Uncertainty & Errors in Measurement

Waterfall by M.C. Escher

Keywords

        Uncertainty Precision Accuracy Systematic errors Random errors Repeatable Reproducible Outliers

Measurements = Errors

 Measurements 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) (2) ____ significant figures in 62cm 3 ____ significant figures in 100.00 g.

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

Measurements Sig. Fig.

1000s Unspecified 1 x 10 3 s 1.0 x 10 3 s 1.00 x 10 3 s 1.000 x 10 3 s 1 2 3 4

Measurements

0.45 mol dm -3 4.5 x 10 -1 s mol dm -3 4.50 x 10 -1 s mol dm -3 4.500 x 10 -1 s mol dm -3 4.5000 x 10 -1 s mol dm -3 4 5 2 3

Sig. Fig.

2

Random (Precision) Errors

 An 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 by

    The 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 Errors

   By 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 Errors

   An 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

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

All the values are ____________.

(a) What is the mean temperature? The temperature is reported as

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 Errors

  Control the variables in your lab.

Design a “perfect” procedure ( not ever realistic)

Errors Systematic errors Apparatus Random errors Equal chance of reading being high or low from 1 measurement to the next Way in which readings are taken

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 20 0 C but the true reading is 19 0 C .

This gives us a

precise

but

inaccurate

reading. If you have consistently obtained a reading of 20 0 C 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 25 0 C.

Scenario I You consistently obtain a pH reading of 6.45 +/- 0.05

Accuracy: Precision:

Scenario II You consistently obtain a pH reading of 8 +/-2 Accuracy: Precision:

Calculations involving addition & subtraction

When 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 & division

When 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

Example

When the temperature of 0.125kg of water is increased by 7.2

0 C. 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 0 C -1 x 7.2

0 C = Since the temperature recorded only has 2 sig fig, the answer should be written as ____________

Multiple math operations

Example:

 5.254 + 0.0016

  2.231×10 -3 34.6

Uncertainties in calculated results

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

Absolute & Percentage Uncertainty

Consider measuring 25.0cm

3 with a pipette that measures to +/- 0.1 cm 3 .

We write Absolute Uncertainty 

cm

3 Percentage Uncertainty 0.1

25.0

 100%  0.4% The uncertainties are themselves approximate and are generally not reported to more than 1 significant fgure.

Percentage Uncertainty & Percentage Error

Percentage uncertainty = absolute uncertainty  100% measured value Percentage error = accepted value-experimental value  100% accepted value

When adding or subtracting measurement , add

the absolute uncertainties Example

Initial temperature = 34.50

0 C Final temperature = 45.21

0 C    0.05 C  0.05 C   Change in temperature, ΔH

When multiplying or dividing measurement, add the

percentage uncertainties Example

volume = 14.1 cm 3     0.05cm

3  What is the density?

Example

Calculate the following: (a)  5.2 0.1

m

  

m

 (b)  

m



m



Example: initial volume from the final volume. The volume delivered is   Final vol = 38.46

Initial vol = 12.15

What is total volume delivered?

Example

The concentration of a solution of hydrochloric 1.00 0.05

cm 3 . 10.0 0.1

 Calculate the number of moles and give the absolute uncertainty.

When multiplying or dividing by a pure number,

multiply or divide the uncertainty by that number Example



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

 4.3 ± 0.5cm

 3

Averaging : repeated measurements can lead to an average value for a calculated quantity.

Example

Average ΔH =[+100kJmol -1 ( 10%)+110kJmol -1 108kJmol -1 = 106kJmol -1

Calculations Add & Subtract Multiply & Divide No. of decimal places No. of significant figures No. with the fewest sig fig used determines the sig fig to be used for the answer.

Factory made thermometers

Assume that the liquid in the thermometer is calibrated by taking the melting point at 0 0 C and boiling point at 100 0 C (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 -10 0 C to 110 0 C with uncertainty +/- 0.2

0 C. Upon trust, we can reasonably state the room temperature is 20.1

0 C +/- 0.2

0 C.

Graphical Technique

  y-axis : values of dependent variable x-axis : values of independent variables

Plotting Graphs

      Give 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

74,0

Change in volume of a fixed gas heated at a constant pressure

Temperature ( 0 C) Volume of Gas (cm 3 ) 20.0

60.0

30.0

40.0

63.0

64.0

50.0

60.0

70.0

67.0

68.0

72.0

72,0 70,0 68,0 66,0 64,0 62,0 60,0 58,0 0,0 10,0 20,0 30,0 40,0 50,0

temperature ( 0 C)

60,0 70,0 80,0

   Graphs can be useful to us in predicting values.

Interpolation – determining an unknown value within the limits of the values already measured.

Extrapolation measured.

– requires extending the graph to determine an unknown value that lies outside the range of the values