Errors during the measurement process

Standard Gaussian tables

A standard Gaussian table, such as that shown in Table 3.1, tabulates F(z) for various values of z, where F(z) is given by:

Thus, F(z) gives the proportion of data values that are less than or equal to z. This proportion is the area under the curve of F(z) against z that is to the left of z. There[1]fore, the expression given in (3.15) has to be evaluated as [F(z₂) – F(z₁)]. Study of Table 3.1 shows that F(z) = 0.5 for z = 0. This confirms that, as expected, the number of data values
 0 is 50% of the total. This must be so if the data only has random errors. It will also be observed that Table 3.1, in common with most published standard Gaussian tables, only gives F(z) for positive values of z. For negative values of z, we can make use of the following relationship because the frequency distribution curve is normalized:

 

                                                        F(-z) - 1 - F(z)                                                              (3.17)

(F(-z) is the area under the curve to the left of (-z), i.e. it represents the proportion of data values ≤ -z.)

Example 3.3

How many measurements in a data set subject to random errors lie outside deviation boundaries of +α and - α, i.e. how many measurements have a deviation greater than j α j? Solution The required number is represented by the sum of the two shaded areas in Figure 3.8. This can be expressed mathematically as:

(This last step is valid because the frequency distribution curve is normalized such that the total area under it is unity.)

Thus

                                    P[E < - α] + P[E > + α] = 0.1587 C 0.1587 = 0.3174 ~ 32%

i.e. 32% of the measurements lie outside the š boundaries, then 68% of the measure[1]ments lie inside.

The above analysis shows that, for Gaussian-distributed data values, 68% of the measurements have deviations that lie within the bounds of š. Similar analysis shows 

that boundaries of š2 contain 95.4% of data points, and extending the boundaries to š3 encompasses 99.7% of data points. The probability of any data point lying outside particular deviation boundaries can therefore be expressed by the following table.