Curve Error

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. 

Note that Laplace (not Gauss) first derived the equation for the Gaussian (normal) error curves, which need not be normal in the sense that they normally apply to errors encountered in practice (text above). 

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. 

FIG. 5 pH-lipophilicity profiles (on the left) and errors (curves in bold on the right) made by using the simplified Eq. (14) instead of the complete Eq. (15) when estimating log/ . The calculation is made for a monobase (top), a monoacid (middle), and a zwitterion (bottom) with a difference of three units between log/ and log/. (Adapted from Ref. 143.)... 

If the right side of this equation is plotted versus dimensionless time for various values of the group Q)JuL (the reciprocal Peclet number), the types of curves shown in Figure 11.8 are obtained. The skewness of the curve increases with 3) JuL and, for small values of this parameter, the shape approaches that of a normal error curve. In physical terms this implies that when 3JuL is small, the shape of the axial concentration profile does not change... 

For small values of the dispersion parameter one may take advantage of the fact that equation 11.1.37 takes the shape of a normal error curve. This implies that for a step function input a plot of (C — Cq)/(Cq — Co) or F(t)... 

Figure 6.2. (a) The effects of salinity on the sensitivity of standard additions of ammonia in laboratory mixed waters ( ) and in waters from the Tamar estuary (A) expressed as percentage of response in river water. For comparison, the salt error curves reported by Loder and Gilbert [3] are also shown (... and —, respectively), (b) Contribution of reactive index and organic absorbance to the optical blacks in the Chemlab Colorimeter. = River water-seawater mixture, o = De-ionized water-seawater mixture. Source [2]... 

A valuable parameter derived from the normal error curve (p. 16) and expressed by l-N 1 i... 

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... 

Normal error curves, (a) Curve (a) shows a normal distribution about the true value. Curve (b) shows the effect of a determinate error on the normal distribution, (b) Curves showing the results of the analysis of a sample by two methods of differing precision. Method A is the more precise, or reliable. 

Figure 2.4(a) shows normal error curves (B and S) with true means pB and ps for blank and sample measurements respectively. It is assumed that for measurements made close to the limit of detection, the standard... 

Normal error curves for blank B and sample S measurements. 

Figure 3 Errors in potential energies for CH4 in a 6-3IG basis. B3LYP and UB3LYP error curves have been shifted up by 70 kcal mof. ...

In Figures 8 and 9 are shown the data for the dependence of the characteristic film buildup time t on Apg and U. In accord with the model, t is found to be independent of U, with only a very weak dependence on Apg indicated. This latter result could in part be a function of experimental inaccuracy. The data reduction for t introduces no assumptions beyond that needed to draw the exponential flux decline curves such as those shown in Figures 2 and 3. However, an error analysis shows that the maximum errors relative to the exponential curve fits occur at the earlier times of the experiment. This is seen in the typical error curve plotted in Figure 10. The error analysis indicates that during the early fouling stage the relatively crude experimental procedure used is not sufficiently accurate or possibly that the assumed flux decline behavior is not exponential at the early times. In any case, it follows that the accuracy of the determination of 6f is greater than that for t. 

Figure 10. Typical error curve for exponential fit of data...

A statistical method for plotting the relative frequency (dN/N) of a probable error in a single measured value X versus the deviation (z) from fi, the mean of the data, in units of standard deviation (o-), such that z = (x -fji)/a. The standard error curve (shown below) does not depend on either the magnitude of the mean or the standard deviation of the data set. The maximum of the normal error curve is poised at zero, indicating that the mean is the most frequently observed value. 

The area beneath the standard error curve can be obtained by integrating the following equation ... 

A statistical term for the deviation from the true value within which lies an experimentally measured value with a probability of 0.50. This corresponds to 0.674 cr (i.e., 0.674 times the standard deviation). See Statistics (A Primer) Normal Error Curve... 

PERMUTATIONS AND COMBINATIONS PROBABILITY DENSITY FUNCTION PROBABLE ERROR NORMAL ERROR CURVE STATISTICS (A Primer)... 

Probable error in a single measured value, NORMAL ERROR CURVE Probability of tunneling,... 

It is well known that the width of the normal error curve at its inflection point (where the... 

We can verify the plausibility of this expression as follows. Recall that z and t are inversely related and that the integral gives the area under one-half of the error curve between its... 

The shape ofthe elution curve for a pulse injection can be approximated by the Gaussian error curve for AT > 100, which is almost the case for column chromatography [2]. The value of N can be calculated from the elution volume Vg (m ) and the peak width W (m ), which is obtained by extending tangents from the sides of the elution curve to the baseline and is equal to four times the standard deviation (Ty (m ) = as shown in Figure 11.7. 

Mathematical analysis of error curves leads to the conclusion that the arithmetic mean x of individual values is the best estimation of the real mean //, (Fig. 21.2). The features and the symmetry of this curve show that ... 

Elution time, 19 EMIT, 340 Energy filter, 295 Enzyme conjugate, 336 Equivalent conductance, 70 Error curve, 388 Exact value, 385 Exchanging anion, 66 Exchanging cation, 66 Experimental error, 385... 

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