Uncertainty in MeasurementsUncertainty in Measurements

Accuracy, Precision and ErrorAccuracy, Precision and Error

- You must be able to make reliable - You must be able to make reliable measurements in the lab. Ideally, measurements in the lab. Ideally, measurements are both correct and measurements are both correct and reproducible.reproducible.

PrecisionPrecision- is a measure of how close a - is a measure of how close a series of measurements are to one series of measurements are to one anotheranother

ExampleExample- Which set of measurements - Which set of measurements is more precise?is more precise?

a.a. 2g, 3g, 4g2g, 3g, 4gb.b. 2.1g, 2.2g, 2.3g2.1g, 2.2g, 2.3g

b.b. is correctis correct because the because the measurements are closer togethermeasurements are closer together

AccuracyAccuracy- is a measure of how close a - is a measure of how close a measurement comes to the actual or measurement comes to the actual or true value of the object measuredtrue value of the object measured

Error in MeasurementsError in Measurements

Percent errorPercent error can be used to evaluate the can be used to evaluate the accuracy of a measurement, it must be accuracy of a measurement, it must be compared to the correct value. compared to the correct value.

% error = % error =

experimental value - accepted valueexperimental value - accepted value accepted valueaccepted value

Significant FiguresSignificant Figures in a measurement in a measurement include all of the digits that are known, include all of the digits that are known, plus a last digit that is estimatedplus a last digit that is estimated..

MeasurementsMeasurements must always be recorded to must always be recorded to the correct number of significant digits.the correct number of significant digits.

Calculated answersCalculated answers depend upon the depend upon the number of significant figures in the values number of significant figures in the values used in the calculationused in the calculation

Making measurements with sig figsMaking measurements with sig figs

Making measurements with sig figsMaking measurements with sig figs

Determining Significant DigitsDetermining Significant DigitsNonzero digitsNonzero digits:: all are considered to be all are considered to be

significantsignificant

Example:Example: 3279g has 4 sig figs 3279g has 4 sig figs

Leading zerosLeading zeros:: none are significantnone are significant. They . They are considered to be place holders and not are considered to be place holders and not part of the measurementpart of the measurement

Example:Example: 0.0045 has 2 sig figs (only the 4 0.0045 has 2 sig figs (only the 4 and the 5)and the 5)

Captive zerosCaptive zeros:: zeros between two zeros between two nonzero digitsnonzero digits

They are considered to be They are considered to be significant.significant.

Example:Example: 5.007 has 4 sig figs 5.007 has 4 sig figs

Trailing zerosTrailing zeros:: zeros at the end of a zeros at the end of a measurementmeasurement

They are cThey are counted onlyounted only if the if the number contains a decimal pointnumber contains a decimal point

Examples:Examples:

1100 has 1 sig fig00 has 1 sig fig

100100. has 3 sig figs. has 3 sig figs

100.0100.0 has 4 sig figs has 4 sig figs

0.00.0100100 has 3 sig figs has 3 sig figs

Scientific notation:Scientific notation: all numbers listed all numbers listed in the coefficient are considered to in the coefficient are considered to be significant.be significant.

Examples:Examples:

1.71.7 x 10 x 10-4-4 has 2 sig figs has 2 sig figs

1.301.30 x 10 x 10-2 -2 has 3 sig figshas 3 sig figs

Exact numbers:Exact numbers: have unlimited have unlimited significant digitssignificant digits

Examples:Examples:

4 chairs (determined by counting)4 chairs (determined by counting)

1 inch = 2.54 cm (determined by 1 inch = 2.54 cm (determined by definition)definition)

How many sig figs are in each How many sig figs are in each measurement?measurement?

1.1. 123 meters123 meters 332.2. 0.123 meters 0.123 meters 333.3. 40,506 meters 40,506 meters 554.4. 9.8000 x 10 9.8000 x 1033 meters meters 555.5. 30.0 meters 30.0 meters 33

More practiceMore practice

6.6. 22 meter sticks 22 meter sticks

unlimitedunlimited sig figs (count sig figs sig figs (count sig figs for measured values only) for measured values only)

7.7. 0.07080 meters 0.07080 meters

4 sig figs4 sig figs

8.8. 98,000 meters 98,000 meters

2 sig figs2 sig figs

Sig figs in calculationsSig figs in calculationsRounding:Rounding: In general, a calculated In general, a calculated

answer cannot be more precise than answer cannot be more precise than the least precise measurement from the least precise measurement from which it was calculated.which it was calculated.

Once you know the number of significant Once you know the number of significant digits your answer should have, digits your answer should have, round to round to that many digitsthat many digits, counting from the left., counting from the left.

RoundingRoundingRound each measurement to the number of Round each measurement to the number of

sig figs shown in parentheses.sig figs shown in parentheses.

1.1. 314.721 meters (round to 4 sig figs)314.721 meters (round to 4 sig figs)

314.7 meters314.7 meters

2.2. 0.001775 meters (round to 2 sig figs) 0.001775 meters (round to 2 sig figs)

0.0018 meters0.0018 meters

3.3. 8792 meters (round to 2 sig figs) 8792 meters (round to 2 sig figs)

8800 meters8800 meters

Rules for Rules for multiplicationmultiplication and and division:division:

When multiplying or dividing with When multiplying or dividing with measurements,measurements, round the answer to round the answer to the same number of sig figs as the the same number of sig figs as the measurement with themeasurement with the least least number of sig figs.number of sig figs.

Rules for Rules for multiplicationmultiplication and and division:division:

Example:Example: 7.557.55 meters x 0. meters x 0.3434 meters meters

= = 2.52.567 meters. 67 meters. Round the answer to Round the answer to 2.62.6 meters (2 sig figs) meters (2 sig figs)

(The position of the decimal point has (The position of the decimal point has nothing to do with the rounding nothing to do with the rounding process when multiplying and process when multiplying and dividing measurements.)dividing measurements.)

Rules for Rules for additionaddition and and subtraction:subtraction:

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

Rules for Rules for additionaddition and and subtractionsubtraction

Example:Example: 7. 7.055055 meters + 0. meters + 0.3535 meters meters

= 7.405 meters. = 7.405 meters.

Round the answer to 7.Round the answer to 7.4141 meters (2 places meters (2 places

to the right of the decimal)to the right of the decimal)

Sample ProblemsSample Problems1.1. 7.55 meters x 0.34 meters =7.55 meters x 0.34 meters = 2.6 m2.6 m22

2.2. 2.10 meters x 0.70 meters = 2.10 meters x 0.70 meters = 1.5 m1.5 m22

3.3. 2.4526 meters / 8.4 seconds= 2.4526 meters / 8.4 seconds= .29 m/s.29 m/s4.4. 0.365 meters – 0.0200 seconds = 0.365 meters – 0.0200 seconds =

0.345 m/s0.345 m/s

More problemsMore problems

5.5. 12.12.52 52 meters + 349.meters + 349.00 meters meters + 8.+ 8.2424 meters =meters =

369.8 m369.8 m

6.6. 74. 74.626626 meters- 28. meters- 28.3434 meters= meters=

46.29 m46.29 m

7.7. 80.0 meters + 0.0002 meters =80.0 meters + 0.0002 meters =

80.0 m80.0 m