Average, standard deviation, normal distribution

Figure 2. Normalized aerosol volume distribution, China Lake, CA (1979 average)—average of 254 measurements. The error bars are standard deviations. The distribution is normalized with respect to total aerosol volume concentration of particles less than 10 lOn in diameter.

Previously we were concerned with estimating and a, which are parameters of an infinite sample set. Now we are dealing with a large but finite set—say 20 or more measurements of X. Consider several such finite sets, each with its average x. The distribution of these averages follows a normal distribution characterized by a sample standard deviation 5. That is, there is a 68.26 percent probability that the true value of lies between X — and + 5 for a sample of 20 measurements. 

To estimate normal parameters p and o of this distribution, we implemented the procedure developed by Fisher and Raman (1996). In this method, the members of a committee (comprised of four buyers in our case), independently provide a forecast of sales for each product. The mean p is set to the average of these forecasts. The standard deviation of demand a is set to 0cTc, where Oo is the standard deviation of the individual committee member forecasts and the factor 0 is chosen so that the average standard deviation of historical forecast errors equals the average standard deviation assigned to new products. In our application, we found 0 to be 1.4. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

It can be shown that the standard deviation (SD) of this distribution is also just yfji. In other words, if one would repeatedly measure the same pixel that on average collects 100 photons during a single dwell time (a normal value for a rather bright confocal image ) one would record less than 100-2x /l00 = 80 photons or more than 100 + 2x /l00 = 120 photons just by coincidence in 5% of the measurements. This uncertainty is expressed as the... 

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. 

It is the distance from the mean to the point of inflexion of the normal distribution curve. In comparison to the average deviation the standard deviation is usually considered to be much more useful and meaningful statistically. For a finite number of values it is normally symbolised as S , and may be expressed as follows ... 

Fig. 6. Distribution of calculated polarizability of F in aqueous solution and the normal distribution with average a= 10.56ao and standard deviation s = 0.66al.

In this example, the likelihood function is the distribution on the average of a random sample of log-transformed tissue residue concentrations. One could assume that this likelihood function is normal, with standard deviation equal to the standard deviation of the log-transformed concentrations divided by the square root of the sample size. The likelihood function assumes that a given average log-tissue residue prediction is the true site-specific mean. The mathematical form of this likelihood function is... 

A total of 254 particle size distributions were measured throughout 1979. The average normalized volume distribution is plotted in Figure 2. The error bars are standard deviations. 

Because the fine aerosol was found to be responsible for the bulk of light scattering at China Lake, this mode was examined to see if its normalized distribution remained constant throughout 1979. Figure 4 shows the 1979 average aerosol volume distribution at China Lake normalized with respect to the total volume of particles smaller than 2 ym. The error bars represent standard deviations in the 254 measurements. The particle volume distribution in the fine mode is seen to preserve its shape rather well. Over half the fine particle volume is due to particles of less than 0.3 ym diameter. 

The averages of random samples of a population are normally distributed. Therefore, the standard deviation of the population of sample means is the standard deviation of the population from which the sample is drawn divided by the square root of sample size. If we standardize the data to have a mean of 0.0 and a standard deviation of 1.0, then the standard deviation of the sample mean is 1.0 divided by the square root of the sample size. To be 95 percent confident that the incidence of insomnia in one group is smaller than the incidence in another group, the incidence in the first must be at least 1.64 standard deviations smaller than the incidence in the second. The sample size required to detect any given difference in means is approximately the square of 1.64 divided by the difference—in this case, (1.64/0.05) or 1,075.84. 

If sufficient history is available on a test method, a reaction, or a process, so that the experimental error is known, the test for differences between averages of two sets of data becomes more sensitive. In each of the situations considered above, the form of the equation is the same, but it is no longer necessary to calculate the standard deviation. The symbol Z is used instead of t and the critical value is found in the t table for infinite degrees of freedom This row of the t table corresponds to the table of the normal distribution. 

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