Gaussian distribution error curve

These skewness factors are widely used to specify the higher-order turbulence components, because the turbulence intensity in gas-liquid two-phase flows in the baths of metallurgical processes is much higher than those of single-phase pipe flows (Tu < 15%) and single-phase jets (Tu < 30%) [21,22], It is known that 5 = 0 and F = 3 for a Gaussian (normal distribution) error curve. 

Emulsion A has a droplet size distribution that obeys the ordinary Gaussian error curve. The most probable droplet size is 5 iim. Make a plot of p/p(max), where p(max) is the maximum probability, versus size if the width at p/p(max) = j corresponds to... 

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. 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Figure 4.51. Distribution of experimental data. Six experimental formulations (strengths 1, 2, resp. 3 for formulations A, respectively B) were tested for cumulative release at five sampling times (10, 20, 30, 45, respectively 60 min.). Twelve tablets of each formulation were tested, for a total of 347 measurements (13 data points were lost to equipment malfunction and handling errors). The group means were normalized to 100% and the distribution of all points was calculated (bin width 0.5%, her depicted as a trace). The central portion is well represented by a combination of two Gaussian distributions centered on = 100, one that represents the majority of points, see Fig. 4.52, and another that is essentially due to the 10-minute data for formulation B. The data point marked with an arrow and the asymmetry must be ignored if a reasonable model is to be fit. There is room for some variation of the coefficients, as is demonstrated by the two representative curves (gray coefficients in parentheses, h = peak height, s = SD), that all yield very similar GOF-figures. (See Table 3.4.)...

Figure 5.25. (A) Quantitative Cu map of an Al-4wt% Cu film at 230 kX, 128 x 128 pixels, probe size 2.7nm, probe current 1.9 nA, dwell time 120 msec per pixel, frame time 0.75 hr. Composition range is shown on the intensity scale (Reproduced with permission by Carpenter et al. 1999). (B) Line profile extracted from the edge-on boundary marked in Figure 5.25a, averaged over 20 pixels ( 55 nm) parallel to the boundary, showing an analytical resolution of 8nm FWTM. Error bars represent 95% confidence, and solid curve is a Gaussian distribution fitted to the data (Reproduced with permission by Carpenter...

Indeterminate errors arise from the unpredictable minor inaccuracies of the individual manipulations in a procedure. A degree of uncertainty is introduced into the result which can be assessed only by statistical tests. The deviations of a number of measurements from the mean of the measurements should show a symmetrical or Gaussian distribution about that mean. Figure 2.2 represents this graphically and is known as a normal error curve. The general equation for such a curve is... 

In a situation whereby a large number of replicate readings, not less than 5 0, are observed of a titrimetric equivalence point (continuous variable), the results thus generated shall normally be distributed around the mean in a more or less symmetrical fashion. Thus, the mathematical model which not only fits into but also satisfies such a distribution of random errors is termed as the Normal or Gaussian distribution curve. It is a bell-shaped curve which is noted to be symmetrical about the mean as depicted in Figure 3.2. 

Tables Va, b, and c attempt to answer these questions. In Table Va percent copper is ranked in increasing order along with the laboratory numbers and the methods. First, are any of the values outliers Should all the values be taken as representative of the copper content, or are some in error Assuming that the data are normally distributed (i.e., that they follow the Gaussian or bell-curve distribution) the V test can be applied to reject anomalous values this was done, and four values were rejected. Three come from one laboratory, and the fourth came from a laboratory which did not detect or take Zn into account on sample 3. Therefore, their calculation for copper was too high.

If an experiment is repeated a great many times and if the errors are purely random, then the results tend to cluster symmetrically about the average value (Figure 4-1). The more times the experiment is repeated, the more closely the results approach an ideal smooth curve called the Gaussian distribution. In general, we cannot make so many measurements in a lab experiment. We are more likely to repeat an experiment 3 to 5 times than 2 000 times. However, from the small set of results, we can estimate the statistical parameters that describe the large set. We can then make estimates of statistical behavior from the small number of measurements. 

Gaussian distribution Theoretical bell-shaped distribution of measurements when all error is random. The center of the curve is the mean, p, and the width is characterized by the standard deviation, a. A nortnalized Gaussian distribution, also called the normal error curve, has an area of unity and is given by... 

If the total number of molecules is very large and the distance increments, Ax, become infinitesimally small (Ax - dx), the distribution will be described by a smooth function (a Gaussian distribution) following the familiar normal curve of error,... 

Tests for gaussian distribution The chi-square test can be used to find out whether an experimental distribution of errors follows the gaussian curve. The method is described here only in principle because only in unusual cases are enough data... 

Since we are not able to find the true value for any parameter, we often make do with the average of all of the experimental data measured for that parameter, and consider this as the most probable value." Measured values, which necessarily contain experimental errors, should lie in a random manner on either side of this most probable value as expressed by the normal or Gaussian distribution. This distribution i. a bell-shaped curve that represents the number of measurements N that have a specific value x (which deviates from the mean or most probable value Xq by an amount x - Xo, representative of the error). Obviously the smaller the value of x - Xo, the higher the probability that the quantity being measured lies near the most likely value xq, which is at the top of the peak. A plot of N against x, shown in Figure 10.1, is called a Gaussian distribution or error curve, expressed mathematically as ... 

Gaussian distribution A symmetrical bell-shaped curve described by the equation y = Aexp(—The value of x is the deviation of a variable from its mean value. The variance of such measurements (the square of the e.s.d.) is fl/2. In many kinds of experiments, repeated measurements follow such a Gaussian or normal error distribution. 

The majority of statistical tests, and those most widely employed in analytical science, assume that observed data follow a normal distribution. The normal, sometimes referred to as Gaussian, distribution function is the most important distribution for continuous data because of its wide range of practical application. Most measurements of physical characteristics, with their associated random errors and natural variations, can be approximated by the normal distribution. The well known shape of this function is illustrated in Figure 1. As shown, it is referred to as the normal probability curve. The mathematical model describing the normal distribution function with a single measured variable, x, is given by Equation (1). 

Normal error curve A plot of a Gaussian distribution of the frequency of results from random errors in a measurement. Normal hydrogen electrode (NHE) Synonymous with standard hydrogen electrode. 

Instead, we mean here the use of experimental data that can be expected to lie on a smooth curve but fail to do so as the result of measurement uncertainties. Whenever the data are equidistant (i.e., taken at constant increments of the independent variable) and the errors are random and follow a single Gaussian distribution, the least-squares method is appropriate, convenient, and readily implemented on a spreadsheet. In section 3.3 we already encountered this procedure, which is based on least-squares fitting of the data to a polynomial, and uses so-called convoluting integers. This method is, in fact, quite old, and goes back to work by Sheppard (Proc. 5th... 

STATISTICAL NATURE OF TURBULENCE. The distribution of deviating velocities at a single point reveals that the value of the velocity is related to the frequency of occurrence of that value and that the relationship between frequency and value is gaussian and therefore follows the error curve characteristic of completely random statistical quantities. This result establishes turbulence as a statistical phenomenon, and the most successful treatments of turbulence have been based upon its statistical nature. ... 

This is the familiar bell-shaped error curve having a maximum at the mean value, y, and a width described by the standard distribution, a. The integral of the gaussian distribution over all values of the argument is unity. Because the function is symmetrical, the integral from the mean value over all values on either side is 0.5. 

Let us suppose that an analytical procedure has been developed in which there is no determinate error. If an infinite number of analyses of a single sample were carried out using this procedure, the distribution of numerical results would be shaped like a symmetrical bell (Fig. 1.5). This bell-shaped curve is called the normal or Gaussian distribution. The frequency of occurrence of any given measured value when only indeterminate error occurs is represented graphically by a plot such as Fig. 1.5. 

The Gaussian distribution curve assumes that an infinite number of measurements of X, have been made. The maximum of the Gaussian curve occurs atx = /r, the true value of the parameter we are measuring. So, for an infinite number of measurements, the population mean is the true value x,. We assume that any measurements we make are a subset of the Gaussian distribution. As the number of measurements, N, increases, the difference between x and /x tends toward zero. For N greater than 20 to 30 or so, the sample mean rapidly approaches the population mean. For 25 or more replicate measurements, the true value is approximated very well by the experimental mean value. Unfortunately, even 20 measurements of a real sample are not usually possible. Statistics allows us to express the random error associated with the difference between the population mean fi and the mean of a small subset of the population, x. The random error for the mean of a small subset is equal tox — fi. 

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