Our statistical analyses are based on the concept that if we measure a sample multiple times, then the results would represent some known distribution. For most of the measurements we make in safeguards, this is a Gaussian (or Normal) Distribution. The Gaussian distribution is given by:

where μ is the mean for the distribution, σ is the standard
deviation of the distribution, and *x* is any measured value. This
distribution can be used to determine the probability that any
additional measurement will fall within some interval from *x _{1}* to

Since the measured value must fall somewhere between -∞ and +∞, we can say that

or that the probability that the next measurement will fall between -∞ and +∞ has to be equal to unity (i.e., it is absolutely certain that it will).

It is convenient to define the cumulative PDF given by

This function is equal to 0 when x=-∞, and it is equal to 1 when
x =+∞. A plot of *p(x)* and *P(x)* for a Gaussian distribution is
shown in Figure 7.

The distribution of measured values considered in the previous section could be well approximated by a Gaussian distribution. Figure 8 shows the frequency plot of measured values (for the higher uncertainty set of 155 measured values) and the Gaussian distribution on the same plot. As can be seen, they match up rather well. The Gaussian distribution was normalized such that the integral of the distribution over the whole domain is equal to the total number of measurements performed.

Using a Gaussian distribution to represent the data allows us to make a number of useful conclusions about the measurement system. The distribution is such that if we make another measurement, we expect that there is a 68% chance that it will fall in the interval (μ-σ,μ+σ). There is a 95% chance that it will fall in the interval (μ-2σ,μ+2σ) and there is a 99.7% chance that it will fall in the interval (μ-3σ,μ+3σ).

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