b a Ratio of deviation x0 to standard deviation a. The standard deviation a is equal to aw, the width parameter of the distribution.

Probability of occurrence of deviation greater than ±x0.

m fluctuation would occur. Such a fluctuation in a single measurement (trial) would be a miracle!

For large values of the mean m, the Poisson distribution approaches in shape the central part of the normal distribution if the width parameter aw of the latter is set to aw = m1/2. (We do not demonstrate this formally.) Poisson and normal distributions are shown in Fig. 8 for a large mean value, m = 100. The normal curve with aw = m1/2 approximates well the Poisson distribution. (In the figure we set aw = a.) The normal distribution thus describes the arrival of random events for large m when aw = m1/2. This then is one example of the physical utility of the normal distribution.

Variance and standard deviation

The width of a measured distribution indicates the range of values obtained from a set of individual measurements of x. This width may be characterized formally as the root-mean-square deviation, that is, as the square root of the average of the square of the deviations from the mean value. This parameter a is called the standard deviation; its square a2 is called the variance,

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