Normalized distributions

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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