Normal error curve

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). 

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)... 

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. 

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. 

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... 

Figure 4-3 A Gaussian curve in which jx = 0 and

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... 

Equation 2.54 has the form of the normal error curve and from the geometrical properties of the curve we can show that... 

A typical distribution of errors. The bar graph represents the actual error frequency distribution 73(e) for 376 measurements the estimated normal error probability function P(e) is given by the dashed curve. Estimated values of the standard deviation cr and the 95 percent confidence limit A are indicated in relation to the normal error curve. 

The statistical parameters generated in the process of fitting the data to the equation are also used to determine the significance of the equation. A common criterion is to retain coefficients if their two-tailed probability is less than 0.05 P(2-tail) < 0.05. A two-tailed probability smaller than 0.05 means that the deviation from the true value lies in the positive or negative regions of the normal error curve corresponding to less than 5% of the area. It... 

Figure 2.4(a) shows normal error curves (B and S) with true means / and ns for blank and sample measurements respectively. It is assumed that for measurements made close to the limit of detection, the standard deviations of the blank and sample are the same, i.e. aB = a% — a. In most cases, a 95 % confidence level is a realistic basis for deciding if a given response arises from the presence of the analyte or not, i.e. there is a 5% risk in reporting the analyte detected when it is not present and vice versa. Thus, point L on curve B represents an upper limit above which only 5% of blank measure-mentswith true mean /tD will lie whilst point L on curve S represents a lower limit below which only 5% of sample measurements with true mean //s will lie. If /is now approaches /iB until points L on each curve coincide (figure... 

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