Random Error Sources

We can get a qualitative idea of the way small undetectable uncertainties produce a detectable random error in the following way. Imagine a situation in which just four small random errors combine to give an overall error. We will assume that each error has an equal probability of occurring and that each can cause the final result to be high or low by a fixed amount ±U.

Table 1.1 shows all the possible ways the four errors can combine to give the indicated deviations from the mean value. Note that only one combination leads to a deviation of +4 U, four combinations give a deviation of +2 U, and six give a deviation of 0 U.

Table 1.1

The negative errors have the same relationship. This ratio of 1:4:6:4:1 is a measure of the probability for a deviation of each magnitude. If we make a sufficiently large number of measurements, we can expect a frequency distribution like that shown in Figure 1.1a. Note that the y-axis in the plot is the relative frequency of occurrence of the five possible combinations. Figure 1.1b shows the theoretical distribution for ten equal-sized uncertainties.

Again we see that the most frequent occurrence is zero deviation from the mean. At the other extreme a maximum deviation of 10 U occurs only about once in 500 results. When the same procedure is applied to a very large number of individual errors, a bell-shaped curve like that shown in Figure 1.1c results. Such a plot is called a Gaussian curve or a normal error curve.

  Figure 1.1 Frequency distribution for measurements containing (a) four random uncertainties, (b) ten random uncertainties, and (c) a very large number of random uncertainties.