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.