Experimental Uncertainties 1 Measurements of any physical quantity can never be exact. One can only know its value with a range of uncertainty. If an experimenter measures some quantity X, the measurement must be written with an uncertainty: X ± ΔX. This expresses the experimenter s judgment that the true value of X lies somewhere between X - ΔX and X + ΔX. I. Uncertainties of Measurements 1. Instrumental uncertainty Every measuring instrument has an inherent uncertainty that is determined by the precision of the instrument. This value is taken as a half of the smallest increment of the instrument scale. For example, if the smallest mark on a ruler is every 1 mm, then 0.5 millimeter is the precision of a ruler. What is the logic of this? Take a look at the two examples below: A measurement of 73 mm Where is the location of the end of the pencil on the ruler? If the smallest unit on the ruler is 1 mm, then we would say that the end is at 73 mm. If the end was further right than 73.5 mm, we would round up to 74 mm since that would be the closest mark on the ruler 70 mm 80 mm 73.5 mm A measurement of 74 mm In this case the end of the pencil is past the middle in between 73 mm and 74 mm. Thus when reading location of the end of the pencil on the ruler, we would round our measurement up to 74 mm since it is the closest mark on the ruler 70 mm 80 mm 73.5 mm
What this means is that if somebody tells you that they measured the end of the pencil at 73 mm, that means that the end could have been located anywhere between 72.5 mm and 73.5 mm. In other words, their measurement (with accompanying uncertainty) was (73 ± 0.5) mm. This is why you can always estimate instrumental uncertainty by taking the smallest unit of the measuring instrument and dividing it by two. So for example, a stopwatch measures time in hundredths of a second. If I measure a time of 10.04 seconds for somebody to run the 100m sprint, then my uncertainty is ± 0.005 seconds, or more clearly written: the measured time is (10.04 ± 0.005) seconds. Instrumental uncertainties are the easiest ones to estimate, but unfortunately they are not the only source of the uncertainty in your measured value. You must be a skillful and lucky experimentalist to get rid of all other sources and to have your measurement uncertainty equal to the instrumental one. 2. Random uncertainty Sometimes when you measure the same quantity, you can get different results each time you measure it. That happens because different uncontrollable factors affect your results randomly. This type of uncertainty, random uncertainty, can be estimated only by repeating the same measurement several times. For example, suppose your lab partner is dropping a ball from 2 m above the floor. You are trying to measure the time for a ball to fall and hit the ground from the moment your lab partner released it using a stopwatch that measures to a precision of 0.01 seconds. You repeat the measurement three times and get the following results: 0.63 seconds, 0.71 seconds, 0.55 seconds. These results are quite possible since your reaction time is about 0.1 seconds. To estimate the random uncertainty you first find the average of your measurements: t =(0.63 s + 0.71 s + 0.55 s) / 3 = 0.63 s You then estimate approximately how much the values are spread with respect to this average in this case we have a spread of about Δt = 0.08 s. That is, our time measurement was t = (0.63 ± 0.08) s. Notice something very interesting here: the random variation in your timing (0.08 s) is much larger than the instrumental uncertainty of the stopwatch (0.005 s). This means that the instrumental uncertainty of the stopwatch is irrelevant to estimating the uncertainty in your measured value! You simply forget about instrumental uncertainty altogether in a case like this. You uncertainty estimate is governed entirely by the random uncertainty inherent in your ability to start and stop the stopwatch precisely. Through multiple trials of the same measurement, you can find the average value and estimate the random uncertainty by looking at the spread of your measurements about that average. 3. The Effect of Assumptions Assumptions inherent in your model may also contribute in uncertainty. For example, an uneven surface may change the speed of a moving car, or energy loss during a calorimetry experiment may cause the measured temperature to drift. Repeating the measurement will not let you get rid of such effects. This type of uncertainty is not easy to recognize and to evaluate. First of all, you have to determine the sign of the effect, i.e. whether the assumption increases the measured value, decreases it, or affects it randomly. Then you have to try to estimate the size of the effect. 2
It is difficult to give strict rules and instructions on how to estimate uncertainties in general. Each case is unique and requires a thoughtful approach. Be ingenious and reasonable. For example, you measure the diameter of the baseball assuming it is a perfect sphere. However, the real size of the ball may differ by 1 or 2 mm if you measure in different dimensions. This difference will determine the uncertainty of your measurement. 3 II. Comparing uncertainties What we have done so far is to discuss two methods by which you could estimate the uncertainty in a measurement. (The assumptions method is a third method, if you like.) If you (lab group A) measure your ball drop time as t A = (0.63 ± 0.08) s, what you have a number that tells you the uncertainty in your measurement (namely 0.08 s). We will call this the absolute uncertainty in your time measurement. Suppose another lab group (lab group B) measured t B = (0.82 ± 0.09) s in their experiment because they dropped their ball from 2.5 m as opposed to 2m in your experiment. By looking at the absolute uncertainty for two quantities (Δt A = 0.08 s and Δt B = 0.09 s) you cannot immediately decide which quantity is more uncertain. This is because the magnitudes of the measured quantities are different. How can we decide which quantity has a larger uncertainty? We need to compare their relative uncertainties. The relative uncertainty of a measurement is the ratio of the absolute uncertainty and the quantity itself. In other words, the relative uncertainty in t A is δt A = Δt A /t A = 0.127. This may be expressed as a fraction, or as a percentage by multiplying the ratio by 100% (12.7%). Likewise the relative uncertainty in t B is δt B = Δt B /t B = 0.11, or 11%. Thus, even though t B has a larger absolute uncertainty it is a more precise measurement than t A because it has a smaller relative uncertainty. To help you understand this rather strange idea, consider the two circles in the figure to the right. Each of them has exactly the same fuzzy edge (9 units of blur) but the larger circle looks sharper. Why? The diameter D of the larger circle is about 90 units (arbitrary units) and on the same scale the smaller circle has a diameter of about 30- units. The absolute uncertainty ΔD is the same for each circle, about 9 units. However, the relative uncertainty δd = ΔD/D is about 10% for the large circle and about 30% for the small one. This
should give you some sense of why relative uncertainty is a better indicator of the uncertainty of the measured quantity than its absolute uncertainty. Note: Common sense and good judgment must be used in representing the uncertainties when stating a result. Consider a temperature measurement with a thermometer known to be reliable to ± 0.5 degree Celsius. Would it make sense to say that this causes a 0.5% uncertainty in measuring the boiling point of water (100 degrees) but a whopping 10% in the measurement of cold water at a temperature of 5 degrees? Of course not! (And what if the temperatures were expressed in degrees Kelvin? That would seem to reduce the relative uncertainty to insignificance!). However in most calorimetry tasks, the value of interest is not temperature itself but only the change of the temperature or the temperature difference. III. Reducing Uncertainties The example with the circles shows a way to reduce relative uncertainty in a measurement. The same absolute uncertainty yields a smaller relative uncertainty if the measured value is larger. Suppose you have a block attached to a spring and want to measure the time interval for it to oscillate up and down, back to its starting position. If you use a watch that displays time in seconds to measure the time interval, the absolute uncertainty of the measurement is about 0.5 s. If you now measure a single time interval of 5 s, you get a relative uncertainty of 10%: [(0.5 s/5 s)*100]. Suppose you measure the time interval for 5 oscillations instead and you measure 25 s. The instrumental uncertainty is still 0.5 s! The relative uncertainty in your measurement of the time interval is now: time interval relative uncertainty = (0.5 s/25 s) *100% =2% By measuring a longer time interval (five oscillations instead of one), you have reduced the uncertainty in your time interval measurement by a factor of 5! Of course you should not forget about the obvious way of reducing relative uncertainties by minimizing absolute uncertainty with a better design, decreasing the effect of assumptions, or increasing the accuracy of instrument if it is possible. IV Finding the Uncertainty in a Final Calculated Value 1. Why do you need to know uncertainty? How can you tell if a value that you calculated from experimental measurements agrees with a predicted value? How can you tell if the data you have collected fits a physical model? If you found the same quantity two different ways by two separate experiments, how can you tell if the two measured values agree with each other? You cannot answer these questions without considering the uncertainties of your measurements and how those uncertainties affect a final value that you calculate using those measurements. Indeed, are the values of two quantities the same if the difference between them is smaller than the uncertainty in their measurements? Thus, to make judgment about two values X and Y, you have to find the ranges where these values lie. If the ranges X ± ΔX and Y ± ΔY overlap, you can claim that the values X and Y agree within your experimental uncertainty. 4
If the two values X and Y do not agree to within their experimental uncertainties, we say they are different. 2. Measured and Calculated Quantities Suppose you want to determine the uncertainty in the final value of a quantity that is calculated from several measured quantities. The uncertainties in these measured quantities propagate through the calculation to produce uncertainty in the final result. Consider the following example. Suppose you know the average mass of one apple m with the uncertainty Δm. If you want to calculate the mass M of the basket of 100 apples, you will get the value M ± ΔM = 100 m ± 100 Δm. The relative uncertainty of calculated value of M remains the same as the relative uncertainty of the single measured value for m ΔM / M = Δm / m. If you have more than one measured quantity, estimating uncertainty becomes a bit more complicated. The way we will handle it is with the weakest link rule. 3. Weakest link rule The percent uncertainty in the calculated value of some quantity is at least as great as the greatest percentage uncertainty of the values used to make calculation. Thus to estimate uncertainty in you calculated value, you have to: 1. Estimate the absolute uncertainty in each measured quantity used to find the calculated quantity. 2. Calculate the relative uncertainty in each measured quantity. 3. Pick the largest relative uncertainty. We call this largest relative uncertainty the weakest link. 4. We say that the relative uncertainty in our calculated value is equal to the weakest link (the largest relative uncertainty in our measured values). We can then apply the relative uncertainty of the weakest link to the calculated quantity to determine its absolute uncertainty. Here s an example: You ve been asked to estimate the volume of your laptop computer. First, you measure the length, width, and thickness with a meter stick (which has an absolute uncertainty of 0.05cm) Measurement Value (with absolute uncertainty) Relative uncertainty Length (39.4 ± 0.05) cm 0.05 cm 39.4 cm =1.27 "10#3 = 0.127% Width Thickness X-!X [ X X-!X [ X+!X ] X (28.7 ± 0.05) cm (4.3 ± 0.05) cm X+!X [ ] ] Y-!Y 0.05 cm 28.7 cm =1.74 "10#3 = 0.174% 0.05 cm 4.3 cm =11.6 "10#3 =1.16% Y Y-!Y Y+!Y Y [ ] Y+!Y 5
6 From this table, you can see that the thickness has by far the largest relative uncertainty - the thickness measurement is our weakest link! The volume of the laptop is V = LWT = (39.4 cm)(28.7 cm)(4.3 cm) = 4862 cm 3 Since the thickness measurement has the largest relative uncertainty (1.16%) we say this is the relative uncertainty in our final calculated volume V. To determine the absolute uncertainty of our calculated volume, we multiply the volume by the relative uncertainty of the weakest link:!v = (4862 cm 3 )(1.16 "10 #2 ) = 56 cm 3 So, the final estimate for the volume of the laptop is V = (4862 ± 56) cm 3. 3. Comparable uncertainties If a final calculated value depends on several measured quantities that each has comparable relative uncertainties, then the rules are more complicated: they depend on the type of the mathematical relationship you use for the calculation. Keep in mind, however, that the overall relative uncertainty cannot be less than the relative uncertainties of the independent measured quantities. In other words, no matter what the relationship is, the relative uncertainty can only increase when doing calculations. Consider a simple example first: Let a calculated value C be the produce of two measured quantities, A and B. In other words: C=AB. Now let s rewrite A and B to include their " uncertainties: A ±!A = A 1±!A % $ ' = A(1±!A). Likewise, B ±!B = B(1±!B). Now we can # A & estimate the product of A and B as AB(1±!A)(1±!B) = AB(1±!A ±!B ±!A!B). If we assume that the relative uncertainties δa and δb are much smaller than 1, then we can neglect the δaδb term and the following statement is approximately true: C(1±!C) = AB(1±!A ±!B) Thus the relative uncertainty δc in the measured quantity C is approximately:!c =!A +!B. The relative uncertainty of a product of measured values is approximately equal to the sum of the individual relative uncertainties if those relative uncertainties are roughly equal. A useful consequence of this is that if C=A 2 =A.A, then:!c =!A +!A = 2!A For instance, if you measure the sides of a square with an uncertainty of 3%, the resulting uncertainty in its area would be 6%. On a side-note, we can observe something very interesting if we do the same analysis in terms of absolute uncertainties instead of relative uncertainties:!c " C #!C = AB(!A +!B) " B #!A + A #!B In other words,!(ab) = B "!A + A "!B, which very much resembles the product rule for derivatives! Here is a trickier example. Suppose you measured the period of a pendulum with an uncertainty 3% and want to calculate the pendulum s frequency. What would be its percent uncertainty? First recall that f=1/t. Now as the period is T(1±δT), the estimate for the frequency will be
1 1 (1 δt ) = f (1 δt ) T(1 ± δt ) T, as you might remember from calculus. Although the sign of the percent uncertainty is reversed on the righthand side, it is practically the same relative uncertainty! (δf = δt) Thus the frequency will also be determined with an uncertainty of 3%. In general, the percent uncertainty of a measured quantity equals the percent uncertainty of its calculated inverse. 7 IV Summary When you are doing a lab and measuring some quantities to determine an unknown quantity: Decide which factors affect your result the most. Wherever possible, try to reduce the effects of these factors that cause uncertainty. Wherever possible, try to reduce uncertainties by measuring longer distances or time intervals, etc. Decide the absolute uncertainty of each measurement. Then, find the relative uncertainty of each measurement. If one relative uncertainty is much larger than the others, you can ignore all other sources and use this uncertainty to write the value of the relative uncertainty of the quantity that you are calculating. If relative uncertainties of several measured quantities are comparable, the overall relative uncertainty can be computed as follows: o Multiplication by a constant does not affect the relative uncertainty o When two or more measured quantities are multiplied (or divided), the relative uncertainties should be added up. In particular, when raising a measured quantity to a power (e.g., n=3), the percent uncertainty should be multiplied by the corresponding factor (e.g., 3). o When two or more measured quantities are added/subtracted, their absolute uncertainties add up. Find the range where your calculated quantity lies. Make a judgment about your results taking into the account this uncertainty.
Exercises 8 1. Is the last result (δf = δt) consistent in terms of units? Explain 2. What is the relative uncertainty δd of the product of three independent measured quantities: D=A.B.C? 3. What is the relative uncertainty δc of a ratio of two independent measured quantities C=A/B? 4. If the uncertainty of the baseball s radius is 2%, what is the percent uncertainty in its volume? 5. What is the percent uncertainty in the length measurement 2.35 ± 0.25 m? 6. Suppose that after a hike in the mountains, a friend asks how fast you walked. You recall that the trail was about 6 miles long and that it took between two and three hours. What is your average speed if you assume the average time of 2.5 h? Assume that absolute uncertainty in your distance measurement is 0.1 mile. Estimate the relative (percentage) uncertainty in your distance measurement? What is an absolute uncertainty in your time measurement? What is its relative uncertainty? Compare the relative uncertainties in time and distance. Which measurements are more accurate? Determine whether you can use the weakest link rule. Determine the relative and absolute uncertainties in the speed estimation. 7. Suppose you want to measure time for the ball falling from a height of 1 m. You took three measurements of the time interval and obtained 0.5 s, 0.6 s, and 0.4 seconds. What is average time of the fall? What is an absolute value of the random uncertainty in the time measurement? What is the relative uncertainty? 8. Suppose now that you measured the time interval for the ball to fall from a 10 m height and got 1.5s, 1.7s, and 1.6s. Estimate the relative uncertainty assuming the absolute value is the same as in the previous task. Compare the relative uncertainties in tasks 3 and 4. Make a conclusion. 9. You drive along highway and want to estimate your average speed. You notice a sign indicating that it is 260 miles to Boston. In 45 min you pass another sign indicating 210 miles to Boston. Make reasonable assumptions for the absolute uncertainties of your time and distance measurements. Estimate relative (percentage) uncertainties. State your average speed with the uncertainty. 10. You want to know how fast your coffee is cooling in your mug. For this you measure temperature with a thermometer. Your first measurement is 76±1 ºC (you use the usual thermometer with the smallest increment 1ºC). In 15 min temperature is 68±1 ºC. What is the temperature drop (state the uncertainty range)? What is the relative uncertainty in your measurement?
Solutions to Exercises 9 1. Relative uncertainties δf and δt are both dimensionless quantities since they are expressed as ratios:! f =!f and!t =!T. So yes, units are indeed consistent since there aren t any. f T 2.!D =!A +!B +!C. 3.!C =!A +!B. 4. V = 4 3! R3. Therefore δv=3δr=6%. 5. The percent uncertainty, also called relative uncertainty, is 0.25 m = 0.106 =10.6%. 2.35 m 6 miles 6. Your average speed is = 2.4 miles per hour. The relative uncertainty in the distance 2.5 hours 0.1 miles measurement is = 0.0167 =1.67%. The absolute uncertainty in the time measurement 6miles is 0.5 hours since the hike might have taken as short as 2 hours or as long as 3 hours. The relative 0.5 hours uncertainty in the time measurement is = 0.2 = 20%. The distance measurement is 2.5 hours more accurate because its relative uncertainty is smaller. Because it is significantly smaller the weakest link may be used to estimate the uncertainty in the speed. The weakest link rule says to use the largest relative uncertainty from the data as the relative uncertainty of anything you calculate from the data. So, the relative uncertainty in the speed is 20%. The absolute uncertainty in the speed is then 2.4 miles per hour " 0.2 = 0.48 miles per hour. 0.5 s + 0.6 s + 0.4 s 7. The average time of the fall is 0.5 s. We can estimate the absolute 3 uncertainty by looking at the range of measured values. The absolute uncertainty is 0.1 s since all the trials lie within 0.1s of the average. The relative uncertainty is 0.1 s = 0.2 = 20%. 0.5 s 8. The average is 1.6 s and the relative uncertainty is 0.1 s = 0.0625 = 6.25%. Conclusion: The 1.6 s measurement of the time of fall from 10 m is more accurate than the time of fall from 1m since the relative uncertainty is lower. 9. This is a little trickier. Each distance measurement could reasonably have an absolute uncertainty of 0.5 miles. That is: (210 ± 0.5) miles and (260 ± 0.5) miles. The distance measured is 260 miles - 210 miles = 50 miles. However, because of the uncertainty in each measurement, the interval could be as large as 260.5 miles " 209.5 miles = 51 miles, and could be as small as 259.5 miles " 210.5 miles = 49 miles. Thus the distance interval measured is (50 ±1) miles. The absolute uncertainty is 1 mile. The relative uncertainty is 1 mile = 0.02 = 2%. A reasonable estimate of the absolute uncertainty in the time 50 miles
0.5 min measurement is 0.5 min. Its relative uncertainty is = 0.0111=1.11%. The average speed 45 min is the distance traveled (50 miles) divided by the time it took (45 min = 0.75 hours) 50 miles v avg = = 66.7 miles/hour. What is the uncertainty in our estimate of the average 0.75 hours speed? Because the two relative uncertainties in the distance and time measurements are comparable to each other we should add them to find the relative uncertainty in the calculated average speed. The relative uncertainty in the speed is 1.11%+2%=3.11%. So the absolute uncertainty is 66.7 miles/hour x 0.0311 = 2.1 miles/hour. The average speed is then (66.7 ± 2.1) miles/hour. 10. The temperature drop is 76 C - 68 C = 8.0 C. The process we need to use here is similar to problem 5. Remember, our measurement is the drop: 8.0 C, but to find the uncertainty in this measurement we need to know the uncertainty in each temperature, 76 C and 68 C. We know that the temperature measurements are (76 ±1) C and (68 ±1) C. Thus our measurement of the temperature drop could have been as large as 77 C " 67 C =10 C and as small as 75 C " 69 C = 6 C. This makes the absolute uncertainty of the temperature drop is 2 C. Based on the above analysis the temperature drop is (8 ± 2) C. 10