Resolution enhancement by dithering

The evaluation of uncertainties and errors is an important topic in introductory physics and chemistry laboratory courses. Errors are usually classified into two groups depending on their source: systematic or bias errors, and random or statistical errors. The first group includes, for instance, miscalibrated scales or bad zero settings. They can be avoided if the meter is checked against a standard, or realistically, against another and hopefully better meter. Random errors may be due, for example, to the inability of the observer or apparatus to determine a quantity exactly ~usually called instrumental uncertainty!, to the fact that the quantity being measured is not precisely defined, to the essentially random nature of the phenomenon, or to fundamental limits of measurement ~for example, the uncertainty principle!. The result of these kinds of errors is a spread of readings on repetition of the measurement under apparently the same conditions. This spread can be used to estimate the inaccuracy or uncertainty of the observation due to random errors. This kind of errors is usually reduced when the experiment is repeated many times and they can be analyzed by using statistical techniques. It also is important to verify that the fluctuations observed in the measurements are random, and not biased or correlated in time. Note that when we talk about errors, we do not include‘‘mistakes,’’ such as the misreading of a scale, the incorrect recording of a number or a mistaken calculation. In some cases each measurement is scale limited, that is, its statistical uncertainty is smaller than the smallest increment we can read on the instrument scale. Such a measurement will yield exactly the same value for repeated measurements of the same quantity. It is usually stated that in these cases, the uncertainty must be quoted as the smallest increment that can be read on the scale ~or some fraction of it!. Most of the instruments that are used in experiments involve a discretization of the measured quantity. This is true, not only for common instruments like rulers, where the measurement is limited by a minimum division, but also for more sophisticated instruments in which analog to digital conversion is involved and a least significant bit ~LSB! always exists. In other cases, where statistical fluctuations are observed in the measured data, it is said that the statistical uncertainty can be reduced to the bound given by the smallest increment that can be read on the scale. This bound is due to the fact that the total error is the square root of the sum of the squares