Page 1 SIGNIFICANT FIGURES ASSIGNED READING: Zumdahal, et.al, Chemistry (10 th ed.), Chapter 1, Sec. 4 and 5. I. Accuracy and Precision It is important to remember, here at the outset of this course, that every measuring device, regardless of what it may be, has limitations in how close it can come to the true value of the property being measured. This is commonly referred to as the accuracy of the instrument. To take full advantage of a given measuring instrument, you should be familiar with the limitations on accuracy posed by its design and manufacture and should learn how to use the instrument properly. One way of improving the reliability of a measurement is to measure the quantity or property several times and to report the arithmetic mean (average) of the individual measurements. While there is some uncertainty in each measurement, we generally assume that we are as likely to measure a value above as below the true value (such uncertainties or errors in measurement are referred to as random errors). If the only uncertainty is due to such random errors, the mean represents the best value of the measured quantity. While good technique cannot improve on the accuracy of a tool, poor technique will negate the accuracy of the best instrument. The most common way of expressing accuracy is to determine the % error in a measurement. Experiment al Value True Value % Error = True Value x 100% The precision of a measurement, on the other hand, is a statement about the internal agreement among repeated results; it is a measure of the reproducibility of a given set of measurements. The simplest measure of precision is the average deviation from the mean. The average deviation is calculated in three steps: 1) the mean value is calculated 2) the difference between each individual measurement and the mean value is determined ) the absolute values of the differences are averaged Sample Data: m1 = 14.98 g m2 = 15.02 g Mean = m = 14.97 g 14.98 g 15.02 g 14.97 g = 14.99 g Deviation for m1 14.98-14.99 = -0.01 = 0.01 Deviation for m2 15.02-14.99 = 0.0 = 0.0 Deviation for m 14.97-14.99 = -0.02 = 0.02 0.01 0.0 0.02 = 0.02 g Final Reported Value for the Mass: 14.99 ± 0.02 g Stating that the mass measured by this balance is 14.99 ±.02 g indicates the limits of precision for the set of measurements made with this instrument. This method is somewhat awkward to use, however, especially when many calculations are involved. For this reason, the more convenient practice of recording measurements with the correct number of significant figures is routinely utilized to convey this same type of information.
Page 14 SIGFIG II. Significant Figures in Recorded Data The significant figures of a quantity are those digits which are known with certainty plus the first digit which is uncertain. In the above example the mass reported would be 14.99 g. The 1, the 4 and the first 9 are certain and uncertainty doesn't show up until the hundredths position with the second 9. This was indicated above by the ±.02. Using the method of significant figures we can also convey that the uncertainty shows up in the hundredths position. If all measured numbers are reported by listing every digit that is certain plus one more that is uncertain, we can tell simply by looking at a measurement where the uncertainty comes in; in other words we can tell the limitation of the measuring device by noting the position of the last digit. A measured value of 72.5 g means that the mass is somewhere between 72. g and 72.4 g. The uncertainty results in the hundredths position and to the eye of the one using the balance it appears to be half way between 72. g and 72.4 g. The last digit written indicates where the uncertainty arises. If another, more reliable balance read 72.46 g, the uncertainty would not appear until the thousandths place. To record a measurement with the correct number of significant figures means to list all the digits you know for sure and then estimate one digit further out. Our top loading balances make this estimate for you rather than you having to interpolate. The flashing digit in the hundredths place indicates that our balance really does have some uncertainty in reporting a mass to the hundredth of a gram. For most measuring devices, however, this means that you must read out to the smallest division for which the device is calibrated and estimate to what extent your measurement is reliable beyond the last division. The measured value in the example below lies between 5 and 6 g. The balance only measured to the nearest gram and all we can be sure of is that our answer is between 5.0 and 6.0 g. But we can estimate one digit beyond the calibration of this balance and say that the mass is 5.8 g. Our last digit known for sure is the 5, but our one estimated digit, the 8, is also said to be significant. So we would report, using this balance, the mass to be 5.8 g. A better balance, one calibrated out further, would give us an answer with more significant digits, or more reliability. Remember, every measurement will have some uncertainty when the limits of the measuring device are reached. To interpret the number of significant figures in a recorded measurement in most cases is very straightforward as seen in the examples above. The one confusing situation is when the recorded measurement contains zeros, especially leading or trailing zeros. If the recorded measurement has a decimal point, the number of significant digits can readily be determined by the following procedure. Draw an arrow from left to right until you come to the first nonzero digit. All the digits that the arrow did not pass through are significant. > 792,00.0 > 0.0050040 > 500. 7 Sig. Fig. 5 Sig. Fig. Sig. Fig. (Note that leading zeros are never significant, but that, when a decimal point is present, trailing zeros are significant.) If a recorded measurement ends with zeros and is written without a decimal point, the zeros at the end of the number may or may not be significant. Thus a mass reported as 500 g may have one significant figure
SIGFIG Page 15 (the mass is closer to 500 g than it is to either 400 g or 600 g), two significant figures (the mass is closer to 500 g than it is to 490 g or to 510 g), or three significant figures (the mass is closer to 500 g than it is to 499 g or to 501 g). The best way to remove this ambiguity is to write the number in exponential (or scientific) notation. The significant digits, including any zeros which are significant are written in the coefficient. In this notation, the 10 and its exponent are ignored in counting significant digits. The three cases considered above would then be written as 5 x 10 2 1 sig. fig. (closer to 500g than to 400g or 600g) 5.0 x 10 2 2 sig. fig. (closer to 500g than to 490g or 510g) 5.00 x 10 2 sig. fig. (closer to 500g than to 499g or 501g) The discussion so far has dealt with measured values, which always involve a level of uncertainty. However, some types of numbers are exact, and thus do not limit the significant figures of calculated results. Most of the exact numbers have defined values. For example, there are exactly 12 eggs in a dozen, exactly 1000 g in a kilogram, and exactly 2.54 cm in an inch. The number 1 in any conversion factor between units, as in 1 m = 100 cm or 1 kg = 2.2046 lb, is also an exact number. Exact numbers can also result from counting numbers of objects. For example, we can count the exact number of marbles in a jar or the exact number of people in a classroom. (Zumdahl and Zumdahl, Chemistry, Seventh Edition; Houghton Mifflin Company: Boston, New York, 2007, page 1.) III. Significant Figures in Calculated Data The importance of recording and interpreting the correct number of significant digits in measurements can be seen in the results of the mathematical computations in which they are used. The reliability of the result depends on the reliability of each quantity, but is affected most by the quantity of lowest reliability. Two separate cases need to be considered. Rule #1 (Addition or Subtraction) For manipulations involving either addition or subtraction the result should be reported with the same number of decimal places as the measurement having the least number of decimal places. Consider the following examples: Example 1. 2.5 g + 4.2 g 6.7 g In the first number the uncertainty is in the tenths place so that the actual mass might be (on the conservative side) somewhere between 2.4 g and 2.6 g. In the second number the uncertainty is in the hundredths place. Once again being conservative, the actual mass might be anywhere from 4.22 g to 4.24 g. The answer your calculator gives for the sum is 6.7 g. To say that the sum of the two measurements is 6.7 g indicates that we know the answer is certain to the tenths place and the only uncertainty is in the hundredths place. But if we examine the extreme combinations of these measurements taking into consideration the conservative uncertainties noted above, we get the following: 2.4 g 2.4 g 2.6 g 2.6 g + 4.22 g + 4.24 g + 4.22 g + 4.24 g 6.62 g 6.64 g 6.82 g 6.84 g Notice that not only is there uncertainty (variation) in the hundredths place in our answer, but even in the tenths place. Thus, we are certain of the value in the units place, but an uncertainty shows up in the tenths
Page 16 SIGFIG place. The best answer to report for this sum would be 6.7 g, because 6.7 g indicates properly that the uncertainty is in the tenths place (rather than in the hundredths place indicated by 6.7 g). Example 2. 112.2 g 5. g 1.68 g 148.91 g Since the second measurement (5 g) was made only to the nearest gram, the answer will only be reliable to the nearest gram, and should be reported as 149 g. Note that the answer is rounded off to the nearest whole gram; the numbers after the decimal are not simply dropped. In this type of calculation it is the number of decimal places rather than the number of significant figures which must be considered in properly reporting the answer. Rule #2 (Multiplication or Division) For manipulations involving either multiplication or division the result should be reported with no more significant figures than the measurement having the least number of significant figures. Now consider the following examples: Example 1 88.45 m x 0.55 m 1.9975 m 2 In the first number the uncertainty is in the hundredths place so that the actual length might be (on the conservative side) somewhere between 88.44 m and 88.46 m. In the second number the uncertainty is in the thousandths place. Once again being conservative, the actual length might be anywhere from 0.54 m to 0.56 m. The answer your calculator gives for the product is 1.9975 m 2, implying that there is no uncertainty in the answer until the hundred thousandths place! But if we examine the extreme combinations of these measurements, taking into consideration the conservative uncertainties noted above, we get the following: 88.44 m 88.46 m 88.44 m 88.46 m x 0.54 m x 0.54 m x 0.56 m x 0.56 m 1.0776 m 2 1.1484 m 2 1.48464 m 2 1.49176 m 2 Notice here that the uncertainty (variation) appears all the way back in the tenths place. Even if we tried to use Rule #1 and rounded the answer off to two decimal places, this would imply that the answer is more accurate than it truly is. The best answer to report for this product would be 1.4 m 2. This actually tells us where the uncertainty comes in our answer (the tenths place). Example 2. (22.4 m)(1.528 m) (0.46 m) = 74.506957 m Since the denominator contains only two significant figures, the answer can have only two significant figures and would be reported as 75 m. In this type of calculation it is the number of significant figures and not the number of decimal places in the measured values which determines how the answer is to be reported. In this the age of the pocket calculator it is easy to fall into the trap of reporting eight and ten digit answers to calculations involving measurements having only three or four significant figures. These answers grossly misrepresent the quality of the measurement and must not be used in reporting values computed with measured quantities.
SIGFIG Page 17 When a computation contains both a multiplication (or division) and an addition (or subtraction) of measured quantities, a decision must be made concerning how many significant figures should be reported for this more involved computation. Consider the following heat transfer calculation. Q = m s T Q = m s (Tf Ti) Q = (46.49g)( J 0.214 g C J Q = (46.49g)( 0.214 )(8. C) g C Q = 82.576 J )(1.7 C - 2.4 C) Although the individual final and initial temperatures (Tf Ti) are the directly measured quantities, it is their difference (Tf Ti) which appears in the formula. The usual rules for the sequence of arithmetic operations require that the expression in parentheses be evaluated, and the appropriate rule for expressing calculated results applied, before carrying out the multiplications. According to the addition/subtraction rule, the temperature difference should be expressed only to the nearest tenth of a degree (8. C). When the multiplication/division rule is then applied, the final answer will be limited to two significant figures by the reliability of this temperature difference and should be reported as 8 J. The final situation that often results in confusion is when a counted or defined quantity is included in the computation. Ex: What is the mean (average) mass of three rubber stoppers with the following masses: 98.87 g, 99.04 g, 98.98 g? Mean = 98.87 g 99.04 g 98.98 g = 98.96 g The addition rule produces 296.89g (to the nearest hundredth of a gram). Because we are dividing by a counted number () which has an infinite number of significant digits, we should report all five significant figures in the average. The rules for significant figures are only a rough guide to the reliability of calculated results, and may in some cases be overly optimistic. Nevertheless, they can help us to avoid grossly over (or under) estimating the reliability of experimentally derived quantities. In this laboratory manual the example calculations will illustrate the above guidelines regarding significant figures. Where the results of intermediate calculations are presented, additional digits are retained. Only the final result is rounded to the appropriate significant digits. (see Example 1 on pages 5-54)
Page 18 SIGFIG
SIGFIG Page 19 HOMEWORK 1. How many significant figures do each of the following measured numbers have? 1.25 m 20.0 ml 0.015 g 25.00 ml.20 cm 0.000120 m 14.200 ms 00 L 2. What answer should be reported from the following calculations involving measured numbers? 2 1.1700 cm 0.000775 cm 48.96 g = (17.8 ml 14.2 ml) = 0.180 m 8.5 m 4.16 m 2.70 m.57 m_ (0.00015 m)(4.002 m)(0.05 m) =. Record the length (in inches) indicated by the position of each arrow on the diagram below. 0 1 2 4 4. Record the temperature (in C) indicated by the position of each arrow on the diagram below. 0 10 20 0 40
Page 20 SIGFIG The following examples are intended to provide practice in the use of exponential notation as well as in the application of the rules for significant figures. Perform the indicated arithmetic operations and give the answer in scientific notation using the appropriate significant figures. 5. a. 6.70 x 10 6 mg + 1.0 x 10 4 mg b. 7.70 x 10 - L.00 x 10-5 L c. 5.00 x 10 11 s+ 9.10 x 10 9 s.50 x 10 10 s d. 56.89 x 10-8 µm + 9.7 x 10-5 µm 6. a. (5.5 x 10 - N)(17.0 x 10-8 m) b. (4.25 x 10-12 dm -2 ) / (4.1 x 10-6 dm) c. (7.60 x 10-1 g + 7.0 x 10 - g)(.00 x 10 8 J) 11.80 x 10 m d. 27-2 7.90 x 10 m 7. a. - 12 6.09 x 10 kg 4.6 x 10 kg b. 4.00 x 10 9.00 x 10 5 1 L L -1 c..2 x 10 2.00 x 10-9 18 mm mm