Frequency Distribution of Observations

In general statistics there is a difference between the parent population of a random variable, e.g. x (sometimes also characterized by capital letters) and a single realization of the parent population expressed, e.g., as single measurements, xh of the variable x. The parent population means an infinity of values which follow a certain distribution function. In the reality of experimental sciences one always has single realizations, x , of the random variable x. 

In the terminology of statistics analytically measured quantities (properties, features, variables) are random variables x. Such a variable may, e.g., be density, absorbance, concentration, or toxicity. Hence, repeated measurements (observations) using the same sample, or measurements of comparable samples, do not result in identical values, x but are single realizations of the random variable x. Using the frequency distributions 

Chemometrics in Environmental Analysis. J. W. Einax, H. W. Zwanziger, S. GeiB Copyright 1997 WILEY-VCH Verlag GmbH ISBN 3-527-28772-8 

Each column of the histogram represents a discrete value or a certain range of values. If the number n of observations is 30 n 400 it is recommended [DIN 53 804] that the class intervals be of width w LJ /n. If/ 400 w Ln/20 is appropriate. L = xmax -xmin is the range of the measurements. 

Starting from such frequency distributions one can easily derive cumulative frequencies which can be smoothed by the integrated distribution function F(x). Fig. 2-2 represents the cumulative frequencies of the example depicted in Fig. 2-1, again with a smoothed distribution function. 

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Figure 12.26 The frequency distribution of observed fracture strengths for a silicon nitride material.

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