Biological Sciences
Significant Figures
Significant figures in biological sciences refer to the digits in a measurement that are known with certainty plus one digit that is estimated. They are used to communicate the precision of a measurement and indicate the reliability of the data. Understanding significant figures is important for accurately reporting experimental results and ensuring the validity of scientific conclusions.
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eBook - ePub- Fiona Dickinson, Andrew McKinley(Authors)
- 2021(Publication Date)
- Royal Society of Chemistry(Publisher)
CHAPTERLearning Points: What We'll Cover- □ Precision, accuracy and errors in the context of experimental measurement
- □ The correct use of Significant Figures in calculating and reporting experimental values
- □ The standard error and its use in combination with the appropriate number of Significant Figures
- □ Propagating experimental uncertainties through mathematical relationships to report the uncertainty correctly in calculated quantities
- □ Statistical tests to qualify any outliers in a data series and to check whether data sets are statistically different
Why This Chapter Is Important
- In chemistry, we validate our theoretical models through experimental measurement; these measurements are subject to laws of probability and we need to know whether the variations in our measurements are significant.
- Significant Figures tell us everything about the measurements made, so it is important that we all use the same rules for reporting these and carrying them through calculations.
- Experimental uncertainty (‘experimental error’) is unavoidable, so we must be able to quantify ‘the bounds of experimental error’ in order to allow for them effectively in our analyses.
- We need to make sure that we have a common sense approach to handling errors; if our calculated values have an uncertainty greater than the value itself, we need to be sure that we have handled the errors correctly.
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Experimental Uncertainty and Significant Figures: What Are the Bounds of Experimental Error?
It's easy not to think about the number of numbers in a number, but in science the number of Significant Figures is representative of how well we know a value. Some values we know to a high precision; for example, the speed of light is 299 792 458 m s−1 . Thousands of measurements went in to knowing this value so accurately, but we will often approximate it to a value with just one significant figure (3 × 108 m s−1 )†
eBook - PDF- Morris Hein, Judith N. Peisen, Robert L. Miner(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
STUDY AID I Significant Figures Every measurement that we make has some inherent error due to the limitations of the measuring instrument and the experimenter. The numerical value recorded for a measure- ment should give some indication of the reliability (precision) of that measurement. In meas- uring a temperature using a thermometer calibrated at one-degree intervals we can easily read the thermometer to the nearest one degree, but we normally estimate and record the temperature to the nearest tenth of a degree (0.1°C). For example, a temperature falling between 23°C and 24°C might be estimated at 23.4°C. There is some uncertainly about the last digit, 4, but an estimate of it is better information than simply reporting 23°C or 24°C. If we read the thermometer as “exactly” twenty-three degrees, the temperature should be reported as 23.0°C, not 23°C, because 23.0°C indicates our estimate to the nearest 0.1°C. Thus in recording any measurement, we retain one uncertain digit. The digits retained in a physi- cal measurement are said to be significant, and are called Significant Figures. Some numbers are exact (have no uncertain digits) and therefore have an infinite number of Significant Figures. Exact numbers occur in simple counting operations, such as 5 bricks, and in defined relationships, such as 100 cm 1 meter, 24 hours 1 day, etc. Because of their infinite number of Significant Figures, exact numbers do not limit or determine the number of Significant Figures in a calculation. Counting Significant Figures. Digits other than zero are always significant. Depending on their position in the number, zeros may or may not be significant. There are several possible situations: 1. All zeros between other digits in a number are significant; for example: 3.076, 4002, 790.2. Each of these numbers has four Significant Figures. 2. Zeros to the left of the first nonzero digit are used to locate the decimal point and are not significant.
eBook - PDFGeneral, Organic, and Biological Chemistry
An Integrated Approach
- Kenneth W. Raymond(Author)
- 2012(Publication Date)
- Wiley(Publisher)
These exact numbers have an unlimited number of Significant Figures. ■ FIGURE 1.14 Accuracy and precision (a) The darts were thrown precisely (they are all close to one another) but not accurately. (b) The darts were thrown accurately (they fall near the bull’s-eye) but not precisely. (c) The darts were thrown precisely and accurately. (a) (b) (c) ■ FIGURE 1.15 Balances Top-loading balances give a digital readout of the mass of an object. BSIP/Photo Researchers, Inc. ■ Precise measurements are grouped together. 1.5 Measurements and Significant Figures 17 SAMPLE PROBLEM 1.8 Determining Significant Figures Specify the number of Significant Figures in each measured value. a. 30.1°C b. 0.00730 m c. 7.30 * 10 3 m d. 44.50 mL STRATEGY All nonzero digits are significant. Zeros, however, are significant only under certain condi- tions (see Table 1.5). SOLUTION a. 3 b. 3 c. 3 d. 4 PRACTICE PROBLEM 1.8 Write each measured value in exponential notation, being sure to give the correct number of Significant Figures. a. 7032 cal b. 88.0 L c. 0.00005 g d. 0.06430 lb Examples Number of Significant Figures Scientific notation a. All digits, including zeros, are significant a 1.55 * 10 9 3 7.0 * 10 -5 2 6.02 * 10 23 3 Ordinary notation a. All nonzero digits are significant 3.4 2 25.85 4 999,999 6 b. Zeros placed between nonzero digits are significant 5.04 3 8.0045 5 20.02 4 c. Zeros placed at the end of a number when a 4.0 2 decimal point is present are significant 40. 2 8500.0 5 d. Zeros placed at the end of a number with 40 1 no decimal point are not significant 6,510 3 103,000 3 e. Zeros placed at the beginning of a number 0.1 1 are not significant 0.453 3 0.0006 1 a When determining Significant Figures for numbers in scientific notation, the power of 10 is not included. TABLE | 1.5 Significant Figures 18 CHAPTER 1 Science and Measurements The percentage of Americans who are obese has risen over the past two decades.
eBook - PDFChemistry
The Molecular Nature of Matter
- James E. Brady, Neil D. Jespersen, Alison Hyslop(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
For example, the improp- erly marked ruler in Figure 1.23 might yield measurements that vary by a hundredth of a centimeter (±0.01 cm), but all the measurements would be too large by 1 cm—a case of good precision but poor accuracy. Significant Figures in Calculations When several measurements are obtained in an experiment they are usually combined in some way to calculate a desired quantity. For example, to determine the area of a rectangular carpet we require two measurements, length and width, which are then multiplied to give the answer we want. To get some idea of how precise the area really is, we need a way to take into account the precision of the various values used in the calculation. To make sure this happens, we follow certain rules according to the kinds of arithmetic being performed. Multiplication and Division For multiplication and division, the number of Significant Figures in the answer should be equal to the number of Significant Figures in the least precise measurement. The least precise measure- ment is the number with the fewest Significant Figures. Let’s look at a typical problem involving some measured quantities. 3 sig. figures 4 sig. figures 3.14 Ž 2.751 0.64 = 13 2 sig. figures Significant Figures: multiplication and division Figure 1.23 | An improperly marked ruler. This improperly marked ruler will yield measurements that are each wrong by one whole unit. The measurements might be precise, but the accuracy would be very poor. How accurate would measurements be with this ruler? 2 1cm 3 4 5 1.5 | The Uncertainty of Measurements 43 44 Chapter 1 | Scientific Measurements The result displayed on a calculator 9 is 13.49709375. However, the least precise factor, 0.64, has only two Significant Figures, so the answer should have only two. The correct answer, 13, is obtained by rounding off the calculator answer. 10 When we multiply and divide measurements, the units of those measurements are multiplied and divided in the same way as the numbers.
eBook - PDF- John Kenkel(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Thus, the correct readings are those indicated . 2.3 Significant Figures As stated in the last section, the number of digits in a given number indicates its precision, more digits implying greater precision . Unfortunately, there are occasions in which some of the digits in a given number are not included in the count of total digits . In other words, not all the digits found in a given 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 (a) (b) (c) (d) (e) FIGURE 2.3 Several illustrations of the application of the fundamen-tal rule of measurement for nondigital measuring devices as stated in Figure 2.1. The correct readings are (a) 3.67 units, (b) 3.3 units, (c) 4.20 units, (d) 4.0 units, and (e) 6.00 units. 31 Significant Figures, the Metric System, and Dimensional Analysis number may be significant . This problem involves only zeros that may be in the number . Sometimes zeros are a part of the measurement and therefore are significant, but sometimes zeros are present merely to locate the decimal point and therefore are not significant . (In such cases, the zeros are call place-holders .) For example, if the number 430,000 is given as the population of a (a) (b) FIGURE 2.4 Additional examples of nondigital measurements, in this case classical thermometers. (a) 22.17° and (b) 32.50°. (a) (b) FIGURE 2.5 Additional examples of nondigital measurements, in this case, graduated cylinders. (a) 66.4 mL and (b) 60.0 mL. Basic Chemistry Concepts and Exercises 32 city, it is not likely that all six digits reflect the true precision of the count . It is more likely that only the 4 and the 3 are significant and that the zeros are present only to locate the decimal point . The zeros in the number 0 .0082 are likewise not significant because they are present only to show that the deci-mal point is located two places to the left of the 8 .
eBook - ePub- W Schofield, Mark Breach, Will Schofield, M C AND MRS L C BREACH, Wilf Schofield(Authors)
- 2007(Publication Date)
- CRC Press(Publisher)
Engineers and surveyors communicate a great deal of their professional information using numbers. It is important, therefore, that the number of digits used, correctly indicates the accuracy with which the field data were measured. This is particularly important since the advent of pocket calculators, which tend to present numbers to as many as eight places of decimals, calculated from data containing, at the most, only three places of decimals, whilst some eliminate all trailing zeros. This latter point is important, as 2.00 m is an entirely different value to 2.000 m. The latter number implies estimation to the nearest millimetre as opposed to the nearest 10 mm implied by the former. Thus in the capture of field data, the correct number of Significant Figures should be used.By definition, the number of Significant Figures in a value is the number of digits one is certain of plus one, usually the last, which is estimated. The number of Significant Figures should not be confused with the number of decimal places. A further rule in Significant Figures is that in all numbers less than unity, the zeros directly after the decimal point and up to the first non-zero digit are not counted. For example:Two Significant Figures: 40, 42, 4.2, 0.43, 0.0042, 0.040 Three Significant Figures: 836, 83.6, 80.6, 0.806, 0.0806, 0.00800Difficulties can occur with zeros at the end of a number such as 83 600, which may have three, four or five Significant Figures. This problem is overcome by expressing the value in powers of ten, i.e. 8.36 × 104 implies three Significant Figures, 8.360 × 104 implies four Significant Figures and 8.3600 × 104 implies five Significant Figures.It is important to remember that the accuracy of field data cannot be improved by the computational processes to which it is subjected. Consider the addition of the following numbers: 155.486 7.08 2183.0 42.0058If added on a pocket calculator the answer is 2387.5718; however, the correct answer with due regard to Significant Figures is 2387.6. It is rounded off to the most extreme right-hand column containing all the Significant Figures, which in the example is the column immediately after the decimal point. In the case of 155.486 + 7.08 + 2183 + 42.0058 the answer should be 2388. This rule also applies to subtraction.
eBook - PDFChemistry for Today
General, Organic, and Biochemistry
- Spencer Seager, Michael Slabaugh, Maren Hansen, , Spencer Seager, Spencer Seager, Michael Slabaugh, Maren Hansen(Authors)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
● In scientific notation, numbers are represented as products of a nonexponential number and 10 raised to some power. ● The nonexponential number is always written with the deci- mal in the standard position (to the right of the first nonzero digit in the number). ● Numbers written in scientific notation can be manipulated in calculations by following a few rules. 1.8 Significant Figures Learning Objective: Can you express the results of measurements and calculations using the correct number of Significant Figures? ● Significant Figures are the numbers representing the part of the measurement that is certain, plus one number represent- ing an estimate. ● The maximum number of Significant Figures possible in a mea- surement is determined by the design of the measuring device. ● Results of calculations made using numbers from measure- ments can be expressed with the proper number of Significant Figures by following simple rules. Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Matter, Measurements, and Calculations 37 1.9 Using Units in Calculations Learning Objective: Can you use the factor-unit method to solve numerical problems? ● The factor-unit method for doing calculations is based on a specific set of steps. ● One crucial step involves the use of factors that are obtained from fixed numerical relationships between quantities. ● The units of the factor must always cancel the units of the known quantity and generate the units of the unknown or de- sired quantity.
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