Section 2.3

How reliable are measurements How reliable are measurements? Standard: I&E (1b) Article: 41 Mastering Concepts: 50(62-68) Terms: 36 Practice Problems: 38(29-30), 39(31-32),41(33-36), 42(37-38) Homework: Cornell Notes: 2.3 Section Assessment: 42(39-42) Mastering Problems: 51(81-85) 11 Stamps

Section 2-3 Define and compare accuracy and precision. Section 2.3 Uncertainty in Data Define and compare accuracy and precision. Describe the accuracy of experimental data using error and percent error. Apply rules for significant figures to express uncertainty in measured and calculated values. experiment: a set of controlled observations that test a hypothesis

Comprehension Verbs Confirm Convert Match Infer Discuss Estimate Predict Explain Relate Describe paraphrase Product Analogy Graph Speech Collage Drama Poster Story Summary Outline Photograph Tape recording Diagram cartoon

Application Verbs Apply Modify Build Construct Solve Report Sketch produce Product Diagram Sculpture Photograph Forecast Illustration List Project Puzzle Cartoon Filmstrip

Section 2-3 accuracy precision error percent error significant figures Section 2.3 Uncertainty in Data (cont.) Section 2-3 accuracy precision error percent error significant figures Measurements contain uncertainties that affect how a result is presented.

Exact Inexact

Accuracy Precision

Section 2-3 Accuracy and Precision Accuracy refers to how close a measured value is to an accepted value. Precision refers to how close a series of measurements are to one another.

good precision & good accuracy poor accuracy but good precision good accuracy but poor precision poor precision & poor accuracy

Accuracy and Precision (cont.) Section 2-3 Error is defined as the difference between and experimental value and an accepted value.

Section 2-3 Accuracy and Precision (cont.) The error equation is error = experimental value – accepted value. Percent error expresses error as a percentage of the accepted value.

Mastering Concepts: 50(62-68) 62. If you report two measurements of mass, 7.42 g and 7.56 g, are the measurements accurate? Are they precise? Explain your answers. (2.3) You must know the accepted value to know if the measurements are accurate. They are fairly precise because there is only 0.14 g difference between the two measurements.

Mastering Concepts: 50(62-68) 63. When converting from meters to centimeters, how do you decide which values to place in the numerator and denominator of the conversion factor? (2.3) Meters will be in the denominator so that the units will cancel when the starting value is multiplied by the conversion factor.

Mastering Concepts: 50(62-68) 64. Why are plus and minus signs ignored in percent error calculations? (2.3) You need to know only the difference between the measured value and the magnitude of the accepted value.

Mastering Concepts: 50(62-68) 65. In 50 540, which zero is significant? What is the other zero called? (2.3) the first one; placeholder 66. Which of the following three numbers will produce the same number when rounded to three significant figures: 3.456, 3.450, or 3.448? (2.3) 3.450 and 3.448

Mastering Concepts: 50(62-68) 67. When subtracting 61.45 g from 242.6 g, which factor determines the number of significant figures in the answer? Explain. (2.3) the number that has the fewest digits to the right of the decimal point; it is less precise. 68. When multiplying 602.4 m by 3.72 m, which factor determines the number of significant figures in the answer? Explain. (2.3) 3.72; it has the smaller number of significant figures.

Practice Problems: 38 (29-30) Density Data Collected by Three Different Students The Accepted Density of Table Sugar is 1.59 g/cm3 Student A Student B Student C Trial 1 1.54 g/cm3 1.4 g/cm3 1.70 g/cm3 Trial 2 1.60 g/cm3 1.68 g/cm3 1.69 g/cm3 Trial 3 1.57 g/cm3 1.45 g/cm3 1.71 g/cm3 Calculate the percent errors for Student B’s trials. Calculate the percent errors for Student C’s trials

Practice Problems: 38 (29-30) Errors for Data in Table 2-3 The Accepted Density of Table Sugar is 1.59 g/cm3 Student A Student B Student C Trial 1 -0.05 g/cm3 -0.019 g/cm3 +0.11 g/cm3 Trial 2 +0.01 g/cm3 +0.09 g/cm3 +0.10 g/cm3 Trial 3 -0.02 g/cm3 -0.14 g/cm3 +0.12 g/cm3 Calculate the percent errors for Student B’s trials. Calculate the percent errors for Student C’s trials

The Accepted Density of Table Sugar is 1.59 g/cm3 29. B/ Density B/ Errors Trial 1 1.4 g/cm3 1.59-1.4 = -0.19 g/cm3 Trial 2 1.68 g/cm3 1.59-1.68 = +0.09 g/cm3 Trial 3 1.45 g/cm3 1.59- 1.45 = -0.14 g/cm3 0.19 x 100 = 11.9% 1.59 The Accepted Density of Table Sugar is 1.59 g/cm3 0.09 x 100 = 5.66% 1.59 0.14 x 100 = 8.80% 1.59

Practice Problems: 38 (29-30) Errors for Data in Table 2-3 Student A Student B Student C Trial 1 -0.05 g/cm3 -0.019 g/cm3 +0.11 g/cm3 Trial 2 +0.01 g/cm3 +0.09 g/cm3 +0.10 g/cm3 Trial 3 -0.02 g/cm3 -0.14 g/cm3 +0.12 g/cm3 Calculate the percent errors for Student B’s trials. Calculate the percent errors for Student C’s trials

The Accepted Density of Table Sugar is 1.59 g/cm3 30. Errors for Data in Table 2-3 C/Density C/ Errors Trial 1 1.70 g/cm3 1.59 - 1.70 = +0.11 g/cm3 Trial 2 1.69 g/cm3 1.59 – 1.69 = +0.10 g/cm3 Trial 3 1.71 g/cm3 1.59 – 1.71 = +0.12 g/cm3 0.11 x 100 = 6.92% 1.59 The Accepted Density of Table Sugar is 1.59 g/cm3 0.10 x 100 = 6.92% 1.59 0.12 x 100 = 7.55% 1.59

measurement (1.4) always has some degree of uncertainty. The uncertainty of a measurement involves estimates and cannot be exactly reproduced. depends on the precision of the measuring device. Precision= how accurate or exact

25 mL vs. 25.00 mL? The quantity 25 mL means that the volume is between 24 mL and 26 mL, whereas the quantity 25.00 mL means that the volume is between 24.99 mL and 25.01 mL

significant figures recording the certain digits and the first uncertain digit (the estimated number) 7.5 are certain digits

Section 2-3 Often, precision is limited by the tools available. Significant Figures Often, precision is limited by the tools available. Significant figures include all known digits plus one estimated digit.

Section 2-3 Rules for significant figures Significant Figures (cont.) Rules for significant figures Rule 1: Nonzero numbers are always significant. Rule 2: Zeros between nonzero numbers are always significant. Rule 3: All final zeros to the right of the decimal are significant. Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation. Rule 5: Counting numbers and defined constants have an infinite number of significant figures.

31.80 0.0020 56430 10000 Pacific or Atlantic? Decimal Present? A little trick for “sig figs” Decimal Present? Count from the Pacific Decimal Absent? Count from the Atlantic 31.80 0.0020 56430 10000

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 31. 508.0 L 820 400.0 L 1.0200 x 105 kg 807 000 kg

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 31. a. 508.0 4 Rule 3: All final zeros to the right of the decimal are significant.

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 31. b. 820 400.0 7 Rule 3: All final zeros to the right of the decimal are significant.

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 31. c. 1.0200 x 105 5 Rule 3: All final zeros to the right of the decimal are significant.

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 31. d. 807 000 3 Rule 1: Nonzero numbers are always significant.

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 32. 0.049 450 s 0.000482 mL 3.1587 x 10-8 g 0.0084 mL

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 32a. 0.049 450 5 Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation

Practice Problems: 39 (31-32) Determine the number of significant figures in each measurement. 32b. 0.000482 3 Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation

Your Turn ... Determine the number of significant figures in each measurement. 32. c. 3.1587 x 10-8 g d. 0.0084 mL

Section 2-3 Calculators are not aware of significant figures. Rounding Numbers Calculators are not aware of significant figures. Answers should not have more significant figures than the original data with the fewest figures, and should be rounded.

Section 2-3 Rules for rounding Rounding Numbers (cont.) Rules for rounding Rule 1: If the digit to the right of the last significant figure is less than 5, do not change the last significant figure. Rule 2: If the digit to the right of the last significant figure is greater than 5, round up to the last significant figure. Rule 3: If the digits to the right of the last significant figure are a 5 followed by a nonzero digit, round up to the last significant figure.

Section 2-3 Rules for rounding (cont.) Rounding Numbers (cont.) Rules for rounding (cont.) Rule 4: If the digits to the right of the last significant figure are a 5 followed by a 0 or no other number at all, look at the last significant figure. If it is odd, round it up; if it is even, do not round up.

Practice Problems: 41(33-34) Round all numbers to four significant figures. Write the answers to problem 34 in scientific notation 33. 84 791 kg 38.5432 g 256.75 cm 4.9356 m

Practice Problems: 41(33-34) Round all numbers to four significant figures. Write the answers to problem 34 in scientific notation 34. 0.000 548 18 g 136 758 kg 308 659 000 mm 2.0145 mL

Section 2-3 Addition and subtraction Multiplication and division Rounding Numbers (cont.) Addition and subtraction Round numbers so all numbers have the same number of digits to the right of the decimal. Multiplication and division Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.

When adding and subtracting, limit and round your answer to the least number of decimal places in any of the numbers that make up your answer. Example 2: 123.25 ml + 46.0 ml + 86.257 ml =225,507 ml The answer is expressed as 225.5 ml since 46.0 ml has only one decimal place.

Practice Problems: 41 (35-36) Complete the following addition and subtraction problems. Round off the answers when necessary. 35. a. 43.2 cm + 51.0 cm + 48.7 cm b. 258.3 kg + 257.11 kg + 253 kg c. 0.0487 mg + 0.058 34 mg + 0.004 83 mg

Practice Problems: 41 (35-36) Complete the following addition and subtraction problems. Round off the answers when necessary. 35a. 43.2 cm + 51.0 cm + 48.7 cm 142.9 cm Addition and subtraction Round numbers so all numbers have the same number of digits to the right of the decimal.

Practice Problems: 41 (35-36) Complete the following addition and subtraction problems. Round off the answers when necessary. 35b. 258.3 kg + 257.11 kg + 253 kg 768 kg Addition and subtraction Round numbers so all numbers have the same number of digits to the right of the decimal.

Your Turn... Complete the following addition and subtraction problems. Round off the answers when necessary. 35c. 0.0487 mg + 0.058 34 mg + 0.004 83 mg

Practice Problems: 41 (35-36) Complete the following addition and subtraction problems. Round off the answers when necessary. 36. a. 93.26 cm - 81.14 cm b. 5.236 cm - 3.14 cm c. 4.32 x 103 cm - 1.6 x 103 cm

Practice Problems: 41 (35-36) Complete the following addition and subtraction problems. Round off the answers when necessary. 36a. 93.26 cm 81.14 cm 12.12 cm Addition and subtraction Round numbers so all numbers have the same number of digits to the right of the decimal.

Practice Problems: 41 (35-36) Complete the following addition and subtraction problems. Round off the answers when necessary. 36b. 5.236 cm - 3.14 cm 2.10 cm Addition and subtraction Round numbers so all numbers have the same number of digits to the right of the decimal.

Your Turn... Complete the following addition and subtraction problems. Round off the answers when necessary. 36. c. 4.32 x 103 cm - 1.6 x 103 cm

Calculations using Significant Figures When multiplying and dividing, limit and round to the least number of significant figures in any of the factors. Example 1: 23.0 cm x 432 cm x 19 cm =188,784 cm3 The answer is expressed as 190,000 cm3 since 19 cm has only two significant figures.

Practice Problems: 42 (37-38) Complete the following calculations. Round off the answers to the correct number of significant figures. 37. a. 24 m x 3.26 m b. 120 m x 0.10 m c. 1.23 m x 2.0 m d. 53.0 m x 1.53 m

Practice Problems: 42 (37-38) Complete the following calculations. Round off the answers to the correct number of significant figures. 37. a. 24 m x 3.26 m 78 m2 Multiplication and division Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.

Practice Problems: 42 (37-38) Complete the following calculations. Round off the answers to the correct number of significant figures. 37b. 120 m x 0.10 m 14 m2 Multiplication and division Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.

Your Turn... Complete the following calculations. Round off the answers to the correct number of significant figures. 37. c. 1.23 m x 2.0 m d. 53.0 m x 1.53 m

Practice Problems: 42 (37-38) Complete the following calculations. Round off the answers to the correct number of significant figures. 38. a. 4.84 m/2.4 s b. 60.2 m/20.1 s c. 102.4 m/51.2 s d. 168 m/58 s

Practice Problems: 42 (37-38) Complete the following calculations. Round off the answers to the correct number of significant figures. 38a. 4.84 m ÷ 2.4 s 4.84 m 2.4 s 2.0 m/s Multiplication and division Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.

Practice Problems: 42 (37-38) Complete the following calculations. Round off the answers to the correct number of significant figures. 38. b. 60.2 m/20.1 s 60.2 m 20.1 s 3.00 m/s Multiplication and division Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.

Your Turn... Complete the following calculations. Round off the answers to the correct number of significant figures. 38. c. 102.4 m/51.2 s d. 168 m/58 s