Errors during the measurement process

3.5.2 Graphical data analysis techniques – frequency distributions

Graphical techniques are a very useful way of analysing the way in which random measurement errors are distributed. The simplest way of doing this is to draw a histogram, in which bands of equal width across the range of measurement values are defined and the number of measurements within each band is counted. Figure 3.5 shows a histogram for set C of the length measurement data given in section 3.5.1, in which the bands chosen are 2 mm wide. For instance, there are 11 measurements in the range between 405.5 and 407.5 and so the height of the histogram for this range is 11 units. Also, there are 5 measurements in the range from 407.5 to 409.5 and so the height of the histogram over this range is 5 units. The rest of the histogram is completed in a similar fashion. (N.B. The scaling of the bands was deliberately chosen so that no measurements fell on the boundary between different bands and caused ambiguity about which band to put them in.) Such a histogram has the characteristic shape shown by truly random data, with symmetry about the mean value of the measurements.

As it is the actual value of measurement error that is usually of most concern, it is often more useful to draw a histogram of the deviations of the measurements

from the mean value rather than to draw a histogram of the measurements them[1]selves. The starting point for this is to calculate the deviation of each measurement away from the calculated mean value. Then a histogram of deviations can be drawn by defining deviation bands of equal width and counting the number of deviation values in each band. This histogram has exactly the same shape as the histogram of the raw measurements except that the scaling of the horizontal axis has to be redefined in terms of the deviation values (these units are shown in brackets on Figure 3.5).

Let us now explore what happens to the histogram of deviations as the number of measurements increases. As the number of measurements increases, smaller bands can be defined for the histogram, which retains its basic shape but then consists of a larger number of smaller steps on each side of the peak. In the limit, as the number of measurements approaches infinity, the histogram becomes a smooth curve known as a frequency distribution curve as shown in Figure 3.6. The ordinate of this curve is the frequency of occurrence of each deviation value, F(D), and the abscissa is the magnitude of deviation, D.

The symmetry of Figures 3.5 and 3.6 about the zero deviation value is very useful for showing graphically that the measurement data only has random errors. Although these figures cannot easily be used to quantify the magnitude and distribution of the errors, very similar graphical techniques do achieve this. If the height of the frequency distribution curve is normalized such that the area under it is unity, then the curve in this form is known as a probability curve, and the height F(D) at any particular deviation magnitude D is known as the probability density function (p.d.f.). The condition that

the area under the curve is unity can be expressed mathematically as:

The probability that the error in any one particular measurement lies between two levels D1 and D2 can be calculated by measuring the area under the curve contained between two vertical lines drawn through D1 and D2, as shown by the right-hand hatched area in Figure 3.6. This can be expressed mathematically as:

articular importance for assessing the maximum error likely in any one measure[1]ment is the cumulative distribution function (c.d.f.). This is defined as the probability of observing a value less than or equal to D₀, and is expressed mathematically as:

Thus, the c.d.f. is the area under the curve to the left of a vertical line drawn through D0, as shown by the left-hand hatched area on Figure 3.6.

The deviation magnitude Dp corresponding with the peak of the frequency distri[1]bution curve (Figure 3.6) is the value of deviation that has the greatest probability. If the errors are entirely random in nature, then the value of Dp will equal zero. Any non-zero value of Dp indicates systematic errors in the data, in the form of a bias that is often removable by recalibration.