Normal distributions

Normal distributions are widely used in modelling repair activities. The failure density function of a normal distribution is  

An application of this type of distribution can be seen in Chapter 8. 

FIGURE 2.9 Normal distribution histogram % mass versus size. 

FIGURE 2.10 linearized normal distritiution Cumulative % less than versus size. 

A size distribution that fits the normal distribution equation can be represented by two parameters, the arithmetic mean size, x, and the standard deviation, cr. The mean size, x, is the size at 50% of the distribution, also written as jcgo. The standard deviation is easily obtained from the cumulative distribution as 

The normal distribution has the disadvantage that finite fractions of the distribution occur at sizes less than zero, which is physically unrealistic. 

The output of most production operations will normally be distributed with a variability of 3cr, or a total of 6cr. A product is normally made with a tolerance of say T, for example x T, which is a total tolerance of IT. This can be expressed as a capability index, Cp  

This is one of the most widely known statistical distributions, sometimes called the Gaussian distribution after Carl Friedrich Gauss (1777-1855), a German mathematician. The probability density function of the distribution is defined by 

Using Equations (2.23) and (2.35), we obtain the following cumulative distribution function  

In the two extreme sectors beyond p 4 A (the tails of the distribution) it falls only 0.26 % of the entire population of experimental data having a true mean given by p. This actually means that just one over 10,000 samples will yield a value below p — 4-A or above p + 4 A. The standard error of the mean (SEM) is the standard deviation of the sample mean estimate of a population mean. SEM is 

The above formula assumes that the sample size n is much smaller than the total population size N, so that population can be considered infinite in size. When the sampling fraction is large (approximately 10 % or more) the estimate of the error must be corrected by multiplying by a finite population correction (FPC) [2] 

P x) represents the probability that in a population of data there is one having a value lower or equal to a given x. In Eq. 4.10 erf is the so called error function defined as 

Integral (4.11) cannot be expressed in closed form, but through a Taylor series expansion 

A distribution in which the logarithms of a variable x follow the normal distribution is known as a log-normal distribution. However, if also the log-normal distribution is not perfectly symmetrical any statistical analysis will be necessarily affected be an error and will not be correct. Unfortunately, the fact that generally in asymmetric distributions the left side is less expanded than the right one results in non-conservative error using the Gaussian distribution since it would assign a probability of survival, though small, to values that in reality have no chance to survive at all. 

The differential logarithmic normal distribution has the same mathematical form as the Gaussian distribution with the small difference that the logarithm of the property occurs in place of the property itself  

the curve is symmetric about nXM- The median of the curve Xm is not identical with the number average (see below). The function corresponds to 

Differential logarithmic normal distributions can be generalized, for example, for the mass distribution of the degree of polymerization  

A molar logarithmic normal distribution as described by Equation (8-25) 

8 Molar Masses and Molar Mass Distributions 

Similar probability models can be used for continuous random variables. The most common, and arguably the most important of these in Statistics, is the normal distribution. As it is encountered so frequently in this book, we spend some time describing its characteristics and uses. 

More precisely, it is one specific kind of symmetrical curve. The precise nature of this curve can be described mathematically by a formula that contains both the mean, p, and the standard deviation, a, of the population that is being represented graphically by the normal curve  

The term population is defined in detail later in the chapter. Until then we can think of a population as the largest group of experimental units (for example, study participants) about which we would like to make a conclusion. 

The precise mathematics of the normal distribution allows quantitative statements of the area under the curve between any two points on the X axis. Of most interest here is the area under the curve between two points that are equidistant from the mean. 

Since there are numerous ways for synthesizing polymers, a molecular weight distribution does not always follow a one-parameter equation. Even the condensation polymers do not necessarily follow the most probable distribution, nor do addition polymers follow the Poisson distribution. In many cases we naturally consider classical statistics, the normal distribution. 

The number average degree of polymerization is identical with the mean  

The height of the curve at some value of x is denoted by f(x) while p, and o are characteristic parameters of the function. The curve is synunetric about p., the mean or average value, and the spread about this value is given by the variance, o, or standard deviation, a. It is common for the curve to be standardized so that the area enclosed is equal to unity, in which case f x) provides the probability of observing a value within a specified range of x values. With 

Equation (1) describes the idealized distribution function, obtained from an infinite number of sample measurements, the so-called parent population distribution. In practice we are limited to some finite number, n, of samples taken from the population being examined and the statistics, or estimates, of mean, variance, and standard deviation are denoted then by x, and s respectively. The mathematical definitions for these parameters are given by equations (2H4), 

A simple example serves to illustrate the use of these statistics in reducing data to key statistical values. Table 1 gives one day s typical laboratory results for 40 mineral water samples analysed for sodium content by flame photometry. In analytical science it is common practice for such a list of replicated analyses to be reduced to these descriptive statistics. Despite their widespread use and analysts familiarity with these elementary statistics care must be taken with 

Standard deviation from the mean, about one in twenty will be more than two standard deviations from the mean, and less than one in 300 will be more than 3cr from p. 

This result is known as the central limit theorem and serves to emphasize the importance and applicability of the normal distribution function in statistical data analysis since non-normal data can be normalized and can be subject to basic statistical analysis.  

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If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

Many companies choose to represent a continuous distribution with discrete values using the p90, p50, plO values. The discrete probabilities which are then attached to these values are then 25%, 50%, 25%, for a normal distribution. 

Therefore an automatic method, which means an objective and reproducible process, is necessary to determine the threshold value. The results of this investigations show that the threshold value can be determined reproducible in the point of intersection of two normal distributed frequency approximations. 

Fig. 1.12 Three normal distributions with different values of a (Equation (1.55)). The functions are normalised, so the area under each curve is the same.

In Figure 1.12 we show three normal distributions that all have zero mean but different values of the variance (cr ). A variance larger than 1 (small a) gives a flatter fimction and a variance less than 1 (larger a) gives a sharper function. 

One option is to first generate two random numbers and 2 between 0 and 1. T1 corresponding two numbers from the normal distribution are then calculated using... 

These two methods generate random numbers in the normal distribution with zero me< and unit variance. A number (x) generated from this distribution can be related to i counterpart (x ) from another Gaussian distribution with mean (x ) and variance cr using... 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

Table 2.26a Ordinates (V) of the Normal Distribution Curve at Values of z 2.121...

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

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

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. 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

Understanding the distribution allows us to calculate the expected values of random variables that are normally and independently distributed. In least squares multiple regression, or in calibration work in general, there is a basic assumption that the error in the response variable is random and normally distributed, with a variance that follows a ) distribution. 

The only two distributions we shall consider are the Gaussian distribution ( normal law ) and the log-normal distribution. 

To obtain the expression for the log-normal distribution it is only necessary to substitute for I and a in Equation (1.52) the logarithms of these quantities. One thus obtains... 

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

Normal Distribution The binomial distribution describes a population whose members have only certain, discrete values. This is the case with the number of atoms in a molecule, which must be an integer number no greater then the number of carbon atoms in the molecule. A molecule, for example, cannot have 2.5 atoms of Other populations are considered continuous, in that members of the population may take on any value. 

The shape of a normal distribution is determined by two parameters, the first of which is the population s central, or true mean value, p, given as... 

The amount of aspirin in the analgesic tablets from a particular manufacturer is known to follow a normal distribution, with p, = 250 mg and = 25. In a random sampling of tablets from the production line, what percentage are expected to contain between 243 and 262 mg of aspirin ... 

Earlier we noted that 68.26% of a normally distributed population is found within the range of p, lo. Stating this another way, there is a 68.26% probability that a member selected at random from a normally distributed population will have a value in the interval of p, lo. In general, we can write... 

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