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7.5 The Central Limit Effect

How can it be that the normal distribution holds such a central position in statistics? There are many other ways that data may be distributed. If we roll a six-sided die, for example, the probability of each score is 1/6. In other words, the frequency distribution of the scores is completely flat, far from the bell-shaped Gaussian curve. The roll of a die is a relatively simple process, however. Measurements are more complex. The error in measurement data is usually combined from a large number of component errors. When discussing the synthesis experiment earlier, we mentioned a few sources of error that could contribute to the overall error: variations in reactant concentrations, mixture homogeneity, temperature, as well as measurement precision and operator repeatability. There are numerous other potential error sources, such as the purity of the reactants. We do not know which distributions these individual errors follow but due to the central limit effect we know something about how the combination of a large number of such errors is distributed. This effect makes the mean of a number of random variables follow a normal distribution when the number of variables becomes large, almost regardless of how each variable is distributed.

Figure 7.9 Histograms showing the scores from throwing a single die (top), two dice (middle) and ten dice (bottom). (When several dice are thrown the mean score is calculated.)

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