. Elementary biophysics: selected topics . Xi xi = the measured value of some property /i = the frequency with which the value Xi is obtained in the set of measurements Fig. 1. A schematic representation of the two kinds of sets of measurements usually obtained experimentally. mental curves are reasonable because their relevance has been proven by the mathematicians. In fact, of course, there is an inseparable mutual interaction between the two approaches. To derive the symmetrical curve, the mathematicians assume that there exists a large number of small, randomly occurring, positive and nega
. Elementary biophysics: selected topics . Xi xi = the measured value of some property /i = the frequency with which the value Xi is obtained in the set of measurements Fig. 1. A schematic representation of the two kinds of sets of measurements usually obtained experimentally. mental curves are reasonable because their relevance has been proven by the mathematicians. In fact, of course, there is an inseparable mutual interaction between the two approaches. To derive the symmetrical curve, the mathematicians assume that there exists a large number of small, randomly occurring, positive and negative contributions of error to the individual measurements. Thereby they derive the following expression for the frequency, fu of deviations from the average value: fi = "4= e~(Xi -a)2/2s2 Here the pi and 2's result from the requirement that the individual fractional frequencies add up to unity. That means that the fractions of cases with all possible deviations from the average must includ 3 all cases, so that the sum is unity (or 100%, if expressed in percentage terms). This distribution of deviation frequencies is called, variously, the Normal Error Curve, the Normal Distribution, or the Gaussian Distribu- tion, after the great 19th-century scientist Gauss. The bell-shaped distri- bution is plotted in Fig. 2, which also indicates the frequencies for deviations equal to zero, ±s, ±2s, and ±3s. We can now begin to answer some of the many questions we have been accumulating. First, what fraction of the deviations lies within ±s of the average value a? This amounts to finding the fraction of cases in the shaded area of Fig. 3. (For those who understand the calculus, we can say that the fraction is found mathematically by integrating the Gaussian Distribution from a — s to a + s.) The result is that about % of the deviations fall in this range; conversely, of course, % fall out- side this range. This means that if the average value of a measurement is a, then by chance
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