Uncertainty in Measurements

1. Uncertainty in Measurements Accuracy, Precision and Error - You must be able to make reliable measurements in the lab. Ideally, measurements are both correct and reproducible.

2. Precision- is a measure of how close a series of measurements are to one another Example- Which set of measurements is more precise? • 2g, 3g, 4g • 2.1g, 2.2g, 2.3g b.is correct because the measurements are closer together

3. Accuracy- is a measure of how close a measurement comes to the actual or true value of the object measured

4. Error in Measurements Percent error can be used to evaluate the accuracy of a measurement, it must be compared to the correct value. % error = experimental value - accepted value accepted value

5. Significant Figures in a measurement include all of the digits that are known, plus a last digit that is estimated. Measurements must always be recorded to the correct number of significant digits. Calculated answers depend upon the number of significant figures in the values used in the calculation

6. Making measurements with sig figs

7. Making measurements with sig figs

8. Determining Significant Digits Nonzero digits:all are considered to be significant Example: 3279g has 4 sig figs Leading zeros:none are significant. They are considered to be place holders and not part of the measurement Example: 0.0045 has 2 sig figs (only the 4 and the 5)

9. Captive zeros: zeros between two nonzero digits They are considered to be significant. Example: 5.007 has 4 sig figs

10. Trailing zeros: zeros at the end of a measurement They are counted onlyif the number contains a decimal point Examples: 100 has 1 sig fig 100. has 3 sig figs 100.0 has 4 sig figs 0.0100 has 3 sig figs

11. Scientific notation: all numbers listed in the coefficient are considered to be significant. Examples: 1.7 x 10-4 has 2 sig figs 1.30 x 10-2 has 3 sig figs

12. Exact numbers: have unlimited significant digits Examples: 4 chairs (determined by counting) 1 inch = 2.54 cm (determined by definition)

13. How many sig figs are in each measurement? • 123 meters 3 2. 0.123 meters 3 3. 40,506 meters 5 4. 9.8000 x 103 meters 5 5. 30.0 meters 3

14. More practice 6. 22 meter sticks unlimited sig figs (count sig figs for measured values only) 7. 0.07080 meters 4 sig figs 8. 98,000 meters 2 sig figs

15. Sig figs in calculations Rounding:In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. Once you know the number of significant digits your answer should have, round to that many digits, counting from the left.

16. Rounding Round each measurement to the number of sig figs shown in parentheses. • 314.721 meters (round to 4 sig figs) 314.7 meters 2. 0.001775 meters (round to 2 sig figs) 0.0018 meters 3. 8792 meters (round to 2 sig figs) 8800 meters

17. Rules for multiplication and division: When multiplying or dividing with measurements,round the answer to the same number of sig figs as the measurement with theleast number of sig figs.

18. Rules for multiplication and division: Example:7.55 meters x 0.34 meters = 2.567 meters. Round the answer to 2.6 meters (2 sig figs) (The position of the decimal point has nothing to do with the rounding process when multiplying and dividing measurements.)

19. Rules for addition and subtraction: The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places.

20. Rules for addition and subtraction Example: 7.055 meters + 0.35 meters = 7.405 meters. Round the answer to 7.41 meters (2 places to the right of the decimal)

21. Sample Problems • 7.55 meters x 0.34 meters = 2.6 m2 2. 2.10 meters x 0.70 meters = 1.5 m2 3. 2.4526 meters / 8.4 seconds= .29 m/s 4. 0.365 meters – 0.0200 seconds = 0.345 m/s

22. More problems • 12.52 meters + 349.0 meters + 8.24 meters = 369.8 m 6. 74.626 meters- 28.34 meters= 46.29 m 7. 80.0 meters + 0.0002 meters = 80.0 m