Errors during the measurement process

 

Gaussian distribution

Measurement sets that only contain random errors usually conform to a distribution with a particular shape that is called Gaussian, although this conformance must always be tested (see the later section headed ‘Goodness of fit’). The shape of a Gaussian curve is such that the frequency of small deviations from the mean value is much greater than the frequency of large deviations. This coincides with the usual expectation in measure[1]ments subject to random errors that the number of measurements with a small error is much larger than the number of measurements with a large error. Alternative names for the Gaussian distribution are the Normal distribution or Bell-shaped distribution. A Gaussian curve is formally defined as a normalized frequency distribution that is symmetrical about the line of zero error and in which the frequency and magnitude of quantities are related by the expression:

where m is the mean value of the data set x and the other quantities are as defined before. Equation (3.11) is particularly useful for analysing a Gaussian set of measurements and predicting how many measurements lie within some particular defined range. If the measurement deviations D are calculated for all measurements such that D – x - m, then the curve of deviation frequency F(D) plotted against deviation magnitude D is a Gaussian curve known as the error frequency distribution curve. The mathematical relationship between F(D) and D can then be derived by modifying equation (3.11) to give:

The shape of a Gaussian curve is strongly influenced by the value of , with the width of the curve decreasing as α becomes smaller. As a smaller α corresponds with the typical deviations of the measurements from the mean value becoming smaller, this confirms the earlier observation that the mean value of a set of measurements gets closer to the true value as α decreases.

 If the standard deviation is used as a unit of error, the Gaussian curve can be used to determine the probability that the deviation in any particular measurement in a Gaussian data set is greater than a certain value. By substituting the expression for F(D) in (3.12) into the probability equation (3.9), the probability that the error lies in a band between error levels D1 and D2 can be expressed as:

Solution of this expression is simplified by the substitution:

                                                     z = D/ α                                                                       (3.14)

The effect of this is to change the error distribution curve into a new Gaussian distri[1]bution that has a standard deviation of one (α = 1) and a mean of zero. This new form, shown in Figure 3.7, is known as a standard Gaussian curve, and the dependent

variable is now z instead of D. Equation (3.13) can now be re-expressed as:

Unfortunately, neither equation (3.13) nor (3.15) can be solved analytically using tables of standard integrals, and numerical integration provides the only method of solu[1]tion. However, in practice, the tedium of numerical integration can be avoided when analysing data because the standard form of equation (3.15), and its independence from the particular values of the mean and standard deviation of the data, means that standard Gaussian tables that tabulate F(z) for various values of z can be used.