Gaussian probability curve

The two pulses are presumed separated by the dashed hne given in Figure E6.7.2. For a large number of tanks-in-series (10 or more) the pulse curve is close to a Gaussian probability curve. From equation (6.12), there are... 

Brownian motion is a characteristic of the movement of single colloidal particles, but this motion has important consequences when many particles are present. Suppose, for example, we consider a thin sheet in which there are initially c° particles in unit volume [Figure 6.3(a)] and examine the distribution of these particles after a time At. They will have spread out in both directions. The chance that a given particle will have reached a distance Ax is proportional to At1 2 the sharp initial concentration peak will spread out into a broad peak, which has the shape of a Gaussian probability curve [Figure 6.3(b)]. 

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. 

Figure 5.1.7 shows the propagator of the motion measured for a clean and a biofilm impacted capillary [14,15] and the residence time distributions calculated for each from these velocity distributions. The clean capillary gives an experimental propagator equal to the theoretical velocity distribution convolved with a Gaussian diffusion curve [14], as shown in Figure 5.1.2. For the flow around the biofilm structure note the appearance of a high velocity tail indicating higher probability of large displacements relative to the clean capillary. The slow flow peak near zero displacement results from the protons trapped within the EPS gel matrix where the...

Many distributions obtained in experimental and observational work are found to have a more or less bell-shaped probability curve. These distributions are described by the normal or gaussian distribution shown in Fig. 2. This theoretical distribution is extremely important in statistics, and its use is not limited to data which are exactly, or very nearly normal. 

When measurements lie on such a Gaussian error curve, they can be analyzed to assess the probability that one observation is significantly... 

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). 

From the Gaussian error curve, what is the probability that a result from a population lies between 0 and -I-1 cr of the mean What is the probability of a result occuiTing that is between +1 and -l-2cr of the mean ... 

The quantity Apx is the reciprocal of the so-called precision index of the Gaussian error curve and is larger than the probable error by the factor 2.10 see R. T. Uirge, Phys. Rev. 40, 207 (1932). 

With all of these parameters, it was possible to calculate the energy levels of the 4f and 4f 5d electron configurations of the Tm " ions in SrCl2. In addition, probabilities of the electric dipole transitions between all possible states were also calculated. Finally, the simulated and measured 4f-5d spectra of SrCl2 Tm are shown together shown in Fig. 5.9 where the pure 4f-5d transition lines are represented by the vertical bars, and the broad bands are reproduced by using the Gaussian-shaped curves with a full width at half maximum width = 400 cm... 

Normal Law Normal Law of Error Normal or Gaussian Distribution Law Gauss Error Curve Probability Curve... 

Statistically, a similar Indication of precision could be achieved by utilising the 95% probability level if the results fell on a "Gaussian" curve, viz., the confidence would lie within two standard deviations of the mean. R 2 x SD = 56.3 24.8... 

The concentration profiles of the solute in both the mobile and stationary phases are depicted as Gaussian in form. In due course, this assumption will be shown to be the ideal elution curve as predicted by the Plate Theory. Equilibrium occurs between the mobile phase and the stationary phase, when the probability of a solute molecule striking the boundary and entering the stationary phase is the same as the probability of a solute molecule randomly acquiring sufficient kinetic energy to leave the stationary phase and enter the mobile phase. The distribution system is continuously thermodynamically driven toward equilibrium. However, the moving phase will continuously displace the concentration profile of the solute in the mobile phase forward, relative to that in the stationary phase. This displacement, in a grossly... 

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. 

The detailed shapes of these curves depend on many factors, and reliable theoretical calculations are probably not yet feasible. However, I believe that the foregoing very simplified curves, where is represented by 0.5 V0 [1 + cos 2(0 + 90°)] and s by Gaussians, give useful pictures of the situations that may be encountered. 

---