DISCUSSION We examined our own radioisotope quality control data to test the assumption of normality and we found that most of our data does not follow a normal distribution, even when the common transformations of data, such as the logarithm and square root, are used. Our data seem to be best described statistically as a heterogeneous normal distribution, also known as a mixture of normal distributions. The following example of cesium-134 is typical of our data. sent the example in the sequence of the data analysis steps. 1346, 12-20-74 I will preThe first These questions % PROBABILITY curve could be composed of one- or two-line segments. 29 30 40 50 60 70 980 step is to plot the cumulative distribution of the data on normal probability paper. The plot of the example data is shown in Figure 1. It is obvious that the data do not fall on one straight line, the requisite for a single normal distribution to describe the data. Note that the slopes of the top and bottom sagments are the same, thus the possibility that they are two pieces of one distribution. The center portion of the of the components of the curve must be resolved by statistical test. The several possible structures were programmed as statistical functions and compared using the maximum likelihood ratio test. In more detail, these steps are as follows. We start with a statistical model of the data distribution. The model for a mixture of three normal distributions is shown in Figure 2. The Parameters denoted as "p” determine the proportion of the data "explained" by each of the component normal distributions. Since the total of all proportions must equal 100 percent of the data, two of these proportions are sufficient to define all three components. The extenSions of this formula to two components, or more than three, should be obvious. A three-component model has eight parameters to be determined from the data. A not so obvious extension is to restrict mean values or Standard deviations to be equal. For example, in Figure 2 we might replace the us with with a u2 so that uz appears twice in the formula 5 giving a seven-parameter model. In about one-fifth of our data sets, though not in the example presented here, this restriction to equal mean values was statistically the best model. The next step is to combine the model with the observed data using the likelihood function. This step is explained in detail in most elementary mathematical statistics textbooks. 300 After we have the likelihood function, we wish to choose those parameter values (in our case the “p's,” “u's,” and “o's") that maximize the likelihood function; this is the statistical principle called maximum likelihood estimation. The discipline of computer sciences has recently provided some generalized functional maximization programs which can act 594 Figure 1. 400 500 DATA VALUES ability Plot Cumulative Norval Proh 595 ay G00