Uncertainty & Errors in Measurement

Waterfall by M.C. Escher

KeywordsUncertaintyPrecisionAccuracySystematic

errorsRandom errorsRepeatableReproducibleOutliers

Measurements = ErrorsMeasurements are done directly by humans

or with the help of

Humans are behind the development of instruments, thus there will always be

associated with all instrumentation, no matter

how precise that instrument is.

Uncertainty

When a physical quantity is taken, the uncertainty should be stated.

Example If the balance is accurate to +/-

0.001g, the measurement is 45.310g

If the balance is accurate to +/- 0.01g, the

measurement is 45.31g

Exercise

A reward is given for a missing diamond, which has a reported mass of 9.92 +/- 0.05g. You find a diamond and measure its mass as 10.1 +/- 0.2g. Could this be the missing diamond?

Significant Figures

(1)____ significant figures in 62cm3

(2)____ significant figures in 100.00 g.

Measurements

Sig. Fig. Measurements Sig. Fig.

1000s Unspecified 0.45 mol dm-3 2

1 x 103s 1 4.5 x 10-1s mol dm-3 2

1.0 x 103s 2 4.50 x 10-1s mol dm-3 3

1.00 x 103s 3 4.500 x 10-1s mol dm-3

4

1.000 x 103s 4 4.5000 x 10-1s mol dm-3

5

The 0s are significant in (2) What is the uncertainty range?

Random (Precision) ErrorsAn error that can based on

individual interpretation.Often, the error is the result of mistakes or errors.Random error is not ______ and can

fluctuate up or down. The smaller your random error is, the greater your ___________ is.

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 repeating measurements. If the same person duplicates the

experiment with the same results, the results are repeatable.

If several persons duplicate the results, they are reproducible.

10 readings of room temperature

19.9 , 20.2 , 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2, 22.3

(a) What is the mean temperature?

The temperature is reported as

as it has a range of

Read example in the notes.

Systematic ErrorsAn error that has a fixed margin,

thus producing a result that differs from the true value by a fixed amount.

These errors occur as a result of poor experimental design or procedure.

They cannot be reduced by repeating the experiment.

10 readings of room temperature

20.0 , 20.3 , 20.1, 20.1, 20.2, 20.0, 20.4, 20.0, 20.3

All the values are ____________.(a) What is the mean temperature?

The temperature is reported as

19.9 , 20.2 , 20.0, 20.0, 20.1, 19.9, 20.3, 19.9, 20.2

Examples of Systematic Errors

Measuring the volume of water from the top of the meniscus rather than the bottom will lead to volumes which are too ________.

Heat losses in an exothermic reaction will lead to ______ temperature changes.

Overshooting the volume of a liquid delivered in a titration will lead to volumes which are too ______ .

Minimizing Systematic ErrorsControl the variables in your lab.Design a “perfect” procedure

( not ever realistic)

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

How trustworthy is your reading?

Accuracy•How close to the accepted(true) value your reading is.

Precision•The reproducibility of your reading•Reproducibility does not guarantee accuracy. It could simply mean you have a very determinate systematic error.

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

Exercise

Putting it together

Example The accurate pH for pure water is 7.00

at 250C.

Scenario IYou consistently obtain a pH

reading of 6.45 +/- 0.05Accuracy:

Precision:

Scenario IIYou consistently obtain a pH

reading of 8 +/-2

Accuracy:

Precision:

Calculations involving addition & subtractionWhen adding and subtracting, the final

result should be reported to the same number of decimal places as the least no. of decimal places.

Example:(a) 35.52 + 10.3 (b) 3.56 – 0.021

Calculations involving multiplication & divisionWhen adding and subtracting, the final

result should be reported to the same number of significant figures as the least no. of significant figures .

Example:(a) 6.26 x 5.8 (b) 5.27

12

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 ____________

Multiple math operationsExample:

-35.254+0.00162.231×10

34.6

Uncertainties in calculated resultsThese uncertainties may be

estimated byfrom the smallest division from a

scalefrom the last significant figure in

a digital measurementfrom data provided by the

manufacturer

Absolute & Percentage Uncertainty

Consider measuring 25.0cm3 with a pipette that measures to +/- 0.1 cm3.We write

325.0 0.1cmAbsolute Uncertainty

0.1100% 0.4%

25.0

Percentage Uncertainty

The uncertainties are themselves approximate and are generally not reported to more than 1 significant fgure.

Percentage Uncertainty & Percentage Error

absolute uncertaintyPercentage uncertainty = 100%

measured value

accepted value-experimental valuePercentage error = 100%

accepted value

When adding or subtracting measurement , add the absolute uncertainties

ExampleInitial temperature = 34.500CFinal temperature = 45.210CChange in temperature, ΔH

00.05 C

00.05 C

When multiplying or dividing measurement, add the percentage uncertainties

ExampleGiven that mass = 9.24 g and volume = 14.1 cm3

What is the density?

0.005g

30.05cm

ExampleCalculate the following:(a)(b)

5.2 0.1 10.2 0.5m m

5.2 0.1 10.2 0.5m m

Example:When using a burette , you

subtract the initial volume from the final volume. The volume delivered is

Final vol = 38.46Initial vol = 12.15What is total volume delivered?

30.02cm

ExampleThe concentration of a solution of

hydrochloric acid = moldm-

3 and the volume = cm3 . Calculate the number of moles and give

the absolute uncertainty.

1.00 0.0510.0 0.1

When multiplying or dividing by a pure number, multiply or divide the uncertainty by that number

Example

4.95±0.05 ×10

Powers : When raising to the nth power,

multiply the percentage uncertainty by n.

When extracting the nth root, divide the percentage uncertainty by n.

Example 34.3±0.5cm

Averaging : repeated measurements can lead to an

average value for a calculated quantity.ExampleAverage ΔH=[+100kJmol-1( 10%)+110kJmol-1 ( 10%)+ 108kJmol-1 ( 10%)] 3= 106kJmol-1 ( 10%)]

Calculations

Add & Subtract

No. of decimal places

Factory made thermometersAssume that the liquid in the

thermometer is calibrated by taking the melting point at 00C and boiling point at 1000C (1.01kPa).

If the factory made a mistake, the reading will be biased.

Instruments have measuring scale identified and also the tolerance.

Manufacturers claim that the thermometer reads from -100C to 1100C with uncertainty +/- 0.20C.

Upon trust, we can reasonably state the room temperature is

20.10C +/- 0.20C.

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 Equation

Temperature (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.