Measurement Accuracy And Uncertainty

Accuracy, Errors, Uncertainties, Resolution, Precision

Measurement accuracy and uncertainty

Accuracy, Errors, Uncertainties, Resolution, Precision

Experimental errors

In any experiment, the recorded measurements will never be exactly the same as their true (actual) value, because no measurement system can be perfect. We use the word error to describe the difference between the recorded value and the underlying, true value (which we can never know exactly: this is a key principle of science which is important to remember). It doesn t necessarily mean any person made a mistake (that would be a human error ). Sometimes we use the word uncertainty instead, recognising that we cannot be certain what is the exact true value of what we measure.

Random and systematic errors

Experimental errors can be classed as random , or systematic . Random errors show no pattern their values may be smaller or bigger, and positive or negative (i.e. above or below the true value). Systematic errors are consistently to one side of the true value. Both kinds may be caused by the apparatus setup, the procedures used, or the measuring instrument itself. Experimental physicists go into great effort to identify and eliminate all possible sources of error. It is important to repeat measurements several times (at least 3, and preferably 5 times for experiments at A level), because this shows how consistent or repeatable they are. Random errors will show up as random variations in the measured value, and the range of values will give an indication of the uncertainty.

However systematic errors will not show up this way you cannot tell if all the readings happen to be too high (or too low) by some factor. Random errors can best be reduced by using instruments with better resolution and accuracy (these are described more fully in the paragraphs below).

Repeating the measurement and taking the mean average will also reduce the effects of random errors. If you are plotting a graph to show a trend, it is not necessary to repeat individual readings, because random errors will show up as variations from the line or curve of best fit, and the fitting process will average out these random deviations. If you can be confident about the pattern that the systematic errors follow, you may be able to apply a correction. Systematic errors often show up as a zero error in results all values are a fixed amount higher (or lower) than expected, and the instrument shows this same fixed value when it should be reading zero. Zero errors can easily be corrected by subtracting this fixed amount from each reading. Sometimes systematic errors cause measurements to be a fraction or multiple (like 95% or 106%) of the true value these are more difficult to correct, unless there is a reliable way to check the instrument against accurately known values (this is called calibration ) in this case a multiplying correction factor could be applied.

Single measurements

The accuracy and precision of measurements have special meanings in science and maths. When talking about a single number or measurement, the word precision refers to the number of digits quoted, or how fine the measurement is made, such as to the nearest centimetre or millimetre. This will depend on the resolution of the instrument, which means what is the smallest difference in values that the instrument can detect. But be careful: just because an instrument gives high resolution does not necessarily mean it has equivalent accuracy! The accuracy is the same as the error or uncertainty described above: it is the difference between the indicated value and the true value.

Sets of measurements

The accuracy of a measurement system (the equipment, and the set of measurements using it) is how close it gets to actual (true) value. The precision of a measurement system is the degree to which repeated measurements give the same results. A measurement system can be precise but not accurate, neither, or both. For example, if an experiment contains an error in the way it is done, then repeating measurements and averaging usually increases precision but does not improve accuracy. The end result would be a consistent, yet inaccurate, set of results from the flawed experiment. Eliminating the systematic error improves accuracy but does not change precision. In physics we are trying to find truth about the world through experiments. So to know if something is one size rather than another, or follows one particular law rather than some others, we must check very carefully that our measurements are precise (in both senses) and accurate.

Common measurement errors and how to reduce them

Timing errors

Timing errors arise because there is a delay between between some event (like an object falling) beginning and the timer starting a delay between the event ending and the timer stopping and these two delays are not exactly the same. This makes the measured duration different from the true duration. If the delays are consistent (start delay = stop delay) then the duration error will be lower, but if the start and stop delays are both long, they will probably be less consistent. Human reaction times for starting and stopping a timer manually are both long and inconsistent, so it is hard to time short durations accurately. If the event is repeated (for example, wave oscilations, or swings of a pendulum), the error can be reduced by timing several repetitions, then dividing to give the time per event. Automatic systems (like light gates which start and stop electronic timers, and multi-flash or multi-image photography) have delays which are very small (and often are more consistent in relation to their length).

Parallax errors

Parallax errors are when the viewing angle changes the apparent measurement value. They can be reduced by putting the measurement scale very close to the thing being measured. Fiducial markers, (any markers that show the exact position of something) can help: for example in the right-hand picture, cardboard triangles have been attached to the spring so their tips are close to the ruler.

Meniscus errors

When reading the level of liquid, always sight along the bottom of the meniscus this is where almost all the volume of liquid lies, and the top of the meniscus also varies with small differences in surface tension (the attraction between the liquid molecules and the container surface).