Firstly, I am not a mathematician. If someone like St.B, with much greater mathematical ability than I, would compile a more informed version of this post, I'd be very grateful
First point: Nothing is absolute. All measurements have errors. Some measuring devices (usually expensive, cumbersome and large) have extremely small errors. Some (usually cheap, small , simple) have much larger errors. The endless advance of technology usually means that cheaper, simpler, more accurate measurement devices supercede older designs.
Second point: Not all errors are equal.
Third point: It's important to distinguish between accuracy and precision. Both are important but very different. The layman usually uses these terms indiscriminately, and thus causes confusion to himself and others
AccuracyFirst we have absolute errors in measurements. These are errors caused by incorrect calibration. An inaccurate reading will always be inaccurate, regardless of how many times the reading is taken. Every reading will have an absolute error to a certain extent. These errors relate to ACCURACY. Readings which are consistently high or consistently low are known as INACCURATE readings. Scientists, and manufacturers of measurement devices often define the extent of inaccuracy. Inaccuracy is often the effect of manufacturer quality control. For example, a thermometer manufacturer may state that a particular type of thermometer is +/- 2% accuracy. Meaning that the reading may be consistently high or low to an extent equivalent to no more than 2%. The manufacturer's spec is usually based on a statistical (yawn!) sampling method, and might, for instance indicate that 95% of all the products will conform to the range indicated (but a large proportion, eg 50% may be much more accurate).
Accuracy errors may be generated by 1)manufacturer's tolerance ie the measuring equipment 2)method in which the equipment is employed/installed 3)environmental effects (eg variations in temperature/pressure/humidity/light levels/vibration levels etc may affect the accuracy of the sensor/measuring device).
Therefore, accuracy errors are not necessarily constant, merely constant under a standard set of conditions.
Measurement accuracy depends entirely on the accuracy of the instrument calibration. I like to think of everything being as simple as a straight line graph - defined by the equation y=(m * x) + c. There are two areas for inaccuracy in this equation. The multiplier 'm' and the constant 'c'. If the multiplier is not 100% accurate, then the calibration line will deviate from the true reading. If the slope of the lines (true reading vs calibration) do not match perfectly, then the higher the reading, the greater the level of inaccuracy. So high readings will be inaccurate, whereas low readings will be relatively much more accurate, even if the error in the slope (m) is quite large. Conversely, errors in the constant 'c' will make very little difference with high readings, so as to become almost negligible. However, at low readings, any error in 'c' turns into a large error in the accuracy of the reading. So if you are measuring temperature, and you have an error of 1C in the 'c' constant, then (assuming the slope m is perfectly calibrated), you will have a 1% error in a reading of 100C, a 10% error in a reading of 10C, a 100% error in a reading of 1C and a 1000% error in a reading of 0.1C!
PrecisionPrecision is usually of much more interest to scientists. It is basically a measure of reproducibility. If you take 10 measurements of the same thing, you would expect to have 10 identical readings. In practice, you will not. There are many effects - eg noise which affect the reproducibility of the reading. This variation is known as Precision. A very good measuring instrument has very high levels of Precision. High levels of Precision lead to a high level of confidence in the readings. A low level of Precision correspondingly leads to a low level of confidence. However, here's the weird thing about Precision. The more readings you take, the more reliable the average becomes. So it is common in sensitive scientific experiments to compensate for limitations of measurement precision by taking multiple readings. For very tiny measurements, scientists sometimes average the results from hundreds or millions of readings to get a meaningful result. The study of precision is very much in the realm of statistics (yawn), so best left to mathematicians and scientists.
Precision and Accuracy errors may vary across the measurement range
In fact, they usually do. For example, a voltmeter may read from 0 - 100V. If you read a voltage of 20V, you may get a reading of 20.1V on one occasion, 20.0V on another and 19.9V on another occasion - The accuracy would be spot on, whereas precision is far from perfect. However, precision is directly related to another concept - background noise. The higher the level of background noise, or to put it another way, the closer the reading is to background noise, the worse the precision.
Background Noise
As you try to take ever-smaller readings, eventually, your readings will descend into the background noise. Instruments (and the design of the set of conditions under which the readings are taken) will dictate at what level the readings become meaningless,as they are overwhelmed by background noise. Highly sensitive scientific equipment is often cooled to very low temperatures to minimise the effects of thermal noise (often on the electronics of the instrument), thus achieving a much higher level of sensitivity than conventional instruments. It is often possible to make meaningful readings of very weak signals by taking many readings thus improving the precision.
Limit of Sensitivity
For any instrument, there exists a useful range throughout which readings can be taken. Most measuring instruments demonstrate working ranges of 2 - 3 decades of measurement. In the past, I have worked with instruments that were able to respond linearly to 6 decades of measurement - something quite unusual. At the lower end of the measurement range we reach a level where the signal disappears into the sea of background noise. This is known as the limit of sensitivity.
Precision Profiles
If you were to take many readings of a variety of measurements across an instrument's working range, you could empirically calculate the precision at different levels. You can also calculate precision using some fairly tedious mathematics (to me, anyway). Precision profiles seem mainly to be of interest to medical professionals - I guess this is because typically there is an inherently low level of precision in the measurements they tend to make, however the concept is valid for all measuring instruments, and is extremely revealing. It also explains why readings can be very misleading at the low end of the working range, even for relatively accurate/precise insturments. Here's a typical precision profile. Note that if as the curve is extrapolated to zero, the precision tends to infinity.
And if you choose to ignore everything above, at least remember this one: Errors multiply and accumulate ('compound')
If you have make a measurement which has a number of areas where errors may be introduced, then these errors multiply. For example, you may be measuring a temperature using a thermocouple attached to a pipe.
Error caused by poor thermal contact with pipe = 10%
Quantum error in the voltage generated by the thermocouple = 1%
Error caused by unaccounted-for resistance of cable = 5%
Calibration error of instrument reading the sensor = 10%
Precision error of reading = 5%
All relatively low errors, but the total (compounded) error of the reading is 34.7% (ie 1.10 x 1.01 x 1.05 x 1.10 x 1.05) (NB See St.B's comments below - I've got this slightly wrong)
Significant FiguresScientifically, it is normal to state your errors when stating a measurement eg 100V +/- 1%. It is also conventional to write readings in a manner which suggests their level of confidence. eg 106V (written to three significant figures) would suggest an error not exceeding +/-1V, but a reading of 106.000V (six significant figures) would suggest that it is measured to a confidence of a thousandth of a Volt.
