Accuracy, Precision, and Significant Figures
Learning Objectives
By the end of this section, you will be able to:- Determine the appropriate number of significant figures in both addition and subtraction, as well as multiplication and division calculations.
- Calculate the percent uncertainty of a measurement.
Figure 1. A double-pan mechanical balance is used to compare different masses. Usually an object with unknown mass is placed in one pan and objects of known mass are placed in the other pan. When the bar that connects the two pans is horizontal, then the masses in both pans are equal. The “known masses” are typically metal cylinders of standard mass such as 1 gram, 10 grams, and 100 grams. (credit: Serge Melki)
Figure 2. Many mechanical balances, such as double-pan balances, have been replaced by digital scales, which can typically measure the mass of an object more precisely. Whereas a mechanical balance may only read the mass of an object to the nearest tenth of a gram, many digital scales can measure the mass of an object up to the nearest thousandth of a gram. (credit: Karel Jakubec)Accuracy and Precision of a Measurement
The precision of a measurement system is refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Consider the example of the paper measurements. The precision of the measurements refers to the spread of the measured values. One way to analyze the precision of the measurements would be to determine the range, or difference, between the lowest and the highest measured values. In that case, the lowest value was 10.9 in. and the highest value was 11.2 in. Thus, the measured values deviated from each other by at most 0.3 in. These measurements were relatively precise because they did not vary too much in value. However, if the measured values had been 10.9, 11.1, and 11.9, then the measurements would not be very precise because there would be significant variation from one measurement to another.
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’s-eye 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.
Figure 3. A GPS system attempts to locate a restaurant at the center of the bull’s-eye. The black dots represent each attempt to pinpoint the location of the restaurant. The dots are spread out quite far apart from one another, indicating low precision, but they are each rather close to the actual location of the restaurant, indicating high accuracy. (credit: Dark Evil)
Figure 4. In this figure, the dots are concentrated rather closely to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (credit: Dark Evil)Accuracy, Precision, and Uncertainty
The factors contributing to uncertainty in a measurement include:
- Limitations of the measuring device,
- The skill of the person making the measurement,
- Irregularities in the object being measured,
- Any other factors that affect the outcome (highly dependent on the situation).
Making Connections: Real-World Connections – Fevers or Chills?
Uncertainty is a critical piece of information, both in physics and in many other real-world 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 3º? If the child’s temperature reading was 37ºC(which is normal body temperature), the "true" temperature could be anywhere from a hypothermic 34º to a dangerously high 40º. A thermometer with an uncertainty of 3º would be useless.Percent Uncertainty
One method of expressing uncertainty is as a percent of the measured value. If a measurement A is expressed with uncertainty, δA, the percent uncertainty (%unc) is defined to be
Example 1: Calculating Percent Uncertainty: A Bag of Apples
A grocery store sells a 5-pound bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:
- Week 1 weight: 4.8 lb
- Week 2 weight: 5.3 lb
- Week 3 weight: 4.9 lb
- Week 4 weight: 5.4 lb
You determine that the weight of the 5-pound bag has an uncertainty of ±0.4 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, δA, is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:
Solution
Plug the known values into the equation:
Discussion
We can conclude that the weight of the apple bag is 5 lb ± 8%. 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 Calculations
Check Your Understanding
Precision of Measuring Tools and Significant Figures
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 36.7 cm. You could not express this value as 36.71 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 36.6 cm and 36.7 cm, and he or she must estimate the value of the last digit. Using the method of significant figures, the rule is that the last digit written down in a measurement is the first digit with some uncertainty. In order to determine the number of significant digits in a value, start with the first measured value at the left and count the number of digits through the last digit written on the right. For example, the measured value 36.7cm has three digits, or significant figures. Significant figures indicate the precision of a measuring tool that was used to measure a value.
Zeros
Check Your Understanding
Determine the number of significant figures in the following measurements:
- 0.0009
- 15,450.0
- 6 × 103
- 87.990
- 30.42
(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 103 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 Calculations
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 = πr2. Let us see how many significant figures the area has if the radius has only two—say, r = 1.2 m. Then,
A = πr2 = (3.1415927...) × (1.2 m)2 = 4.5238934 m2
is what you would get using a calculator that has an eight-digit output. But because the radius has only two significant figures, it limits the calculated quantity to two significant figures or A = 4.5m2, even though π 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 7.56-kg of potatoes in a grocery store as measured with a scale with precision 0.01 kg. Then you drop off 6.052-kg 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:
7.56 kg − 6.052 kg + 13.7 kg = 15.208 kg = 15.2kg
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 Text
Check Your Understanding
(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 0.0255 m/s2, what is the force on the wagon? (The unit of force is called the newton, and it is expressed with the symbol N.)
(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: Estimation
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Summary
- Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value.
- Precision of measured values refers to how close the agreement is between repeated measurements.
- The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool.
- Significant figures express the precision of a measuring tool.
- When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value.
- When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value.
Conceptual Questions
2. Prescriptions 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 & Exercises
Express your answers to problems in this section to the correct number of significant figures and proper units.
3. (a) A car speedometer has a 5.0% uncertainty. What is the range of possible speeds when it reads 90 km/h? Convert this range to miles per hour. (1 km = 0.6214 m)
4. An infant’s pulse rate is measured to be 130 ± 5 beats/min. What is the percent uncertainty in this measurement?
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?
6. A can contains 375 mL of soda. How much is left after 308 mL is removed?
7. State how many significant figures are proper in the results of the following calculations: (a) (106.7)(98.2) / (46.210)(1.01) (b) (18.72) (c) (1.60 × 10–19) (3712).
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?
9. (a) If your speedometer has an uncertainty of 2.0 km/h at a speed of 90 km/h, what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h, what is the range of speeds you could be going?
10. (a) A person’s blood pressure is measured to be 120 ± 2 mm Hg. What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg?
11. A person measures his or her heart rate by counting the number of beats in 30s. If 40± 1 beats are counted in 30 ± 0.5 s, what is the heart rate and its uncertainty in beats per minute?
12. What is the area of a circle 3.102 in diameter?
13. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22-mi marathon?
14. A marathon runner completes a 42.188-km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1s 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?
16. When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 11bm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?
17. The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m. Calculate the area of the room and its uncertainty in square meters.
18. A car engine moves a piston with a circular cross section of 7.500 ± 0.002 cm diameter in a distance of 3.250 ± 0.001 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.
Glossary
- accuracy:
- the degree to which a measured value agrees with correct value for that measurement
- method of adding percents:
- the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation
- percent uncertainty:
- the ratio of the uncertainty of a measurement to the measured value, expressed as a percentage
- precision:
- the degree to which repeated measurements agree with each other
- significant figures:
- express the precision of a measuring tool used to measure a value
- uncertainty:
- a quantitative measure of how much your measured values deviate from a standard or expected value
Selected Solutions to Problems & Exercises
1. 2 kg3. (a) 85.5 to 94.5 km/h (b)53.1 to 58.7 mi/h
5. (a) 7.6 × 107 beats (b) 7.57 × 107 beats (c) 7.57 × 107 beats
7. (a) 3 (b) 3 (c) 3
9. (a) 2.2% (b) 59 to 61 km/h
11. 80 ± 3 beats/min
13. 2.6 h
15. 11 ± 1 cm3
17. 12.06 ± 0.04 m2