Means of distributions

An analytical determination usually consists of two or three replicate analyses, and the mean is reported as the best value. Thus, the mean is om most vital concern. We want to know how the number of replicate measurements affects the goodness of the mean and how much confidence can be placed in the value we report. 

A single analytical measurement, X, is a random variable. When several measurements of X are taken and the mean is calculated, and then several more are taken and a second mean is calculated, we usually find that the means are not exactly the same. If we follow this procedure a number of times, we can plot a frequency distribution of means in the same way as for the distribution of individual measurements. Thus, the mean, X is also a random variable. 

Perhaps the most important question in chemical measurements is how accurate the mean of several repeated measurements is likely to be. This is not an easy question to answer with absolute certainty, but by using simple statistical principles, we can get a numerical answer that is very likely to be correct. 

It can be shown that the variance of the distribution of sample means is equal to the variance of the underlying distribution divided by the number of individual measurements making up the sample means, n. 

In Equation 14.10, is called the standard deviation of the mean. This is a very significant equation because it teUs us that the standard deviation of the mean becomes smaller and smaller as the number of individual measurements, n, increases. For example, the standard deviation of the mean is cut in half by a fourfold increase in n. 

---

The standard deviation of the distribution of means equals cr/N. Since cr is not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance and the reliability of using 5- in... 

Hollow Sprays. Most atomizers that impart swid to the Hquid tend to produce a cone-shaped hoUow spray. Although swid atomizers can produce varying degrees of hoUowness in the spray pattern, they aU seem to exhibit similar spray dynamic features. For example, detailed measurements made with simplex, duplex, dual-orifice, and pure airblast atomizers show similar dynamic stmctures in radial distributions of mean droplet diameter, velocity, and Hquid volume flux. Extensive studies have been made (30,31) on the spray dynamics associated with pressure swid atomizers. Based on these studies, some common features were observed. Test results obtained from a pressure swid atomizer spray could be used to iUustrate typical dynamic stmctures in hoUow sprays. The measurements were made using a phase Doppler spray analyzer. 

We studied vacancy segregation near interphase and antiphase boundaries using the MFA and PCA approaches described in Sec. 6 below. For the A-B alloy with vacancies, the stationary distribution of mean vacancy occupations =< > can be explicitly... 

Table III. Distribution of Mean Dust Lead Levels by Study Area...

James and Guth showed rigorously that the mean chain vectors in a Gaussian phantom network are affine in the strain. They showed also that the fluctuations about the mean vectors in such a network would be independent of the strain. Hence, the instantaneous distribution of chain vectors, being the convolution of the distribution of mean vectors and their fluctuations, is not affine in the strain. Nearly twenty years elapsed before his fact and its significance came to be recognized (Flory, 1976,... 

In 47 houses, Rn-222 measurements were performed by Track Etch detectors from Terradex Inc. during two periods, November-May and June-October. From these measurements, the normal seasonal variations were assessed. In Fig. 2 a plot of the cumulative distribution of mean Rn-222 concentrations during summer and winter are shewn. From this figure, the mean concentration during sunnier is about 50 % of the mean concentration during the winter months. 

Figure 36. The scaled distributions of mean, P(H/Y ) (a), and Gaussian, P(K/Y]2) (b), curvatures scaled with the inteface area density, computed at several time intervals of the spindal decomposition of a symmetric blend. There is no scaling at the late times because the amplitude of the thermal undulations does not depend on the average growth of the domains, and therefore the scaled curvature distributions functions broaden with rescaled time.

Fig. 1 Spatial distribution of mean water temperatures in the Taillon catchment, French Pyrenees [33]...

Figure 2. Distribution of mean concentrations with wind direction for 4 crustal elements measured on a streaker sampler at ground level during one week in June 1978. Radial bars indicate I standard error of the mean. The numbers at the end of the bars indicate the number of 2-h samples from each direction.

The results presented a variety of evidence for the identity of Ca sources near our rural sampling site. The distribution of mean crustal element concentrations as a function of wind direction in summer and fall, from the streaker data, suggest a combination of road and soil sources. This agrees with a comparison of crustal abundances in aerosols and source materials. The comparison showed that most of the elements examined had abundances in the aerosol that often fell between those characteristic of roads and soil. This was not the case for Si, but Si may be expected to be less abundant in aerosol samples than in bulk surficial materials because of the preponderance of quartz (Si02) in the larger particles. 

The spin-lattice relaxation time map (discussed in Section II.A.2) yields information about the spatial distribution of mean pore size within a given image pixel. Lighter shades in the image correspond to larger mean pore size. Even at this coarse... 

Poisson distribution of mean n =0.0121 for events with NHIT 2 30. Hence the rate of occurrence of 6 events in a 10 second time interval due to a statistical fluctuation is less than one in 7 x 10 years in our experiment. 

Constant-Stress Layer in Flowing Fluids. In the boundary layer of a fluid flowing over a solid wall. Ihe shear stress varies with distance from Ihe wall bul ii may be considered nearly constant within a small fraction of the layer thickness. The concept is of particular importance in turbulent flow where it leads lo a theoretical derivation of the law of ihe wall," the logarithmic distribution of mean velocity. The constant stress layer is ihe best-known example of the equilibrium flow s near a wall. 

Figure 3. (a) Spatial distribution of mean annual moist static energy. The station locations (crosses) indicate the spatial coverage of the data set. (b) Spatial distribution of the function G=T+ y, Z where T is the mean annual temperature, Z is the station elevation, and y, = 5.9 K/km. The distributions of each variable as functions of latitude are shown in the insets. [Used by permission of Geological Society of America, from Forest et al. (1999), Geol. Soc. Am. Bull., Vol. Ill, Fig. 3, p. 502.]... 

Statistical evaluations of data are warranted by the fact that the true mean concentration // (the population mean) will never be known and that we can only estimate it with a sample mean x. As a reflection of this fact, there are two parallel systems of symbols. The attributes of the theoretical distribution of mean concentrations are called parameters (true mean p, variance a2, and standard deviation sample results are called statistic (sample mean x, variance s2, and standard deviation 5). 

Consider a conical hopper flow where there is no-slip between the gas and the particle. Determine the distribution of mean stress of particles in the hopper flow. The particles can be assumed to be in a moving bed condition. 

Another piece of evidence to suggest that content uniformity data is not normally distributed can also be seen in the content uniformity data of Rohrs et al.4 They report the content uniformity assay results for 11 batches of tablets. In every case except one, the actual mean potency is below the target value. If the content uniformity data were normally distributed, as assumed by the Yalkowsky and Bolton approach,2 one might expect to see a more even distribution of mean values above and below the target. The one exception was for the lowest potency batch for which one tablet was assayed to be 292% of intented value, as mentioned above. This characteristic is... 

A second facility that is sometimes useful is the random number generator function. There are several possible distributions, but the most usual is the normal distribution. It is necessary to specify a mean and standard deviation. If one wants to be able to return to the distribution later, also specify a seed, which must be an integer number. Figure A. 15 illustrates the generation of 10 random numbers coming from a distribution of mean 0 and standard deviation 2.5 placed in cells A1 -A10 (note that the standard deviation is of the parent population and will not be exactly the same for a sample). This facility is very helpful in simulations and can be employed to study die effect of noise on a dataset. 

Figure 3 The distribution of mean annual salinity in the surface waters of the oeean (Adapted from The Open University, 1989. )...

The combustion stability limits, size of the recirculation zone, completeness of combustion, and the radial distribution of mean temperatures have been measured. Mean temperatures were measured with a 0.076-mm Pt vs. Pt-13% Rh thermocouple coated with silica to reduce catalytic eflFects. The completeness of combustion was measured with gas chromatography using a Pye Unicam isothermal gas chromatograph. A detailed description of the measurement technique used is given in Refs. 5 and 9. 

Radial distribution of mean temperatures were measured in the main... 

Figure 13. Radial distribution of mean temperatures corresponding to flames...

I-orius C. and Merlivat L. (1977) Distribution of mean surface stable isotope values in East Antarctica. Observed changes with depth in a coastal area. In Isotopes and Impurities in Snow and Ice. Proceedings of the Grenoble Symposium Aug./Sep. 1975 (eds. lAHS), lAHS, vol. 118, pp. 125-137. 

The mean of the sampling distribution of means is equal to the mean of the population from which the samples were drawn. 

If the original population is distributed normally (i.e., it is bell shaped), the sampling distribution of means will also be normal. Even if the original population is not normally distributed, the sampling distribution of means will increasingly approximate a normal distribution as sample size increases. 

Second, consider the quantity s(k ). This is an estimate of the standard error of k based on ( —2) degrees of freedom. We use the quantity standard error to describe the width of the distribution of k values which we would observe if we were able to make a vast number of separate determinations we use the term standard error rather than standard deviation to remind us that the distribution under consideration is a distribution of mean values, each of which is itself derived from a population. The term ( —2) degrees of freedom signifies that, out of our nt experimental pairs of observations, only ( —2) are available to give us an estimate of the precision of the measurement, the other two having been lost in fixing the two parameters, /j and k, of our fitted line. Now, clearly we can never obtain a vast number of determinations of a statistic such as k in order to obtain a value for its standard error necessarily, the number of observations which we can make is limited and so we cannot do any better than make an estimate of what the width of the distribution would b6 for a yast population. This estimate is extremely useful for it enables us to calculate from the sample mean, i.e. k[, the limits between which the population mean is likely to lie. Suppose we designate the population mean by the symbol. A). We now introduce the statistic, t,... 

---