Uncertainty In Measurement

1. Uncertainty In Measurement Accuracy, Precision, Significant Figures, and Scientific Notation

2. ACCURACY • A measure of how close a measurement comes to the accepted or true value of whatever is being measured • Accepted valueis a quantity used by general agreement of the scientific community (usually found in a reference manual)

3. PRECISION • Measure of how close a series of measurements are to one another • Measurements can: • Be very precise without being accurate • Have poor precision and poor accuracy • Have good accuracy and good precision

4. ERROR • Difference between experimental value and accepted value • Do you recall what accepted value is? • Ea = | Observed – Accepted|

5. PERCENT ERROR • Since it is close to impossible to measure (through experimentation) anything and reach the accepted value, there must be some way to determine just how close you actually got – that is called percent error. • Percent error is simply a mathematical formula. • % Error = (Ea ÷ Accepted Value) ×100

6. SIGNIFICANT FIGURES

7. SIGNIFICANT FIGURES • Measurement that includes all of the digits that are known PLUS a last digit that is estimated.

8. SIGNIFICANT FIGURE RULE #1 • Every nonzero digit is significant • Examples: 24.7 meters has 3 significant figures 0.473 meter has 3 significant figures 714 meters has 3 significant figures 245.4 meters has 4 significant figures 4793 meters has 4 significant figures

9. SIGNIFICANT FIGURES RULE #2 • Zeros between nonzero digits are significant • Examples: 7003 meters has 4 significant figures 40.79 meters has 4 significant figures 0.40093 meters has 5 significant figures

10. SIGNIFICANT FIGURE RULE #3 • Zeros appearing in front of nonzero digits are not significant • Examples: 0.032 meters has 2 significant figures 0.0003 meters has 1 significant figure 0.0000049 meters has 2 significant figures

11. SIGNIFICANT FIGURES RULE #4 • Zeros at the end of a number and to the right of the decimal place are always significant. • Examples: 43.00 meters has 4 significant figures 1.010 meters has 4 significant figures 9.000 meters has 4 significant figures

12. SIGNIFICANT FIGURES RULE #5 • Zeros at the end of a number but to the left of the decimal are not significant UNLESS they were actually measured and not rounded. • To avoid ambiguity, use scientific notation to show all significant figures if measured amounts with no rounding. • THIS IS A DIFFICULT RULE TO UNDERSTAND SO LET’S TALK FOR A BIT.

13. RULE #5 continued 300 meters (actually measured at 299) has 1 significant figure, but 300. meters (actually measured at 300.) has 3 significant figures. The actual (not rounded) amount should be shown as 3.00 x 102 meters. The rounded 300 meters (299) can also be shown in scientific notation but with only 1 significant figure: 3 x 102 meters.

14. CALCULATIONS USING SIGNIFICANT FIGURES • In all cases, round to the correct number of significant figures as the LAST step. • Your final answer cannot be more precise than the measured values used to obtain it. • Scientific notation is often helpful in rounding your final answer to the correct number of significant figures.

15. ADDITION/SUBTRACTION RULE • Answers will always be reported with the same number of decimal places as the measurement with the least numberof decimal places. • Example: 12.52 m + 349.0 m + 8.24 m • The “math” answer would be 369.76 m • However, the precise answer can only have one decimal place: 369.8 m or 3.698 x 102 m

16. ADDITION/SUBTRACTION EXAMPLES • 560.12 grams + 278.1 grams = 838.22 grams Precise Answer would be 838.2 or 8.382 x 102 grams 454 cm + 2.15 cm + 200 cm = 656.15 cm Precise Answer would be 656 or 6.56 x 102 cm 0.0010 meters – 0.123 m = - 0.122 m Precise Answer would be -0.122 or -1.22 x 10-1 m 2.321 L – 1.1145 L = 1.2065 L Precise Answer would be 1.207 or 1.207 x 100 m

17. MULTIPLICATION/DIVISION RULE • Round the final answer to the same number of significant figures as the measurement with the least numberof significant figures. • Example: 7.55 m x 0.34 m • “Math” answer will be 2.567 m2 • But, the precise answer will be 2.6 m2 because the measurement 0.34 m only has 2 significant figures.

18. MULTIPLICATION/DIVISION EXAMPLES 2.3 g/mL x 12.335 mL = 28.3705 g Precise answer would be 28 or 2.8 x 101 grams 5.45 g/mL x 15.145 mL = 82.54025 g Precise answer would be 82.5 or 8.25 x 101 grams 35.6 g / 2.3 mL = 15.47826087 g/mL Precise answer would be 15 or 1.5 x 101 g/mL 15.565 g / 3.56 mL = 4.372191011 g/mL Precise answer would be 4.37 or 4.37 x 100 g/mL

19. MEASUREMENTSWriting them out! • Scientific Notation: the product of two numbers; a coefficient and 10 raised to a power “Product”: means multiplication Coefficient always has one digit followed by a decimal and then the rest of the significant figures

20. Numbers to Scientific Notation • To change any number to scientific notation, move the decimal point directly behind the very first digit, counting how many places you move. Look at these examples: • 36,000 meters = 3.6 x 104 meters: I moved the “understood” decimal 4 places to the left   

21. 245,000,000 buttons = 2.45 x 108 buttons: I moved the understood decimal 8 places to the left. • 150. Grams = 1.50 x 102 grams: I moved the decimal 2 places to the left. Note: I also put a zero on the end of my scientific notation. • These examples are all BIG numbers (or numbers greater than one) so the exponents are positive.

22. Numbers to Scientific Notation • 0.036 meters = 3.6 x 10-2 meters: I moved the decimal 2 places to the right  • 0.0000245 liters = 2.45 x 10-5 liters: I moved the decimal 5 places to the right

23. Small to Scientific Notation • 0.150 Grams = 1.50 x 10-1 grams: I moved the decimal 1 place to the right. Note: I also put a zero on the end of my scientific. • These examples are all small numbers (or numbers less than one) so the exponents are negative.

24. Determine the number of significant figures: Work-out these problems in your notes: 1) 0.502 6) 1362205.2 7) 450.0 x 103   2) 0.0000455 8) 1000 x 10-3 3) 0.000984 4) 0.0114 x 104 9) 1.29 5) 2205.2 10) 0.982 x 10-3

25. Bell Ringer • Please take out a sheet of paper and number down to 10 • You will have 8 minutes

26. Bell Ringer To Scientific Notation: To decimal: 1) 3427  3.427 x 103 6) 1.56 x 104 15600 4.56 x 10-3 7) 0.56 x 10-2  2) 0.00456 0.0056 8) 0.000459 x 10-1 3) 123,453 1.23453x 105 0.0000459 4) 3100.0 x 102 9) 0.0209 x 10-3 3.1000 x 105 0.0000209 5) 1362205.2 10) 0.00259 x 103 1.3622052 x 106 2.59

27. Bell Ringer #2 • Please take out a sheet of paper and number down to 10 • You will have 8 minutes

28. Bell Ringer #2 To Scientific Notation: To decimal: 1) 4005  6) 4.58 x 104 7) 0.321 x 10-4  2) 0.000698 8) 0.000895 x 10-3 3) 25,514 4) 814,524 9) 0.0114 x 103 5) 23,564.12 10) 5.124 x 103

29. Work-out these problems in your notes: Addition and subtraction rule 1) 6.18 x 10-4 + 4.72 x 10-4 2) 9.10 x 103 + 2.2 x 106 3) 1913.0 - 4.6 x 103 4) 4.25 x 10-3 - 1.6 x 10-2 5) 2.34 x 106  + 9.2 x 106

30. Work-out these problems in your notes: Multiplication and Division rule 1) 8.95 x 107/ 1.25 x 105 2) (4.5 x 102)(2.45 x 1010) 3) 3.9 x 6.05 x 420 4) 14.1 / 5  5) (1.54 x 105)(3.5 x 106)