2.3 The Gaussian or Normal Distribution
The Gaussian or normal distribution plays a central role in all of statistics and is the most ubiquitous distribution in all the sciences. Measurement errors, and in particular, instrumental errors are generally described by this probability distribution. Moreover, even in cases where its application is not strictly correct, the Gaussian often provides a good approximation to the true governing distribution.
The Gaussian is a continuous, symmetric distribution whose density is given by
The two parameters µ and ^{2} can be shown to correspond to the mean and variance of the distribution by applying (8) and (9).
Fig. 3. The Gaussian distribution for various . The standard deviation determines the width of the distribution. |
The shape of the Gaussian is shown in Fig. 3 which illustrates this distribution for various . The significance of as a measure of the distribution width is clearly seen. As can be calculated from (19), the standard deviation corresponds to the half width of the peak at about 60% of the full height. In some applications, however, the full width at half maximum (FWHM) is often used instead. This is somewhat larger than and can easily be shown to be
This is illustrated in Fig. 4. In such cases, care should be taken to be clear about which parameter is being used. Another width parameter which is also seen in the Literature is the full-width at one-tenth maximum (FWTM).
Fig. 4. Relation between the standard deviation a and the full width at half-maximum (FWHM). |
The integral distribution for the Gaussian density, unfortunately, cannot be calculated analytically so that one must resort to numerical integration. Tables of integral values are readily found as well. These are tabulated in terms of a reduced Gaussian distribution with µ = 0 and ^{2} = 1. All Gaussian distributions may be transformed to this reduced form by making the variable transformation
where µ and are the mean and standard deviation of the original distribution. It is a trivial matter then to verify that z is distributed as a reduced Gaussian.
Fig. 5. The area contained between the limits µ ± 1, µ ± 2 and µ ± 3 in a Gaussian distribution. |
An important practical note is the area under the Gaussian between integral intervals of . This is shown in Fig. 5. These values should be kept in mind when interpreting measurement errors. The presentation of a result as x ± signifies, in fact, that the true value has 68% probability of lying between the limits x - and x + or a 95% probability of lying between x - 2 and x + 2, etc. Note that for a 1 interval, there is almost a 1/3 probability that the true value is outside these limits! If two standard deviations are taken, then, the probability of being outside is only 5%, etc.