Measurements & Uncertainties

Measurements & Uncertainties Experiment AA Objective Uncertainties, random and systematic errors, precision and accuracy, noise, calibrations and In this experiment you will make measure- standards, signi¯cant ¯gures, propagation of ments and explore aspects of determining how errors, and linear regression are concepts and well those measurements were made. tools used to interpret measurements. In this ¯rst experiment, the measurement Introduction process itself is explored. The second experiment focuses on aspects of statistical analy- What justi¯es the faith scientists place in sis used in interpreting measurements and in the theories, mathematical models, or \laws" comparing measurements and predictions. they invent to describe natural phenomena? Ultimately, experimentation is the only way Measurements to establish the validity and importance of such constructions. Quantitative measure- A measurement is an attempt to discover a ments and mathematical descriptions are the quantity's ideal or \true" value. Generally, bricks and mortar with which science is built. the true value is unknown. We will use the A theory should make predictions about the subscript u (for unknown) to denote true val- outcomes of measurements. A set of measure- ues and the subscript m to denote measured ments should be able to test those predictions. values. Taking the length y of a carefully ma- The simplest question to ask is: Is the theo- chined rod as an example, yu would represent retical model in agreement with the measure- its true length and ym would represent a mea- ments designed to test it? If the answer is surement of its length. no, the theory has to be abandoned or at least A measurement ym made with a real instru- modi¯ed. If theory and experiment do agree, ment is never expected to give the exact, true the theory is not proven correct, it is simply value. The measurement might be a little low validated. The degree to which the theory is or it might be a little high. The di®erence ¢y validated depends on the quality of the mea- (¢ is the capital Greek letter delta) between surements used to test it. To know to what de- the true value and the measured value is the gree the theory is validated, one must consider measurement error. the quantities being measured, the measuring ¢y = yu ¡ ym (1) instruments, and the conditions under which the measurements were taken. One must also ¢y cannot be determined without already consider the theoretical model and the condi- knowing yu. However, knowing how big ¢y tions under which it is expected to be valid. might be is very important; it would allow AA 1 AA 2 Introductory Physics Laboratory 2.63 cm cation for estimating two extra digits because one is usually not certain that even the single estimated digit is correct. Suppose by critically examining the measurement process you estimate you could be 0 1 2 3 high or low by about 0.03 cm. The measure- cm ment would be recorded as (2:63 § 0:03) cm and indicates you believe the true value is probably between 2.60 and 2.66 cm. The Figure 1: Measuring the length of a rod with a meter stick. §0:03 cm is called the random uncertainty (or simply the uncertainty) of the measurement. It gives a range for the measurement error; ¢y you to determine a range of true values that is probably in the range from -0.03 to 0.03 cm. are consistent with the measured value. This We will use the symbol σy for such an esti- range could then be compared with theoretical mate of the uncertainty. (σ is the Greek letter predictions. sigma.) Thus, the notation (2:63 § 0:03) cm represents two numbers: ym = 2:63 cm and Random Errors; Uncertainties σy = 0:03 cm. Note that: Random errors are measurement errors arising in just about all measurements. They are just ² σy is not the measurement error ¢y; it is as likely to be positive as negative. When ran- an indicator of possible ¢y's. dom errors are likely to be small, one says the measurements have a high precision or they ² σy should not give the maximum possible are very precise. A very precise instrument is ¢y. There should be about 32% proba- capable of distinguishing very small changes bility (around 1 chance in 3) that the er- in the quantity it measures. ror ¢y will actually be outside the range between ¡σy and +σy. Consider a measurement ym of a rod using a simple meter stick (see Fig. 1). The scale's Why not make σy extra large and report smallest marked division is 0.1 cm. We line your measurement as (2:63 § 0:05) cm? Do- up one end with the 0 cm mark. (Though not ing so would give you a better chance of be- true in practice, to simplify the discussion we ing \within error" since you would be giving will assume this can be done perfectly.) Then, a larger range (2.58 to 2.68 cm) within which a reading is made at the other end. The ¯g- the true value can lie and still be in agree- ure shows the true length appears to be be- ment with your measurement. The only thing tween 2.6 and 2.7 cm. Since it's closer to the wrong with this argument is that when you 2.6 cm mark, you might report the length as specify an uncertainty you must also specify ym = 2:6 cm. As is obviously the case in this its level of con¯dence. The con¯dence level is example, one can and should estimate one ex- an estimate of the probability that the true tra digit. It would be reasonable to record the value will lie within the range speci¯ed by the length as ym = 2:63 cm | estimating the end measurement value and its uncertainty. Were is about three-tenths of the way between the you to use the larger range, you should be hon- marks. There is seldom, if ever, any justi¯- est and say that you are closer to 90 or 95% Rev. 1.2 Measurements & Uncertainties AA 3 con¯dent that the true value is in this range. In statistical analysis, if one does not explicitly specify the con¯dence level of an uncertainty, it is generally agreed to be 68% and this is the con¯dence level we will use in the physics laboratory. Eye It will not be critical that you estimate ex- Position actly the 68% con¯dence level for your uncer- Object tainties. We only point this out so you can appreciate why you should be realistic in your Scale uncertainty estimate and not choose an overly large value to improve your chances of being Figure 2: Parallax errors can be personal, random, or systematic errors. \in range." Assignment or estimation of the uncertainty depends on many factors and there is no one back and forth continuously. (High-precision \rule" which will work all the time. You balances often have windshields around them should be forewarned that the uncertainty in to minimize this problem.) These fluctuations reading the scale on a measuring instrument would have to be considered when determin- does not always determine the uncertainty as- ing the uncertainty of the mass measurements. sociated with making the measurement. When the noise level is high, it may become For example, you may need to take a read- the determining factor for the uncertainty of ing when some prescribed condition has been the measurement. If your measurements fluc- met. Such is the case with a beam balance. tuate, you should adjust the uncertainty ac- (See Fig. 4.) The object to be measured is cordingly. Whenever it is reasonable to do so, placed on a pan. Its mass is determined by the repeat the measurement several times. If the mass settings on the beam when the pointer readings fluctuate, the uncertainty of a single lines up properly. In such cases, the uncer- reading is approximately the size of the fluc- tainty might be taken as the amount by which tuations. you can change the setting before you can be fairly certain the condition is no longer met. Personal and Systematic Errors You may be able to read the beam balance to 0.1 g, but if you can change it by 0.5 g be- Besides random errors, there are two other fore there is any noticeable misalignment of kinds of measurement errors: personal errors the pointer, the uncertainty of the measure- and systematic errors. ment is 0.5 g, not 0.1 g. Personal errors arise from incorrectly us- Another factor that can contribute to uncer- ing or reading an instrument. They are of- tainty is noise. The quantity being measured ten called illegitimate errors or simply mis- or the conditions under which it is being mea- takes. A typical example might be a parallax sured may produce fluctuations that cannot be error incurred when lining up a marker with eliminated. For example, in measuring mass a scale. (See Fig. 2.) When you move your with a high-precision beam balance, random eye, the scale reading varies. You must know pu®s of air might cause the pointer to jiggle where your eye should be positioned to read Rev. 1.2 AA 4 Introductory Physics Laboratory the scale properly. (Sometimes there is a mir- Accuracy refers to the level of agreement ror on the scale to aid in determining where that can be expected between a measured your eye should be placed.) This kind of er- value and a standard.

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