Standard deviation of a normal distribution

You need not spend much time attempting to master rigorous statistical theory. Because EVOP was developed to be used by nontechnical process operators, it can be applied without any knowledge of statistics. However, be prepared to address the operators tendency to distrust decisions based on statistics. Concepts that you should understand quantitatively include the difference between a population and a sample the mean, variance, and standard deviation of a normal distribution the estimation of the standard deviation from the range standard errors sequential significance tests and variable effects and interactions for factorial designs having two and three variables. Illustrations of statistical concepts (e.g., a normal distribution) will be valuable tools. 

The traditional method of identifying a magnitude threshold has been accomplished by a variety of techniques. These include (1) the mean plus two standard deviations of a normally-distributed data set (2) arbitrarily selecting the 90 percentile or 95 percentile, etc., of the data (3) identifying the inflection point on a cumulative frequency plot that deviates from a straight line (Sinclair, 1976). 

Note that the standard deviation can be used to compute the percentile rank associated with a given data point (if the mean and standard deviation of a normal distribution are known). In such a normal distribution, about 68% of the data points are within one standard deviation of the mean and about 95% of the data points are within two standard deviations of the mean. 

Given the mean /ij, and the standard deviation of a normal-distributed random variable y we obtain the pdf of a lognormal-distributed random variable X by... 

Parameter p3- ram-3-t9r [NL, fr. para- + Gk metron measure] (1656) n. (1) Loosely, a system factor or variable that may take on a range of values as decided by the observer or operator of the system. Example hydraulic-line pressure and cylinder temperature are parameters in injection molding. (2) A defining constant of a statistical distribution, such as the mean or standard deviation of a normal distribution, and distinct from estimates of same calculated from sample measurements. (3) An independent variable through whose functions relations between other factors may conveniently be expressed. 

Figure 2-3 Effect of the standard deviation on a normal distribution with a mean of 0. The distribution becomes more pronounced around the mean as the standard deviation decreases.

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

The GUM approach described here has the advantage that each uncertainty component is designed to have the properties of a standard deviation, and so the rules for combining standard deviations of the normal distribution can be followed. The complete equation will be given, but it may be simplified to useable equations for the majority of applications. 

A quantity used to calibrate or specify a model, such as parameters of a probability model (e.g. mean and standard deviation for a normal distribution). Parameter values are often selected by fitting a model to a calibration data set. 

Wb = peak width at the base of each peak equivalent to 4 a, o being the standard deviation in a normal distribution curve i.e., obtained by drawing tangents to the inflection points as shown in Figure 1. 

The mean and standard deviation of the normal distribution are T and a, respectively. Since the normal distribution is designed for continuous data, the cumulative distribution function is more practical than the probability density function. For a particular data population, the cumulative distribution [2] is as follows ... 

Sigma g (geometric standard deviation, ug is the arithmetic antilog of the standard deviation of a normal population of logs of aerosol diameters, surface areas, volumes, or masses. Each of these different measures of aerosol size is log-normaUy distributed, o-g is dimensionless. Using antilogs to transform them to arithmetic numbers makes the distribution easier to understand. 

CSDs may be conveniently classified by the median size and the coefficient of variation. The CV, which quantifies the size spread, is a statistical property related to the standard deviation of a Gaussian distribution and is normally expressed as a percentage by... 

In order to investigate the effect of catalyst pore structure and reactor operating pressure upon the pellet effectiveness factor, simulations were carried out over the range of variables shown in Table 23.2. The PSD of the pore network was assigned from a Normal distribution of radii in a random fashion. The mean and standard deviation of the Normal distribution are given in Table 23.2. The random network was generated 10 times for each set of conditions, and the results of these simulations then averaged. 

Figure 14.1 Effect of standard deviation on a normal distribution curve.

The variables x and y represent the components of the distance vector between the center of the laser beam and a point of the planar environment. The value of o stands for the standard deviation of the normal distribution q Q,- and characterizes the variance of the distribution. The variable i identifies the different planes and allows for a volumetric heat input (depth effect) due to melt pool dynamics. In particular holds for Q(i) in Eq. (28) ... 

Quantifying the main uncertainty contributions. Each of the input quantities receives a standard uncertainty m(x ) either in the form of a standard deviation of a measurement series (type A evaluation) or a standard deviation of a reasonable distribution of values (type B evaluation). In case of a normal... 

There are statistical, and perhaps systematic, uncertainties associated with the primary measured quantities used to define the thermodynamic state these should be considered in the reporting of experimental uncertainties of the measured quantity and accounted for in the regressions used to determine a correlation. All the uncertainties are likely to have state point dependence, so that the assessment in the dilute gas regime will differ markedly firom that in the critical region or in the compressed liquid (Perkins et al. 1991a). The uncertainty reported for a correlation should be a function of the fluid state point. The coverage factor (based on, perhaps, two standard deviations in a normal distribution) should be applied to the appropriate distribution associated with the combined standard uncertainty of the correlation. 

This method employs the classical statistics to calculate the variance with a set of model outputs from a set of input parameters that are randomly generated. The number of runs depends on the model and the assumed input parameter distribution. According to Harr (25), the required number of MC simulations, N, for, m, independent variables is estimated as, N=(h /4 r, where h is the standard deviation in a normal distribution corresponding to the confidence interval, and is the maximum allowable system error in estimating the confidence interval. For example, if a required confidence interval is 99% with 1% system error, h is 2.58, e is 0.01, and (16,641)" is estimated. Therefore computing time is the major disadvantage of this method. Cawlfield and Wu (24) required over 400,000 computer runs to achieve a good level of accuracy for a one-dimensional transport code for a reactive contaminant. 

P(x, t) dx has the familiar bell shape of a normal distribution function [Eq. (1.39)], the width of which is measured by the standard deviation o. In Eq. (9.83), t takes the place of o. It makes sense that the distribution of matter depends in this way on time, with the width increasing with t. 

The ciimnlative prohahility of a normally distributed variable lying within 4 standard deviations of the mean is 0.49997. Therefore, it is more than 99.99 percent (0.49997/0.50000) certain that a random value will he within 4<3 from the mean. For practical purposes, <3 may he taken as one-eighth of the range of certainty, and the standard deviation can he obtained ... 

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

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. 

In analytical chemistry one of the most common statistical terms employed is the standard deviation of a population of observations. This is also called the root mean square deviation as it is the square root of the mean of the sum of the squares of the differences between the values and the mean of those values (this is expressed mathematically below) and is of particular value in connection with the normal distribution. 

The principle of Maximum Likelihood is that the spectrum, y(jc), is calculated with the highest probability to yield the observed spectrum g(x) after convolution with h x). Therefore, assumptions about the noise n x) are made. For instance, the noise in each data point i is random and additive with a normal or any other distribution (e.g. Poisson, skewed, exponential,...) and a standard deviation s,. In case of a normal distribution the residual e, = g, - g, = g, - (/ /i), in each data point should be normally distributed with a standard deviation j,. The probability that (J h)i represents the measurement g- is then given by the conditional probability density function Pig, f) ... 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

IQR is the difference between the third and the first quartile, and thus is not influenced by up to 25% of the lowest and 25% of the largest data. In the case of a normal distribution the theoretical standard deviation cr can be estimated from IQR by... 

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