Measurement and Measurement ErrorPHYS 1301 F99Prof. T.E. CoanVersion: 8 Sep ’99IntroductionPhysics makes both general and detailed statements about the physicaluniverse. These statements are organized in such a way that they provide a model or akind of coherent picture about how and why the universe works the way it does.These sets of statements are called “theories” and are much more than a simple list of“facts and figures” like you might find in an almanac or a telephone book. (Eventhough almanacs and telephone books are quite useful!) A good physics theory is farmore interested in principles than simple “facts.” Noting that the moon appearsregularly in the night sky is far less interesting than understanding why it does so.We have confidence that a particular physics theory is telling us somethinginteresting about the physical universe because we are able to test quantitatively itspredictions or statements about the universe. Indeed, all physics (and scientific)theories have this “put up or shut up” quality to them. For something to be called aphysics “theory “ in the first place, it must make quantitative statements about theuniverse that can be then quantitatively tested. These tests are called “experiments”.The statement, “My girlfriend is the most charming woman in the world,” howevertrue it may be, has no business being in a physics theory because it simply cannot bequantitatively tested. If the experimental measurements are performed correctly andthe observations are inconsistent with the theory, then at least some portion of thetheory is wrong and needs to be modified. Yet, consistency between the theoreticalpredictions and the experimental observations never decisively proves a theory true. Itis always possible that a more comprehensive theory can be developed which willalso be consistent with these same experimental observations. Physicists, as well asother scientists, give a kind of provisional assent to their theories, howeversuccessfully they might agree with experimental observation. Comfortably workingwith uncertainty is a professional necessity for them. Making careful quantitative measurements are important if we want to claimthat a particular physics theory explains something about the world around us.Measurements represent some physical quantity. For example, 2.1 meter represents adistance, 7 kilograms represents a mass and 9.3 seconds represents a time. Notice thateach of these quantities has a number like 9.3 and a unit, “seconds”. The number tellsyou the amount and the unit tells you the thing that you are talking about, in this caseseconds. Both the number and the unit are required to specify a measured quantity.1PrecisionThere is a certain inherent inaccuracy or variation in the measurements wemake in the laboratory. This inherent inaccuracy or variation is called experimental“error” and the word is not meant to imply incompetence on the part of theexperimenter. Error merely reflects the condition that our measuring instruments areimperfect. This lack of perfection in our measuring procedure is to be contrasted withmistakes like adding two numbers incorrectly or incorrectly writing down a numberfrom an instrument. Those mistakes have nothing to do with what I mean byexperimental error and everything to do with the competence of the experimenter!Understanding and quantifying measurement error is important in experimentalscience because it is a measure of how seriously we should believe (or not believe)our theories abut how the world works. If I measure my mass to be 120.317 kilograms, that is a very precisemeasurement because it is very specific. It also happens to be a very inaccuratemeasurement because I am not quite that fat. My mass is considerably less, somethinglike 85 kilograms. So, when we say that we have made a precise measurement we canalso say that we have made a very specific measurement. When we say we have madean accurate measurement we can also say that we have made a correct measurement.When we make measurements in the laboratory we should therefore distinguishbetween the precision and the accuracy of these measurements.The so-called number of significant figures that it contains indicates theprecision of a measurement. In our mass example, the quantity 120.317 kilograms has6 significant figures. This is rather precise as measurements go and is considerablymore precise than anything you will measure in this course. Again, the fact that ameasurement is precise does not make it accurate, just specific. In any event, it isuseful to be familiar with how to recognize the number of significant figures in anumber. The technique is this: 1. The leftmost non-zero digit is the most significant digit;2. If there is no decimal point, the rightmost non-zero digit is the least significant;3. If there is a decimal point, the rightmost digit is the least significant, even if it iszero.4. All digits between the least significant and the most significant (inclusive) arethemselves significant.For example, the following four numbers have 4 significant figures:2,314 2,314,000 2.314 9009 9.009 0.000009009 9.000No physical measurement is completely exact or even completely precise. Theapparatus or the skill of the observer always limits accuracy and precision. Forexample, often physical measurements are made by reading a scale of some sort(ruler, thermometer, dial gauge, etc.). The fineness of the scale markings is limitedand the width of the scale lines is greater than zero. In every case the final figure ofthe reading must be estimated and is therefore somewhat inaccurate. This last figuredoes indeed contain some useful information about the measured quantity, uncertainas it is. A significant figure is one that is reasonably trustworthy. One and only oneestimated or uncertain figure should be retained and used in a measurement. This2figure is the least significant figure. It is also important to understand approximatelythe magnitude of the uncertainty so that you can state it with the measurement.For example, a length measurement of 2.500 meters (4 significant figures) isone digit less uncertain that a measurement of 2.50 meters (3 significant figures). Ifyour measuring instrument is capable of 4 digit precision then you can distinguishbetween an item 2.500 meters long and one 2.501 meters long. If your instrumentwere precise to only 3 significant figures, then both items would appear to be