14. Accuracy, Precision, and Significant FiguresLearning Objectives
Accuracy and Precision of a MeasurementScience is based on observation and experiment—that is, on measurements. The The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider an example of a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull’seye target, and think of each GPS attempt to locate the restaurant as a black dot. In Figure 3, you can see that the GPS measurements are spread out far apart from each other, but they are all relatively close to the actual location of the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in Figure 4, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system. Accuracy, Precision, and UncertaintyThe degree of accuracy and precision of a measuring system are related to the The factors contributing to uncertainty in a measurement include:
In our example, such factors contributing to the uncertainty could be the following: the smallest division on the ruler is 0.1 in., the person using the ruler has bad eyesight, or one side of the paper is slightly longer than the other. At any rate, the uncertainty in a measurement must be based on a careful consideration of all the factors that might contribute and their possible effects. Making Connections: RealWorld Connections – Fevers or Chills?Uncertainty is a critical piece of information, both in physics and in many other realworld applications. Imagine you are caring for a sick child. You suspect the child has a fever, so you check his or her temperature with a thermometer. What if the uncertainty of the thermometer were $3\text{.}\mathrm{0\xba}\text{C}$? If the child’s temperature reading was $\text{37}\text{.}\mathrm{0\xba}\text{C}$ (which is normal body temperature), the “true” temperature could be anywhere from a hypothermic $\text{34}\text{.}\mathrm{0\xba}\text{C}$ to a dangerously high $\text{40}\text{.}\mathrm{0\xba}\text{C}$. A thermometer with an uncertainty of $3\text{.}\mathrm{0\xba}\text{C}$ would be useless. Percent UncertaintyOne method of expressing uncertainty is as a percent of the measured value. If a measurement $A$ is expressed with uncertainty, $\mathrm{\delta A}$, the Example 1: Calculating Percent Uncertainty: A Bag of ApplesA grocery store sells $\text{5lb}$ bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:
You determine that the weight of the $\text{5lb}$ bag has an uncertainty of $\pm 0\text{.}4\phantom{\rule{0.25em}{0ex}}\text{lb}$. What is the percent uncertainty of the bag’s weight? Strategy First, observe that the expected value of the bag’s weight, $A$, is 5 lb. The uncertainty in this value, $\mathrm{\delta A}$, is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight: $$\text{}\text{\% unc =}\frac{\mathrm{\delta A}}{A}\times \text{100\%}\text{}\text{.}$$Solution Plug the known values into the equation: $$\text{}\text{\% unc =}\frac{0\text{.}4\text{lb}}{5\text{lb}}\times \text{100\%}\text{}=8\text{\%}\text{.}$$Discussion We can conclude that the weight of the apple bag is $5\phantom{\rule{0.25em}{0ex}}\text{lb}\pm 8\text{\%}$. Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value. Uncertainties in CalculationsThere is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the Check Your UnderstandingA high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of $\pm 0\text{.}\text{05}\phantom{\rule{0.25em}{0ex}}\mathrm{s}$. Runners on the track coach’s team regularly clock 100m sprints of $\text{11.49 s}$ to $\text{15.01 s}$. At the school’s last track meet, the firstplace sprinter came in at $\text{12}\text{.}\text{04 s}$ and the secondplace sprinter came in at $\text{12}\text{.}\text{07 s}$. Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not? Show/Hide Solution SolutionNo, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times. Precision of Measuring Tools and Significant FiguresAn important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be. When we express measured values, we can only list as many digits as we initially measured with our measuring tool. For example, if you use a standard ruler to measure the length of a stick, you may measure it to be $\text{36}\text{.}7\phantom{\rule{0.25em}{0ex}}\text{cm}$. You could not express this value as $\text{36}\text{.}\text{71}\phantom{\rule{0.25em}{0ex}}\text{cm}$ because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between $\text{36}\text{.}6\phantom{\rule{0.25em}{0ex}}\text{cm}$ and $\text{36}\text{.}7\phantom{\rule{0.25em}{0ex}}\text{cm}$, and he or she must estimate the value of the last digit. Using the method of ZerosSpecial consideration is given to zeros when counting significant figures. The zeros in 0.053 are not significant, because they are only placekeepers that locate the decimal point. There are two significant figures in 0.053. The zeros in 10.053 are not placekeepers but are significant—this number has five significant figures. The zeros in 1300 may or may not be significant depending on the style of writing numbers. They could mean the number is known to the last digit, or they could be placekeepers. So 1300 could have two, three, or four significant figures. (To avoid this ambiguity, write 1300 in scientific notation.) Zeros are significant except when they serve only as placekeepers. Check Your UnderstandingDetermine the number of significant figures in the following measurements:
Show/Hide Solution Solution(a) 1; the zeros in this number are placekeepers that indicate the decimal point (b) 6; here, the zeros indicate that a measurement was made to the 0.1 decimal point, so the zeros are significant (c) 1; the value ${\text{10}}^{3}$ signifies the decimal place, not the number of measured values (d) 5; the final zero indicates that a measurement was made to the 0.001 decimal point, so it is significant (e) 4; any zeros located in between significant figures in a number are also significant Significant Figures in CalculationsWhen combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and the other for addition and subtraction, as discussed below. 1. For multiplication and division: The result should have the same number of significant figures as the quantity having the least significant figures entering into the calculation. For example, the area of a circle can be calculated from its radius using $A={\mathrm{\pi r}}^{2}$. Let us see how many significant figures the area has if the radius has only two—say, $r=1\text{.}2\phantom{\rule{0.25em}{0ex}}\text{m}$. Then, $$A={\mathrm{\pi r}}^{2}=\left(3\text{.}\text{1415927}\text{.}\text{.}\text{.}\right)\times {\left(1\text{.}2\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}=4\text{.}\text{5238934}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$$is what you would get using a calculator that has an eightdigit output. But because the radius has only two significant figures, it limits the calculated quantity to two significant figures or $$A\text{=}4\text{.}5\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\text{,}$$even though $\pi $ is good to at least eight digits. 2. For addition and subtraction: The answer can contain no more decimal places than the least precise measurement. Suppose that you buy 7.56kg of potatoes in a grocery store as measured with a scale with precision 0.01 kg. Then you drop off 6.052kg of potatoes at your laboratory as measured by a scale with precision 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with precision 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction: $$\begin{array}{}\phantom{+1}7.56\phantom{0}\phantom{\rule{0.25em}{0ex}}\text{kg}\phantom{=15.2\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}}\\ \phantom{1}6.052\phantom{\rule{0.25em}{0ex}}\text{kg}\phantom{=15.2\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}}\\ \frac{+13.7\phantom{00}\phantom{\rule{0.25em}{0ex}}\text{kg}}{\phantom{+}15.208\phantom{\rule{0.25em}{0ex}}\text{kg}}=15.2\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}\end{array}$$Next, we identify the least precise measurement: 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer is rounded to the tenths place, giving us 15.2 kg. Significant Figures in this TextIn this text, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits, for example. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, particularly in optics, more accurate numbers are needed and more than three significant figures will be used. Finally, if a number is exact, such as the two in the formula for the circumference of a circle, $c=\mathrm{2\pi}r$, it does not affect the number of significant figures in a calculation. Check Your UnderstandingPerform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. What is the total weight of the bags? (b) The force $F$ on an object is equal to its mass $m$ multiplied by its acceleration $a$. If a wagon with mass 55 kg accelerates at a rate of $0\text{.}\text{0255}{\text{m/s}}^{2}$, what is the force on the wagon? (The unit of force is called the newton, and it is expressed with the symbol N.) Show/Hide Solution Solution(a) 37.2 pounds; Because the number of bags is an exact value, it is not considered in the significant figures. (b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures. PhET Explorations: EstimationExplore size estimation in one, two, and three dimensions! Multiple levels of difficulty allow for progressive skill improvement. Summary
Conceptual QuestionsExercise 1What is the relationship between the accuracy and uncertainty of a measurement? Exercise 2Prescriptions for vision correction are given in units called diopters (D). Determine the meaning of that unit. Obtain information (perhaps by calling an optometrist or performing an internet search) on the minimum uncertainty with which corrections in diopters are determined and the accuracy with which corrective lenses can be produced. Discuss the sources of uncertainties in both the prescription and accuracy in the manufacture of lenses. Problems & ExercisesExpress your answers to problems in this section to the correct number of significant figures and proper units. Exercise 1Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)? Show/Hide Solution Solution2 kg Exercise 2A goodquality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty? Exercise 3(a) A car speedometer has a $5.0\text{\%}$ uncertainty. What is the range of possible speeds when it reads $\text{90}\phantom{\rule{0.25em}{0ex}}\text{km/h}$? (b) Convert this range to miles per hour. $\left(\text{1 km}=\text{0.6214 mi}\right)$ Show/Hide Solution Solution
Exercise 4An infant’s pulse rate is measured to be $\text{130}\pm 5$ beats/min. What is the percent uncertainty in this measurement? Exercise 5(a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y? Show/Hide Solution Solution(a) $7\text{.}6\times {\text{10}}^{7}\text{beats}$ (b) $7\text{.}\text{57}\times {\text{10}}^{7}\text{beats}$ (c) $7\text{.}\text{57}\times {\text{10}}^{7}\text{beats}$ Exercise 6A can contains 375 mL of soda. How much is left after 308 mL is removed? Exercise 7State how many significant figures are proper in the results of the following calculations: (a) $\left(\text{106}\text{.}7\right)\left(\text{98}\text{.}2\right)/\left(\text{46}\text{.}\text{210}\right)\left(1\text{.}\text{01}\right)$ (b) ${\left(\text{18}\text{.}7\right)}^{2}$ (c) $\left(1\text{.}\text{60}\times {\text{10}}^{\text{19}}\right)\left(\text{3712}\right)$. Show/Hide Solution Solution
Exercise 8(a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties? Exercise 9(a) If your speedometer has an uncertainty of $2\text{.}0\phantom{\rule{0.25em}{0ex}}\text{km/h}$ at a speed of $\text{90}\phantom{\rule{0.25em}{0ex}}\text{km/h}$, what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads $\text{60}\phantom{\rule{0.25em}{0ex}}\text{km/h}$, what is the range of speeds you could be going? Show/Hide Solution Solutiona) $2\text{.}2\text{\%}$ (b) $\text{59 to 61 km/h}$ Exercise 10(a) A person’s blood pressure is measured to be $\text{120}\pm 2\phantom{\rule{0.25em}{0ex}}\text{mm Hg}$. What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of $\text{80}\phantom{\rule{0.25em}{0ex}}\text{mm Hg}$? Exercise 11A person measures his or her heart rate by counting the number of beats in $\text{30}\phantom{\rule{0.25em}{0ex}}\text{s}$. If $\text{40}\pm 1$ beats are counted in $\text{30}\text{.}0\pm 0\text{.}5\phantom{\rule{0.25em}{0ex}}\text{s}$, what is the heart rate and its uncertainty in beats per minute? Show/Hide Solution Solution$\text{80}\pm 3\phantom{\rule{0.25em}{0ex}}\text{beats/min}$ Exercise 12What is the area of a circle $3\text{.}\text{102}\phantom{\rule{0.25em}{0ex}}\text{cm}$ in diameter? Exercise 13If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22mi marathon? Show/Hide Solution Solution$2\text{.}8\phantom{\rule{0.25em}{0ex}}\text{h}$ Exercise 14A marathon runner completes a $\text{42}\text{.}\text{188}\text{km}$ course in $2\phantom{\rule{0.25em}{0ex}}\text{h}$, 30 min, and $\text{12}\phantom{\rule{0.25em}{0ex}}\text{s}$. There is an uncertainty of $\text{25}\phantom{\rule{0.25em}{0ex}}\text{m}$ in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed? Exercise 15The sides of a small rectangular box are measured to be $1\text{.}\text{80}\pm 0\text{.}\text{01}\phantom{\rule{0.25em}{0ex}}\text{cm}$, $$$2\text{.}\text{05}\pm 0\text{.}\text{02}\phantom{\rule{0.25em}{0ex}}\text{cm, and 3}\text{.}1\pm 0\text{.}\text{1 cm}$ long. Calculate its volume and uncertainty in cubic centimeters. Show/Hide Solution Solution$\text{11}\pm 1\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{3}$ Exercise 16When nonmetric units were used in the United Kingdom, a unit of mass called the poundmass (lbm) was employed, where $1\phantom{\rule{0.25em}{0ex}}\text{lbm}=0\text{.}\text{4539}\phantom{\rule{0.25em}{0ex}}\text{kg}$. (a) If there is an uncertainty of $0\text{.}\text{0001}\phantom{\rule{0.25em}{0ex}}\text{kg}$ in the poundmass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in poundmass has an uncertainty of 1 kg when converted to kilograms? Exercise 17The length and width of a rectangular room are measured to be $3\text{.}\text{955}\pm 0\text{.}\text{005}\phantom{\rule{0.25em}{0ex}}\text{m}$ and $3\text{.}\text{050}\pm 0\text{.}\text{005}\phantom{\rule{0.25em}{0ex}}\text{m}$. Calculate the area of the room and its uncertainty in square meters. Show/Hide Solution Solution$\text{12}\text{.}\text{06}\pm 0\text{.}\text{04}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ Exercise 18A car engine moves a piston with a circular cross section of $7\text{.}\text{500}\pm 0\text{.}\text{002}\phantom{\rule{0.25em}{0ex}}\text{cm}$ diameter a distance of $3\text{.}\text{250}\pm 0\text{.}\text{001}\phantom{\rule{0.25em}{0ex}}\text{cm}$ to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.
