6/25/20081Measurement Estimating the value or magnitude of some
physical quantity relative to a unit of measurement Described in
terms of accuracy accuracy and precision
precisionLengthMassTimeTemperaturemeter, footkilogram,
poundsecondCelsius, Fahrenheit6/25/2008 2 Measurement, Uncertainty
and DeviationAccuracy versus PrecisionAccuracy PrecisionDegree of
conformity of a measurement to the actual valueDegree of agreement
among a series of repeated measurementsMeasured in terms of
deviation Measured in terms of uncertaintydeviation, accuracy
uncertainty, precisionHigh accuracy Low precisionLow accuracy High
precision6/25/2008 3 Measurement, Uncertainty and
DeviationUncertainty Dictated by the sensitivity of the measuring
device or the fineness of the scale:1 cm (least precise)0.1 cm0.01
cm (most precise)Uncertaintyuncertainty, precision6/25/2008 4
Measurement, Uncertainty and DeviationReporting UncertaintyThe
uncertainty (precision) of a measurement can be reported in terms
of Order-of-Magnitude Estimate Significant Figures Maximum
Pessimism Statistical Treatment6/25/2008 5 Measurement, Uncertainty
and DeviationSignificant FiguresAs we improve the sensitivity of
the measuring device, the number of SFs increases:Postage Scale3 1
g 1 SF (least precise)2 SF3 SF (most precise)Reporting
UncertaintySFs, precision6/25/2008 6 Measurement, Uncertainty and
Deviation6/25/20082Counting SFs1. All non-zero digits are
considered significant. 123.452. Zeros appearing anywhere between
two non-zero digits are significant. 101.123. Leading zeros are not
significant. 0.000124. Trailing zeros in a number containing a
decimal point are significant.12.23005. A bar may be placed over
the last significant digit; any trailing zeros following this are
insignificant. 1300Reporting Uncertainty(5 SF)(5 SF)(2 SF)(6 SF)(3
SF)6/25/2008 7 Measurement, Uncertainty and DeviationAddition and
Subtraction with SFs When adding or subtracting measurements,the
answer can contain no more decimalplaces than the least precise
measurement.Reporting Uncertainty(1 DP) (3 DP)150.0 + 0.507 =
150.507(1 DP)= 150.5(2 DP) (1 DP)6.56 3.1 = 3.46(1 DP)=
3.56/25/2008 8 Measurement, Uncertainty and DeviationAddition and
Subtraction with SFs When one or both measurements has nodecimal
place, convert convert first first to to scientific
scientificnotation notation, then proceed as usual150 + 5.07 = 1.5
102 (1 DP)+ 0.0507 102 (4 DP)= 1.5507 102= 1.6 102 (1 DP) =
160Reporting Uncertainty6/25/2008 9 Measurement, Uncertainty and
DeviationMultiplication and Division with SFs When multiplying or
dividing, the answer cancontain no more significant figures than
theleast precise measurement.Reporting Uncertainty(4 SF) (2
SF)0.005580 67 = 0.37386(2 SF)= 0.37(3 SF) (2 SF)2.53 / 450 =
0.005622(2 SF)= 0.00566/25/2008 10 Measurement, Uncertainty and
DeviationMaximum Pessimism Measurement is expressed as a range of
valueswherein the true value lies:x = xx = x(%)Reporting
UncertaintyBest estimateExpectation valueAbsolute uncertaintyBest
estimateExpectation valueRelative uncertainty6/25/2008 11
Measurement, Uncertainty and DeviationGetting the best
estimateReporting UncertaintyTrial 1 2 3 4 5T ( C) 20.03 20.12
20.04 20.10 20.09Expectation value = (20.03 + 20.12 + 20.04 + 20.10
+ 20.09) /5 = 20.08 CAbsolute uncertainty:T = max{|20.03 20.08|;
|20.12 20.08|}T = 0.05 CRelative uncertainty:T(%) = 0.05/20.08
100%T(%) = 0.2 %6/25/2008 12 Measurement, Uncertainty and
Deviation6/25/20083Getting the best estimateReporting
UncertaintyTrial 1 2 3 4 5T ( C) 20.03 20.12 20.04 20.10 20.09Best
Estimate = 20.08 0.05 C or 20.08 C 0.2%= 20.08 C (expectation
value) T = 0.05 C (absolute uncertainty) T(%)= 0.2% (relative
uncertainty)6/25/2008 13 Measurement, Uncertainty and DeviationWhen
measuring a quantity, obtain as many trials as possible1, then get
the best estimate1Minimum of five trials6/25/2008 14 Measurement,
Uncertainty and DeviationAddition and SubtractionWhen adding or
subtracting measurements, the absolute uncertainty of the result is
the sum of the absolute uncertainties of the individual
measurementsMarky: 56.6 0.5 kgBogart: 71.65 0.05 kgTotal mass=
(56.6 0.5) + (71.65 0.05 kg)= (56.6 + 71.65) (0.5 + 0.05) kg=
128.25 0.55 kgTotal mass= 128.3 0.6 kg (1 DP)Uncertainty
Propagation6/25/2008 15 Measurement, Uncertainty and
DeviationMultiplication and DivisionWhen multiplying or dividing
measurements, the relative uncertainty of the result is the sum of
the relative uncertainties of the individual measurementsArea=
(29.25 0.05) (36.7 0.5) cm2= (29.25 0.171%) (36.7 1.36%) cm2=
(29.25 36.7) cm2 (0.171 + 1.36)%= 1073.475 cm2 1.531%Area= 1070 cm2
2% (relative uncertainty)orArea= 1070 20 cm2 (absolute
uncertainty)Uncertainty Propagation29.2 cm 0.5 cm36.75 cm 0.05
cm6/25/2008 16 Measurement, Uncertainty and DeviationDeviationCan
be measured in terms of Absolute deviation abs. dev. =
|experimental value theoretical value| Relative deviation (percent
error)% error = |experimental value theoretical value|
100%theoretical value6/25/2008 17 Measurement, Uncertainty and
DeviationAcceptability of MeasurementA measurement is valid or
acceptable if:1. Absolute deviation < Absolute uncertainty2. %
error < % errormax6/25/2008 18 Measurement, Uncertainty and
Deviation6/25/20084Summary6/25/2008 Measurement, Uncertainty and
Deviation 19MeasurementUncertainty(Precision)Significant
FiguresBest Estimatex = xDeviation(Accuracy)Absolute
Deviation=|xexpt- xtheo|Percent Error=|xexpt- xtheo| 100%xtheo
