Uncertainty In Uncertainty In MeasurementMeasurement

Accuracy, Precision, Accuracy, Precision,

Significant Figures, and Significant Figures, and Scientific NotationScientific Notation

ACCURACYACCURACY

A measure of how close a measurement A measure of how close a measurement comes to the comes to the accepted or true value accepted or true value of of whatever is being measuredwhatever is being measured

Accepted valueAccepted value is a quantity used by general is a quantity used by general agreement of the scientific community agreement of the scientific community (usually found in a reference manual)(usually found in a reference manual)

PRECISIONPRECISION

Measure of how close a series of Measure of how close a series of measurements are to one anothermeasurements are to one another

Measurements can:Measurements can: Be very precise without being accurateBe very precise without being accurate Have poor precision and poor accuracyHave poor precision and poor accuracy Have good accuracy and good precisionHave good accuracy and good precision

ERRORERROR

Difference between experimental Difference between experimental value and accepted valuevalue and accepted value

Do you recall what Do you recall what accepted valueaccepted value is? is?

EEaa = | Observed – Accepted = | Observed – Accepted ||

PERCENT ERRORPERCENT ERROR

Since it is close to impossible to Since it is close to impossible to measure (through experimentation) measure (through experimentation) anything and reach the accepted anything and reach the accepted value, there must be value, there must be some way to some way to determine just how close you actually determine just how close you actually got – that is called percent error.got – that is called percent error.

Percent error is simply a mathematical Percent error is simply a mathematical formula.formula.

% Error = (E% Error = (Eaa ÷ Accepted Value) ×100 ÷ Accepted Value) ×100

SIGNIFICANT FIGURESSIGNIFICANT FIGURES

SIGNIFICANT FIGURESSIGNIFICANT FIGURES

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

SIGNIFICANT FIGURE RULE #1SIGNIFICANT FIGURE RULE #1

Every nonzero digit is significantEvery nonzero digit is significant Examples: Examples:

24.7 meters has 3 significant figures24.7 meters has 3 significant figures0.473 meter has 3 significant figures0.473 meter has 3 significant figures714 meters has 3 significant figures714 meters has 3 significant figures245.4 meters has 4 significant figures245.4 meters has 4 significant figures4793 meters has 4 significant figures4793 meters has 4 significant figures

SIGNIFICANT FIGURES RULE #2SIGNIFICANT FIGURES RULE #2

Zeros between nonzero digits are Zeros between nonzero digits are significantsignificant

Examples:Examples:

7003 meters has 4 significant figures7003 meters has 4 significant figures

40.79 meters has 4 significant figures40.79 meters has 4 significant figures

0.40093 meters has 5 significant 0.40093 meters has 5 significant figuresfigures

SIGNIFICANT FIGURE RULE #3SIGNIFICANT FIGURE RULE #3

Zeros appearing in front of Zeros appearing in front of nonzero digits nonzero digits are not are not significantsignificant

Examples:Examples:

0.032 meters has 2 significant figures0.032 meters has 2 significant figures

0.0003 meters has 1 significant figure0.0003 meters has 1 significant figure

0.0000049 meters has 2 significant 0.0000049 meters has 2 significant figuresfigures

SIGNIFICANT FIGURES RULE #4SIGNIFICANT FIGURES RULE #4

Zeros at the end of a number and to Zeros at the end of a number and to the right of the decimal place the right of the decimal place are are alwaysalways significant. significant.

Examples:Examples:

43.00 meters has 4 significant figures43.00 meters has 4 significant figures

1.010 meters has 4 significant figures1.010 meters has 4 significant figures

9.000 meters has 4 significant figures9.000 meters has 4 significant figures

SIGNIFICANT FIGURES RULE #5SIGNIFICANT FIGURES RULE #5

Zeros at the end of a number but to Zeros at the end of a number but to the left of the decimal are not the left of the decimal are not significant significant UNLESSUNLESS they were they were actually measured and not rounded.actually measured and not rounded.

To avoid ambiguity, use scientific To avoid ambiguity, use scientific notation to show all significant figures if notation to show all significant figures if measured amounts with no rounding.measured amounts with no rounding.

THIS IS A DIFFICULT RULE TO THIS IS A DIFFICULT RULE TO UNDERSTAND SO LET’S TALK FOR A BIT.UNDERSTAND SO LET’S TALK FOR A BIT.

RULE #5 continuedRULE #5 continued

300 meters (actually measured at 299) 300 meters (actually measured at 299) has 1 significant figure, but 300. has 1 significant figure, but 300. meters (actually measured at 300.) meters (actually measured at 300.) has 3 significant figures. The actual has 3 significant figures. The actual (not rounded) amount should be (not rounded) amount should be shown as 3.00 x 10shown as 3.00 x 1022 meters. meters.

The rounded 300 meters (299) can The rounded 300 meters (299) can also be shown in scientific notation also be shown in scientific notation but with only 1 significant figure: 3 x but with only 1 significant figure: 3 x 101022 meters. meters.

CALCULATIONS USING CALCULATIONS USING SIGNIFICANT FIGURESSIGNIFICANT FIGURES

In all cases, round to the correct In all cases, round to the correct number of significant figures as the number of significant figures as the LAST step.LAST step.

Your final answer cannot be more Your final answer cannot be more precise than the measured values precise than the measured values used to obtain it.used to obtain it.

Scientific notation is often helpful in Scientific notation is often helpful in rounding your final answer to the rounding your final answer to the correct number of significant figures.correct number of significant figures.

ADDITION/SUBTRACTION RULEADDITION/SUBTRACTION RULE

Answers will always be reported with Answers will always be reported with the same number of decimal places as the same number of decimal places as the measurement with the measurement with the least the least numbernumber of decimal places.of decimal places.

Example: 12.52 m + 349.0 m + 8.24 mExample: 12.52 m + 349.0 m + 8.24 m The “math” answer would be 369.76 mThe “math” answer would be 369.76 m However, the precise answer can only However, the precise answer can only

have one decimal place: have one decimal place: 369.8 m or 3.698 x 10369.8 m or 3.698 x 1022 m m

ADDITION/SUBTRACTION ADDITION/SUBTRACTION EXAMPLESEXAMPLES

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 cmPrecise Answer would be 656 or 6.56 x 102 cm

0.0010 meters – 0.123 m =

- 0.122 mPrecise Answer would be -0.122 or -1.22 x 10-1 m

2.321 L – 1.1145 L =

1.2065 LPrecise Answer would be 1.207 or 1.207 x 100 m

MULTIPLICATION/DIVISION MULTIPLICATION/DIVISION RULERULE

Round the final answer to the same Round the final answer to the same number of significant figures as the number of significant figures as the measurement measurement with the least with the least numbernumber of significant figures.of significant figures.

Example: 7.55 m x 0.34 mExample: 7.55 m x 0.34 m ““Math” answer will be 2.567 mMath” answer will be 2.567 m22

But, the precise answer will be 2.6 But, the precise answer will be 2.6 mm22 because the measurement 0.34 because the measurement 0.34 m only has 2 significant figures.m only has 2 significant figures.

MULTIPLICATION/DIVISION MULTIPLICATION/DIVISION EXAMPLESEXAMPLES

2.3 g/mL x 12.335 mL =

28.3705 gPrecise answer would be 28 or 2.8 x 101 grams

5.45 g/mL x 15.145 mL =

82.54025 gPrecise answer would be 82.5 or 8.25 x 101

grams 35.6 g / 2.3 mL =

15.47826087 g/mLPrecise answer would be 15 or 1.5 x 101 g/mL

15.565 g / 3.56 mL = 4.372191011 g/mLPrecise answer would be 4.37 or 4.37 x 100 g/mL

MEASUREMENTSWriting them out!

Scientific Notation: the product of Scientific Notation: the product of two numbers; a coefficient and 10 two numbers; a coefficient and 10 raised to a powerraised to a power

““Product”:Product”: means multiplication means multiplication

CoefficientCoefficient always has one digit always has one digit

followed by a decimal and then followed by a decimal and then the the

rest of the significant figuresrest of the significant figures

Numbers to Scientific NotationNumbers to Scientific Notation

To change any number to scientific To change any number to scientific notation, move the decimal point notation, move the decimal point directly behind the very first digit, directly behind the very first digit, counting how many places you counting how many places you move. Look at these examples:move. Look at these examples:

36,000 meters = 3.6 x 1036,000 meters = 3.6 x 1044 meters: I meters: I moved the “understood” decimal 4 moved the “understood” decimal 4

places to the left 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.

 

0.036 meters = 3.6 x 100.036 meters = 3.6 x 10-2-2 meters: I meters: I moved the decimal 2 places to the moved the decimal 2 places to the

right right    0.0000245 liters = 2.45 x 100.0000245 liters = 2.45 x 10-5-5 liters: I liters: I

moved the decimal 5 places to the moved the decimal 5 places to the

rightright

Numbers to Scientific NotationNumbers to Scientific Notation

Small to Scientific NotationSmall to Scientific Notation

0.150 Grams = 1.50 x 100.150 Grams = 1.50 x 10-1-1 grams: I grams: I moved the decimal 1 place to the moved the decimal 1 place to the right. Note: I also put a zero on the right. Note: I also put a zero on the end of my scientific.end of my scientific.

   These examples are all small These examples are all small

numbers (or numbers less than one) numbers (or numbers less than one) so the exponents are negative.so the exponents are negative.

Work-out these problems in Work-out these problems in your notes:your notes:

Determine the number of significant figures:Determine the number of significant figures:

3) 0.000984

4) 0.0114 x 104

5) 2205.2

6) 1362205.2

7) 450.0 x 103  

8) 1000 x 10-3

9) 1.29

10) 0.982 x 10-3

2) 0.0000455

1) 0.502

Bell RingerBell Ringer

Please take out a sheet of paper and Please take out a sheet of paper and number down to 10number down to 10

You will have 8 minutesYou will have 8 minutes

Bell RingerBell Ringer

To Scientific Notation:To Scientific Notation: To decimal:To decimal:

1) 3427 

2) 0.00456

3) 123,453

4) 3100.0 x 102

5) 1362205.2

6) 1.56 x 104

7) 0.56 x 10-2 

8) 0.000459 x 10-1

9) 0.0209 x 10-3

10) 0.00259 x 103

3.427 x 103

4.56 x 10-3

1.23453x 105

3.1000 x 105

1.3622052 x 106

15600

0.0056

0.0000459

0.0000209

2.59

Bell Ringer #2Bell Ringer #2

Please take out a sheet of paper and Please take out a sheet of paper and number down to 10number down to 10

You will have 8 minutesYou will have 8 minutes

Bell Ringer #2Bell Ringer #2

To Scientific Notation:To Scientific Notation: To decimal:To decimal:

1) 4005 

2) 0.000698

3) 25,514

4) 814,524

5) 23,564.12

6) 4.58 x 104

7) 0.321 x 10-4 

8) 0.000895 x 10-3

9) 0.0114 x 103

10) 5.124 x 103

Work-out these problems in Work-out these problems in your notes:your notes:

Addition and subtraction ruleAddition and subtraction rule

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                                         

2) 9.10 x 103 + 2.2 x 106                                                              

1) 6.18 x 10-4 + 4.72 x 10-4                                                        

Work-out these problems in Work-out these problems in your notes:your notes:

Multiplication and Division ruleMultiplication and Division rule

3) 3.9 x 6.05 x 420

4) 14.1 / 5 

5) (1.54 x 105)(3.5 x 106)                                          

2) (4.5 x 102)(2.45 x 1010)                                                              

1) 8.95 x 107/ 1.25 x 105