Uncertainty in Uncertainty in MeasurementMeasurement

Accuracy vs. PrecisionAccuracy vs. Precision

UncertaintyUncertainty Basis for significant figures Basis for significant figures All measurements are uncertain to All measurements are uncertain to

some degreesome degree The last estimated digit represents The last estimated digit represents

the uncertainty in the measurementthe uncertainty in the measurement

Each Person may estimate a measurement differently

Person 16.63mls

Person 2 6.64mls

Person 3 6.65 mls

Rules for Counting Rules for Counting Significant FiguresSignificant Figures

1. 1. Non-zerosNon-zeros always count always count as significant figures:as significant figures:

34563456 hashas

44 significant figuressignificant figures

Rules for Counting Rules for Counting Significant FiguresSignificant Figures

2. 2. LeadingLeading zeroes do not zeroes do not count as significant count as significant figures:figures:

0.04860.0486 has has

33 significant figures significant figures

Rules for Counting Rules for Counting Significant FiguresSignificant Figures

3. 3. CaptiveCaptive zeroes always zeroes always count as significant figures:count as significant figures:

16.0716.07 hashas

44 significant figures significant figures

Rules for Counting Rules for Counting Significant FiguresSignificant Figures

4. 4. TrailingTrailing zeros (or zeros after a zeros (or zeros after a non-zero digit)non-zero digit) are significant are significant only if the number contains a only if the number contains a written written decimaldecimal point: point:

9.3009.300 has has 44 significant figures significant figures

100100 has has 11 significant figure significant figure

100100.. has has 33 significant figures significant figures

Sig Fig Practice #1Sig Fig Practice #1How many significant figures in the following?

1.0070 m 5 sig figs

17.10 kg 4 sig figs

100,890 L 5 sig figs

3.29 x 103 s 3 sig figs

0.0054 cm 2 sig figs

3,200,000 mL 2 sig figs

These all come from some measurements

Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations

Addition and SubtractionAddition and Subtraction: The : The number of decimal places in number of decimal places in the result equals the number the result equals the number of decimal places in the of decimal places in the least least preciseprecise measurement. measurement.

6.8 + 11.934 =6.8 + 11.934 =18.734 18.734 18.7 18.7 ((3 sig figs3 sig figs))

Rules for Significant Figures in Rules for Significant Figures in Mathematical OperationsMathematical Operations

Multiplication and DivisionMultiplication and Division:: # # sig figs in the result equals the sig figs in the result equals the number in the number in the least preciseleast precise measurement used in the measurement used in the calculation.calculation.

6.38 x 2.0 =6.38 x 2.0 = 12.76 12.76 13 13 (2 sig figs)(2 sig figs)

Precision vs. AccuracyPrecision vs. Accuracy PrecisionPrecision-- how repeatable how repeatable

Precision is determined by the uncertainty in Precision is determined by the uncertainty in the instrument used to take a measurement.the instrument used to take a measurement.

So . . . The precision of a measurement is So . . . The precision of a measurement is also how many decimal places that can be also how many decimal places that can be recorded for a measurement.recorded for a measurement.

1.476 grams has more precision than 1.5 1.476 grams has more precision than 1.5 grams.grams.

AccuracyAccuracy-- how correct - closeness to how correct - closeness to true value.true value.

Measurement ErrorsMeasurement Errors Random errorRandom error - equal chance of being - equal chance of being

high or low- addressed by averaging high or low- addressed by averaging measurements - expectedmeasurements - expected

Systematic error-Systematic error- same direction each same direction each timetime Want to avoid thisWant to avoid this Bad equipment or bad technique.Bad equipment or bad technique.

Better precision implies better accuracyBetter precision implies better accuracy You can have precision without accuracyYou can have precision without accuracy You can’t have accuracy without You can’t have accuracy without

precision (unless you’re really lucky).precision (unless you’re really lucky).

Percent ErrorPercent Error

Percent Error compares a measured Percent Error compares a measured value to its true value.value to its true value.

It measures the accuracy in your It measures the accuracy in your measurement.measurement.

%Error = %Error = Measured value – accepted Measured value – accepted value value x 100x 100

accepted valueaccepted value

Average DeviationAverage Deviation Average Deviation – measures the Average Deviation – measures the

repeatability (or precision) of your repeatability (or precision) of your measurements.measurements.

Deviation = measured value – average Deviation = measured value – average valuevalue

You calculate the deviation for each You calculate the deviation for each measurement and then take the average of measurement and then take the average of those deviations to get the “Average those deviations to get the “Average Deviation”Deviation”

Measurement is then reported as the Measurement is then reported as the average average ++ average deviation average deviation

For example: 6.64mls For example: 6.64mls ++ 0.01mls 0.01mls

Each Person may estimate a measurement differently

Deviation Person 1 6.63mls 0.01 mls Person 2 6.64mls 0.00 mls Person 3 6.65 mls 0.01 mls Average 6.64 mls +/- 0.01 mls