Measurements

1. Measurements Error Analysis Exact & Inexact Measurements Significant Figures

2. Error Analysis • Error Analysis is the study and evaluation of uncertainty in a measurement • No measurement can be free of uncertainties • The whole structure and application of science depend on measurements • Evaluating and keeping uncertainties to a minimum is crucial

3. Error Analysis • Error (in science) does not mean mistake • Error refers to the fact that ALL measurements are uncertain to some degree • Any measurement that we take is inexact and subject to some error

4. Error Analysis • Error can be reduced • Ex. Carpenter is asked to determine height of door • Rough estimate: 210 cm (could be 205 cm or 215 cm) • Tape measure: 211.3 cm (more precise) • Carpenter is still subject to error. His measurement cannot tell you if the door’s height is 211.300 cm or 211.301 cm

5. Accuracy vs. Precision • Accuracy is how close a measurement is with its actual/accepted value • Precisionis the degree to which repeated measurements give the same result • If you measure the length of a sheet of paper using a ruler with a lot of “marks” you will get similar results each time (more precise) • If your ruler had fewer “marks” or you just estimated the length of the paper, you would get varying results (less precise)

6. Accuracy vs. Precision The gravitational acceleration on earth is 9.8 m/s2 Group 1 makes the following measurements: 9.7 m/s2 9.5 m/s2 9.6 m/s2 9.8 m/s2 Group 2 makes the following measurements: 8.8 m/s2 8.6 m/s2 8.9 m/s2 8.7 m/s2 Are the groups as accurate, precise, both or neither?

7. Accuracy vs. Precision The density of gold is 19.3 g/mL Group 1 makes the following measurements: 15.2 g/mL 20.1 g/mL 24.1 g/mL 18.5 g/mL Group 2 makes the following measurements: 27.1 g/mL 30.5 g/mL 28.2 g/mL 29.9 g/mL Are the groups as accurate, precise, both or neither?

8. Significant Figures Communicating precision and determining which digits are significant are important in science • We do not want to lie. • Using a $0.35 ruler, we couldn’t measure the width of a textbook to be 4.02341736 cm • Significant figures are the digits in a measurement that are presented to communicate how precise a measurement is

9. Determining Significant Figures • All nonzero digits are significant 754 cm 12 mL 9.8 m/s2 • Zeros between significant digits are significant 1001 mm 34.03 g 9108.3 m3

10. Determining Significant Figures • All nonzero digits are significant 754 cm 3 s.f. 12 mL2 s.f. 9.8 m/s2 2 s.f. • Zeros between significant digits are significant 1001 mm 4 s.f. 34.03 g4 s.f. 9108.3 m3 5 s.f.

11. Determining Significant Figures • Leading zeros and zeros to the left of the first nonzero digit are NOT significant 0.0034 in. 0.21 kL • When a number ends in zeros that are to the left of an understood decimal point, those zeros are NOT significant 7600 nm 20 V 870 ft.

12. Determining Significant Figures • Leading zeros and zeros to the left of the first nonzero digit are NOT significant 0.0034 in. 2 s.f. 0.21 kL2 s.f. • When a number ends in zeros that are to the left of an understood decimal point, those zeros are NOT significant 7600 nm 2 s.f. 20 V 1 s.f. 870 ft. 2 s.f.

13. Determining Significant Figures • When a number ends in zeros that are to the right of a decimal point, those zeros are significant. 2.00 L 501.00 mg 1.00 s • When a number ends in zeros that are to the left of an understood decimal point, a bar over a zero or an added decimal point can indicate significance. 7Ō0 km 70Ō km 700. km

14. Determining Significant Figures • When a number ends in zeros that are to the right of a decimal point, those zeros are significant. 2.00 L 3 s.f. 501.00 mg 5 s.f. 1.00 s 3 s.f. • When a number ends in zeros that are to the left of an understood decimal point, a bar over a zero or an added decimal point can indicate significance. 7Ō0 km 2 s.f. 70Ō km 3 s.f. 700. km 3 s.f.

15. Exact Measurements • Some measurements are exact and have an infinite amount of significant figures. • None of these are physical measurements. • Counting numbers • Fractions • Numbers written out in plain English • English to English conversions (12 in. = 1 ft.) • Metric to metric conversions (1000 mg = 1 g)

16. Inexact Measurement • All measurement that we make are inexact • All metric to English conversions are inexact • Except: There are exactly 2.54 cm in 1 inch

17. Practice • Determine the number of significant figures in a measurement • Round measurements to specified amount of significant figures

18. Adding/Subtracting Measurements • When adding or subtracting measurements, the sum or difference can only be as precise as the least precise measurement 2.003251 cm -> goes out 6 places + 1.20 cm -> goes out 2 places 3.203251 cm wrong – too much precision is being communicated

19. Adding/Subtracting Measurements 2.003251 cm -> goes out 6 places + 1.20 cm -> goes out 2 places 3.20 cm CORRECT • The least precise measurement here is the 1.20 cm • In addition or subtraction, sum or difference can only be as precise as the least precise measurement

20. Check Yourself • Put a checkmark over the rightmost significant digit • After addition or subtraction, round the sum or difference to the leftmost check 43.25 mm 43.25 mm + 21 mm+ 21 mm 64.25 mm 64 mm check round

21. Examples 35.7 m 760 mL 1 000 A 5.0 s + 2.354 m+ 240 mL+900 A+ 5 s

22. Practice • Practice adding and subtracting measurements and rounding to display the appropriate number of significant figures • SHOW THE CORRECT UNIT

23. Multiplying & Dividing Measurements • When mult. & dividing, the result should contain as many significant figures as the measurement with the fewest significant figures • Ex. (10.0 m)(4.0 m) = 40. m2 3 s.f. 2 s.f. Answer must have 2 s.f. • Ex. (3.000 m)(9 m) = 30 m2 4 s.f. 1 s.f. Answer must have 1 s.f.

24. Examples (526.2 cm)(401 cm) = (2.00 m) _ = (1.00 s)(2.0 s) (2x103 J) _ = (1.00 mol)(273 K)

25. Practice • Practice multiplying and dividing measurements and rounding to display the appropriate number of significant figures • SHOW THE CORRECT UNIT