Quantitative evaluation is central to scientific investigation (some would even say the quantitative nature of the investigation is the defining aspect of a science). The result of any measurement process is a value for the parameter measured. This is obvious, but what is often overlooked is that there is some uncertainty in the measurement process and thus some uncertainty in the value reported. There is a certain irony in the fact that chemists (and all scientists) who are involved in quantitative investigations must be so concerned about uncertainty!

By way of example, consider the measurement of the length of a ball point pen. Certainly there is one and only one correct length of the pen. This correct length is the true value. The task before us is to obtain this true value.

Perhaps a likely approach would be to use a ruler graduated in centimeters (Figure 1a), but would such an operation give the true value? An alternative would be to use a ruler which was marked in millimeters (Figure 1b). Is the true value obtained with this device? An even more sophisticated method would be to employ a caliper, a device capable of measuring to the nearest tenth of a millimeter (Figure 1c). Certainly the point is clear. Even though a more sophisticated measuring device can provide a better estimate of the length of the pen, it is not possible to determine the true value of the length.

Figure l. Measurement of Pen

As was seen in the previous example, it is quite unlikely that in the measurement process, the marks on the measuring instrument coincide exactly with the dimensions of the ball point pen. It is much more common for the quantity being measured to fall between two of the marks on the measuring device. In such a situation it is necessary to estimate the fractional distance between the marks. In fact, to fail to make such an estimate would be to under-represent the measuring process. When a centimeter ruler is used as in Figure 1a, one can say with good confidence that the pen is closer to 12.5 cm in length than it is to 13 cm. One has estimated between the marks to the nearest 0.1 cm.  In Figure 1b, one could say that the length is closer to 12.46 cm than it is to 12.50 cm, and so on (estimating to the nearest 0.1 mm). In these two situations, the choice of the measuring system employed (rulers marked off in centimeters or millimeters) defines something fundamental about the measurement process. Further, the manner in which the data are reported (12.5 or 12.46) depends upon the capabilities of the measurement system.  When reading any analog device, always estimate and record one more significant figure than the smallest marked division.

Many authors divide numbers into two types - exact and inexact. Exact numbers are those derived from counting or definition. For example, the number of students in a class is found by counting, and there is no uncertainty in the value reported by the Registrar (ignore for the moment any uncertainty in the definition of a student). Likewise there is no confusion that a dozen eggs will contain twelve eggs.

Inexact numbers, on the other hand, are the result of some measurement process. As was discussed above, the degree of uncertainty in the measurement depends upon the nature of the measurement process. In Figure 1a the value reported would have been 12.5 cm, and the uncertainty would have been one or two tenths of a centimeter. For the case shown in Figure 1 b, 12.46 cm would be reported, and the uncertainty would have been about 0.01 cm. The measurement process affects the magnitude of the uncertainty. It is often assumed that there is an uncertainty of plus or minus 1 unit in the last reported digit of a measured number.  The uncertainty in a measurement or scatter in a set of replicate measurements is described in terms of the precision of the measurement.  High precision means low uncertainty or low scatter.  The next question is how do we communicate this uncertainty unambiguously.

Precision may be expressed quantitatively by means of the relative average deviation, which is calculated as follows.  First, the mean,  xavg , of the individual results is calculated:

            xavg  = (x1 + x2 + x3 + ...)/n                                                                                                     (2)

 where xi are the individual results and n is the total number of results.  The relative average deviation is then calculated:

                             [|x1- xavg |  +  |x2- xavg |  +  |x3- xavg |  + ...]
           r.d.    =                                n ·  xavg                                        x 100%                                       (3)

 Note that the individual deviations |xi -  xavg| are expressed as absolute values, i.e., positive values,  in the numerator (this is indicated by the vertical bars).  A small percentage for the relative average deviation indicates very low scatter, i.e., good precision.

 Example: A data set is composed of the four values 3.4, 3.5, 3.8, and 3.9.  The mean value is 3.7.  The relative average deviation is

                    [0.3 + 0.2 + 0.1 + 0.2]
  r.d.    =                 4 x 3.7                    x 100%        =     5.4%                                                               (4)

Thus, the result could be reported as 3.7 ± 5.4%.

Error and Accuracy

Accuracy refers to the comparison of a measured value with the "true" value.  Low error means that the experimental value corresponds very closely with the true value.  The absolute error is the difference between the measured value and the true value, X(experimental) - X(true), which has units and a sign.  Usually one presents the relative error, which is calculated by dividing the absolute error by the true value and then multiplying by 100.  This is a percentage, with no units.

Note that accuracy and precision are not necessarily related.  It would be possible to have several highly scattered results (low precision), but by chance have the average value fall very close to the true value (high accuracy).  It would also be possible to have several closely-spaced results (high precision), yet, due to a calibration error in an instrument, have all of them be very far from the true value (low accuracy).  Of course, the ideal is to have both high precision and high accuracy.
Last revised 09/04/2006