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

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

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

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

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

SIGNIFICANT FIGURES

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

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

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

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

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

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.

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.

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.

ADDITION/SUBTRACTION RULE Answers will always be reported with the same number of decimal places as the measurement with the least number of 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

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

MULTIPLICATION/DIVISION RULE Round the final answer to the same number of significant figures as the measurement with the least number of 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.

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

MEASUREMENTS Writing 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

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     

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.

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

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.

Work-out these problems in your notes: Determine the number of significant figures: 1) 0.502 6) 1362205.2 2) 0.0000455 7) 450.0 x 103   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

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

Bell Ringer To Scientific Notation: To decimal: 1) 3427 3.427 x 103 1) 3427  3.427 x 103 6) 1.56 x 104 15600 2) 0.00456 4.56 x 10-3 7) 0.56 x 10-2  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

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

Bell Ringer #2 To Scientific Notation: To decimal: 1) 4005 1) 4005  6) 4.58 x 104 2) 0.000698 7) 0.321 x 10-4  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

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                                         

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)