Distributions of standard deviations

Figure 8.23 Monte Carlo simulations of the distribution of standard deviations.

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

The standard deviation can also be evaluated from the same classified data it measures the width of the distribution. The standard deviation a is defined as... 

The degree of data spread around the mean value may be quantified using the concept of standard deviation. O. If the distribution of data points for a certain parameter has a Gaussian or normal distribution, the probabiUty of normally distributed data that is within Fa of the mean value becomes 0.6826 or 68.26%. There is a 68.26% probabiUty of getting a certain parameter within X F a, where X is the mean value. In other words, the standard deviation, O, represents a distance from the mean value, in both positive and negative directions, so that the number of data points between X — a and X -H <7 is 68.26% of the total data points. Detailed descriptions on the statistical analysis using the Gaussian distribution can be found in standard statistics reference books (11). 

Fig. 26. The effect of pore size distribution spread (standard deviation) on the predicted tensile failure [jrobability distribution for grade H-451 graphite.

Population parameters were shown in Table 11. Guiyu was considered Coastal Region to obtain the Daily intake from Li et al. [23], If no data of distribution was found, normal distribution with standard deviation equal to the 10% of the mean was assumed (Table 10). 

The zero-field spin Hamiltonian parameters, D and E, are assumed to be distributed according to a normal distribution with standard deviations oD and aE, which we will express as a percentage of the average values (D) and (E). -Strain itself is not expected to be of significance, because the shape of high-spin spectra in the weak-field limit is dominated by the zero-field interaction. 

The other key assumption that we sort of implied was that the comparison of standard deviation is constant. Of course we know that as n changes, the comparison value changes as the square root of n. This is on top of and in addition to the changes caused by the use of the t rather than the Normal (Z) distribution. 

Figure 53-34 Family of curves of the Energy-Distribution product, corresponding to various truncation points. The numbers indicate the truncation point of the Normal distribution, as the number of standard deviations from the peak of the Normal distribution.

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. 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

Problems arise to get informations about the diffusion coeffients Ky and Kz. If equation (3.4) is interpreted as Gaussian distribution, a lot of available dispersion data can be taken into consideration because they are expressed in terms of standard deviations of the concentration distribution. Though there is no theoretical justification the Gaussian plume formula is converted to the K-theory expression by the transformation /11/... 

It can be shown that if the uncertainties associated with the measurements of the response are approximately normally distributed (see Equation 3.8), then parameter estimates obtained from these measurements are also normally distributed. The standard deviation of the estimate of a parameter will be called the standard uncertainty, s, of the parameter estimate (it is usually called the standard error ) and can be calculated from the matrix if an estimate of is available. 

In addition to a tight distribution of the thickness variation within a wafer, the average of a group of individual thicknesses must also be targeted within a certain range. Statistically, the control of the WIWNU is the control of standard deviation of individual thicknesses, and the control of final thickness post CMP is the control of the mean. The variation of the mean from wafer to wafer is called wafer-to-wafer nonuniformity (WTWNU). All the thicknesses mentioned in this section are actually the means of many individual thickness measurements in each wafer. Control is not easy, for reasons discussed in the following. 

Just as an aside, look back at the formula for the 95 per cent confidence interval. Where does the 1.96 come from It comes from the normal distribution 1.96 is the number of standard deviations you need to move out to, to capture 95 per cent of the values in the population. The reason we get the so-called 95 per cent coverage for the confidence interval is directly linked to this property of the normal distribution. 

The simulation says that the maximum value is. 5188, which is less than the expected value. Remember that for the resistor with the 5% Gaussian distribution, the standard deviation was 1.25%, and the absolute limits on the distribution were 4o = 5%. In the Worst Case analysis, a device with a Gaussian distribution is varied by only 3cr. Had we calculated the maximum value with a 3.75% resistor variation, we would have come up with a maximum gain of 0.51875, which agrees with the PSpice result. To obtain the worst case limits, I prefer to use the uniform distribution. Type CTRL-F4 to close the output file and display the schematic. 

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

Now, everything falls into place We set out to study the laws of random walk by using the simple model of Fig. 18 and found the Bernoulli coefficients. We then saw that for large n (which is equivalent to large times), the Bernoulli coefficients can be approximated by a normal distribution whose standard deviation, a, grows in proportion to the square root of time, tm (Eq. 18-3). And now it turns out that the solution of the Fick s second law for unbounded diffusion is also a normal distribution. In fact, the analogy between Eqs. 18-3b and 18-17 gave the basis for the law by Einstein and Smoluchowski (Eq. 18-17) that we used earlier (Eq. 18-8). The expression (2Dt)U2 will also show up in other solutions of the diffusion equation. 

---