Uncertainty & Errors in Measurement
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Transcript of Uncertainty & Errors in Measurement
Uncertainty & Errors in Measurement
Waterfall by M.C. Escher
ObjectivesDifference between random errors
(uncertainties) and systematic errors
Difference between precision and accuracy
RepeatableReproducibleOutliers
Calculations involving addition & subtractionWhen adding and subtracting quantities,
the absolute uncertainties are added. Example:(a) Mass of 1st zinc = 1.21g ± 0.01g Mass of 2nd zinc = 0.56g ± 0.01g Total mass of the 2 pieces of zinc =
(b) Final burette reading = 38.46 cm3 ± 0.05 cm3 Initial burette reading = 12.15 cm3 ± 0.05
cm3
Volume titrated =
WS
Calculations involving multiplication & divisionWhen multiplying or dividing quantities,
then the percent (or fractional) uncertainties are added.
Example:Molarity of NaOH(aq) = 0.20 M (± 0.05 M) Percentage uncertainty = Volume of NaOH(aq) = 25.00 cm3 (± 0.10 cm3)
Percentage uncertainty =
Therefore, the no. of moles of NaOH =
May convert % uncertainty back to absolute uncertainty. copy
ExampleWhen the temperature of 0.125kg of
water is increased by 7.20C. Find the heat required.
Heat required = mass of water x specific heat capacity x
temperature rise= 0.125 kg x 4.18 kJ kg-1 0C-1 x 7.20C=
Since the temperature recorded only has 2 sig fig, the answer should be written as ____________ WS
Multiple math operationsExample:
-35.254+0.00162.231×10
34.6
copy
Quoting values with uncertaintiesMeasured value ± uncertainty
Value you should quote
253.4 ± 0.3253.56 ± 0.10.06200 ± 0.0001261.4 ± 8261.4 ± 20261.4 ± 100
The uncertainty is usually quoted to one significant figure.Your measurement should be stated so that the significant is in the last significant figure.
Errors (uncertainties) in raw data
When a physical quantity is taken, the uncertainty should be stated
These uncertainties may be estimated by
from the smallest division from a scale
from the last significant figure in a digital measurement
from data provided by the manufacturer
.
Digital Instruments If the balance is accurate to +/- 0.001g, the measurement is
45.310g
If the balance is accurate to +/- 0.01g, the
measurement is 45.31gUncertainty for digital instrument : +/- the smallest division
Analogue InstrumentsA burette of value 34.1cm3
becomes 34.10cm3 (±0.05cm3)
Note: the volume is cited to 2 decimal places so as to be consistent with the uncertainty.
Uncertainty for analogue instrument:half of the smallest division.
Higher levels of uncertainty is normally indicated by an instrument manufacturer.
WS:Practice
Errors
Systematic errors
Apparatus
Way in which readings are taken
Random errors
Equal chance of reading being high or low from 1 measurement to the
next
Random ErrorsArise from the imprecision of
measurements and lead to readings being above or below the ‘true’ value.
Random Errors are caused byThe readability of the measuring
instrument.The effects of changes in the
surroundings such as temperature variations and air currents.
Insufficient data.The observer misinterpreting the
reading.
Minimizing Random ErrorsBy using more precise measuring
equipment repeating measurements so that
te random errors cancel out.
Systematic ErrorsArise from a problem in the
experiment set-up that results in the measured values deviating from the ‘true’ value in the same direction, that is always higher or always lower.
Examples of Systematic ErrorsMiscalibration of a measuring device.Measuring the volume of water from
the top of the meniscus rather than the bottom will lead to volumes which are too ________.
Overshooting the volume of a liquid delivered in a titration will lead to volumes which are too ______ .
Poor insulation in calorimetry experiments
Minimizing Systematic ErrorsControl the variables in your lab.Design a “perfect” procedure
( not ever realistic)
Percentage Uncertainty & Percentage Error
absolute uncertaintyPercentage uncertainty = 100%measured value
accepted value-experimental valuePercentage error = 100%accepted value
Systematic error can be identified by comparison with accepted literature values.
Practice Qn(a) Density =(b)Percentage uncertainty of(i) Mass(ii) Volume
(iii)Density
(c) Percentage error
Comment on the errorThe percentage error (4.5%) is
greater than the percentage uncertainty (2.9%)
The literature value does not fall within the range 0.63 +/- 0.02 g/ml.
Since random error is estimated by the uncertainty and it is smaller than the percentage error, systematic errors are at work making the measured data inaccurate.
Data from Preparation of a Standard Solution( Electronic Balance is accurate to ) Mass of anhydrous Na2CO3 =Titration ( Burette is accurate to )
( Measuring cylinder is accurate to ) of Na2CO3 is titrated withHCl.
1.104 0.001g g
Initial Volume
Final Volume
Volume of Acid
60.00 53.50 6.50 53.50 47.00 6.5047.00 40.00 7.00
Average : 6.70
0.001g
30.05cm
30.05cm 30.05cm 30.10cm
30.5cm3 310.0 0.5cm cm
3 36.70 0.10cm cm
Percentage uncertainties due to measurements
Mass of Na2CO3 =
Volume of HCl =
Volume of Na2CO3 =
Total percentage uncertainty
0.001 100% 0.0906%1.104
0.05 100% 0.7463%6.70
0.5 100% 5%10.0
0.0906% 0.7463% 5% 5.837%
How do we quote the value in the report?Molarity of HCl from experiment
=
Absolute uncertainty of molarity of HCl
Therefore the concentration of HCl is
1.104 1000 10 2 1000 0.3109106 100 1000 1 6.70
30.31 0.02moldm
5.837 0.3109 0.02 one significant figure100
Comparing % error & % random uncertaintySince the percentage error
(55.45%) is greater than the percentage random uncertainty (5.837%), it is suggested that the experiment involves some systematic errors.
How trustworthy is your reading?
Accuracy•How close a measured value is to the correct value.
Precision•The reproducibility of your reading.•How many significant figures there are in a measurement.
ExampleA mercury thermometer could
measure the normal boiling temperature of water as 99.50C (±0.50C) whereas
A data probe recorded it as 98.150C (±0.050C) .
Which is more accurate? more precise?
If all the temperature reading is 200C but the true reading is 190C .
This gives us a precise but inaccurate reading.
If you have consistently obtained a reading of 200C in five trials. This could mean that your thermometer has a large systematic error.
systematic error accuracy
random error precision
systematic error accuracy
random error precision
Calculations
Add & Subtract
No. of decimal places
Graphical Techniquey-axis : values of dependent
variablex-axis : values of independent
variables
Plotting GraphsGive the graph a title.Label the axes with both quantities and
units.Use sensible linear scales – no uneven
jumps.Plot all the points correctly.A line of best fit should be drawn clearly. It
does not have to pass all the points but should show the general trend.
Identify the points which do not agree with the general trend.
Line of Best EquationTemperature (0 C) Volume of Gas (cm3)
20.0 60.0
30.0 63.0
40.0 64.0
50.0 67.0
60.0 68.0
70.0 72.0
10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.054.0
56.0
58.0
60.0
62.0
64.0
66.0
68.0
70.0
72.0
74.0
Change in volume of a fixed gas heated at a constant pressure
temperature (0C)
Volu
me
(cm
3)
Graphs can be useful to us in predicting values.
Interpolation – determining an unknown value within the limits of the values already measured.
Extrapolation – requires extending the graph to determine an unknown value that lies outside the range of the values measured.