Section 2.3 1 2 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),

Section 2.3

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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 2-3Section 2.3 Uncertainty in Data

• Define and compare accuracy and precision.

experiment: a set of controlled observations that test a hypothesis

• Describe the accuracy of experimental data using error and percent error.

• Apply rules for significant figures to express uncertainty in measured and calculated values.

ComprehensionVerbs

• 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

ApplicationVerbs• Apply• Modify• Build• Construct• Solve• Report• Sketch• produce

Product• Diagram• Sculpture• Photograph• Forecast• Illustration• List• Project• Puzzle• Cartoon• Filmstrip

Section 2-3Section 2.3 Uncertainty in Data (cont.)

accuracyprecisionerror

Measurements contain uncertainties that affect how a result is presented.

percent errorsignificant figures

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Exact Inexact

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Accuracy Precision

Section 2-3Accuracy 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.

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good precision & good accuracy

poor accuracy but good precision

good accuracy but poor precision poor precision & poor accuracy

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Section 2-3Accuracy and Precision (cont.)

• Error is defined as the difference between and experimental value and an accepted value.

Section 2-3Accuracy and Precision (cont.)

• The error equation is error = experimental value – accepted value.

• Percent error expresses error as a percentage of the accepted value.

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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)

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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.


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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.


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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


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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 StudentsThe Accepted Density of Table Sugar is 1.59 g/cm3

Student A Student B Student CTrial 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

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29.Calculate the percent errors for Student B’s trials.30.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 CTrial 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

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B/ Density B/ ErrorsTrial 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

0.09 x 100 = 5.66%1.59

0.14 x 100 = 8.80%1.59

29.


Practice Problems: 38 (29-30)Errors for Data in Table 2-3

Student A Student B Student CTrial 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

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Errors for Data in Table 2-3

C/Density C/ ErrorsTrial 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

0.10 x 100 = 6.92%1.59

0.12 x 100 = 7.55%1.59

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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 2-3Significant Figures

• Often, precision is limited by the tools available.

• Significant figures include all known digits plus one estimated digit.

Section 2-3Significant 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.

Pacific or Atlantic?• Decimal Present?• Count from the Pacific

• Decimal Absent?• Count from the Atlantic

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A little trick for “sig figs”

31.80

564300.0020

10000

Practice Problems: 39 (31-32)

Determine the number of significant figures in each measurement.

31.a. 508.0 Lb. 820 400.0 Lc. 1.0200 x 105 kgd. 807 000 kg

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Practice Problems: 39 (31-32)Determine the number of significant figures in each

measurement.31. a.

508.0

4 30

– Rule 3: All final zeros to the right of the decimal are significant.


measurement.31. b.

820 400.0

7 31



measurement.31. c.

1.0200 x 105

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measurement.31. d.

807 000

3 33

Rule 1: Nonzero numbers are always significant.



32.a. 0.049 450 sb. 0.000482 mLc. 3.1587 x 10-8 gd. 0.0084 mL

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measurement.32a.

0.049 450

5 35

Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation


measurement.

32b. 0.000482

3 36

Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation

Your Turn ...


32.c. 3.1587 x 10-8 gd. 0.0084 mL

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Section 2-3Rounding 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 2-3Rounding 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.


• 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.a. 84 791 kgb. 38.5432 gc. 256.75 cmd. 4.9356 m

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Practice Problems: 41(33-34)Round all numbers to four significant figures. Write

the answers to problem 34 in scientific notation34.a. 0.000 548 18 gb. 136 758 kgc. 308 659 000 mmd. 2.0145 mL

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• 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.

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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

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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

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• 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

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• Addition and subtractionRound 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

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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

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problems. Round off the answers when necessary.36a.

93.26 cm - 81.14 cm 12.12 cm

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Addition and subtraction



problems. Round off the answers when necessary.36b.

5.236 cm - 3.14 cm 2.10 cm

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• Addition and subtraction


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

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Calculations using Significant FiguresWhen 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.

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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

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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

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Multiplication and division

Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.



37b.

120 m x 0.10 m 14 m2

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Your Turn...Complete the following calculations. Round off the


37. c. 1.23 m x 2.0 m d. 53.0 m x 1.53 m

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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

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38a. 4.84 m ÷ 2.4 s

4.84 m 2.4 s2.0 m/s

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38. b. 60.2 m/20.1 s

60.2 m20.1 s3.00 m/s

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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

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Your Turn...

Documents

Section 2.3 1 2 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),