3. Schematic frequency distribution of measured values y (a), GAUSsian normal distributions of measured values y (b) as well as of analytical values x(c), and standard normal distribution (d)...

Frequently, the measurement error distributions arising in a practical data set deviate from the assumed Gaussian model, and they are often characterized by heavier tails (due to the presence of outliers). A typical heavy-tailed noise record is given in Fig. 7, while Fig. 8 shows the QQ-plots of this record, based on the hypothesized standard normal distribution. 

A widely used a = 5 percent significance level produces a 95 percent confidence interval extending over t91 confidence interval for a standard normal distribution. Therefore, the normal approximation of the t-distribution is correct to 12 percent for m> 10 and to 4 percent for m> 30. 

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. 

Data values x following a normal distribution N(p, a2) can be transformed to a standard normal distribution by the so-called z-transformation... 

Figure 1.8 explains graphically how probabilities and quantiles are defined for a normal distribution. For instance the 1 %-percentile (p = 0.01) of the standard normal distribution is —2.326, and the 99%-percentile (p 0.99) is 2.326 both together define a 98% interval. 

FIGURE 1.9 -Distributions with 3 and 20 DF, respectively, and standard normal distribution corresponding to a -distribution with DF = oo (left). Chi-square distribution with 3, 10, and 30 DF, respectively (right). 

FIGURE 2.4 Probability density function of the uniform distribution (left), and the logit-transformed values as solid line and the standard normal distribution as dashed line (right). 

Using a standard normal distribution table, Z = Zoos = 1.96 Zp = Zo.i = 1.65. Substituting into the equation, we have... 

Figure 21.3 The standard normal distribution curve. The shaded area gives the probability that the standard normal variate, Z, lies between z and infinity, i.e. P(Z>z).

Because (a) is the area in the lower tail of the normal distribution, Za is called the ath quantile of the standard normal distribution, (or the (100)(a)th percentile). A useful identity follows directly from the symmetry of the Gaussian distribution in Equation A-2 (4). 

The number 0.128 is the largest acceptable true CV p for which the net error would not exceed +25% at the 95% confidence level. The number 1.96 is the appropiate Z-statistic (from tables of the standard normal distribution) at the same confidence level. 

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