Measurement and Uncertainties (HL/SL) – Guided Notes

Slide 3: Why does reporting data accurately matter?

• Science is communal
• Science is open to scrutiny through peer review
• All measurements have uncertainties
• Uncertainties limit legitimate conclusions that are drawn from data
• Recognition of uncertainties allow science to progress
• Given the uncertainty within the practice of science, what behavior characteristics should scientists exhibit when conducting and publishing scientific research?
• “One aim of the physical sciences has been to give an exact picture of the material world. One achievement…has been to prove that this aim is unattainable.” (J. Bronowski)
• What are the implications of this claim for the aspirations of science?

Slide 4: Uncertainty in measurement

• Different laboratory ___________________ have different levels of uncertainty.
• The ___________________ of an analogue scale is (±) half of the smallest division.
• The last digit of a digital scale is _________________.
• ___________________ of a digital scale is (±) the smallest scale division.

• 41.9 ± 0.5 mL
• Report to the tenths place

• 1.40 ± 0.05 mL
• Report to the __________________ place

• *________________ to read from the bottom of the meniscus

• 100.00 ± 0.01 g
• Report to the __________________ place

Slide 5: Other sources of uncertainty

• All _____________________ should be noted, even if not quantifiable
• Time ____________________
• Changes in _________________ colors
• Voltages of _______________________ cells
• Temperatures during _____________________ reactions

Slide 7: Significant figures in measurements

• Show the amount of ___________________ in a measurement.

• The digits in a ___________________ up to and including the first uncertain digit

• *Remember to use scientific notation to help report ____________________ to the correct number of sig figs.
• *____________________ ALWAYS have units!

Slide 8: Let’s Practice!

• How would you report the level of the liquid in the graduated cylinder shown in the image? (remember to include the level of uncertainty)
• 2. A reward is given for a missing diamond, which has a reported mass of 9.92 ± 0.05 g. You find a
• diamond and measure its mass as 10.2 ± 0.2 g. Could this be the missing diamond? Why or why not?
• 3. What is the number of significant figures in each of the following measurements?
• (a) 15.500 cm³ (b) 50 s (c) 0.00123 g (d) 150.00 g
• 4. Express the following measurements in scientific notation (keeping the correct number of significant
• figures).
• (a) 0.040 g (b) 220.0 cm³ (c) 0.00300 g (d) 0.300 °C

Slide 10: Experimental errors

• Two __________________:
• Random errors:
• Have multiple causes, including changes in surroundings (e.g. temperature, air currents, etc.), _______________________ the measurement readings, insufficient data
• have an equal ___________________ of being too high or too low
• Can be reduced by repeating ___________________
• 2. __________________ errors:
• occur as a result of poor ____________________ design or procedure
• Cannot be reduced by repeating ___________________
• Can be reduced by changing ____________________ design

• Occurs from the __________________ between the recorded value and the literature (accepted) value

Slide 11: Random error

• It is good practice to duplicate experiments in order for the results to be repeatable and reproducible.

• The same person duplicates the experiment and obtains the same results.

• Several different experimenters duplicate the experiment and obtain the same results.

• Example:
• The mass of a piece of magnesium ribbon is measured several times and the following results obtained:
• 0.1234 g 0.1232 g 0.1233 g 0.1234 g 0.1235 g 0.1236 g
• Average mass Mg: [0.1234 + 0.1232 + 0.1233 + 0.1235 + 0.1236]/5 = 0.1234 g
• Average mass Mg: 0.1234 ± 0.0002 g
• Why is the uncertainty ± 0.0002 g?

• Affects the precision of an experiment.

• Uncertainty: Range/2
• [0.1236-0.1232]/2 = 0.0002 g

• The range of data spans ±0.0002 g from the average!

Slide 12: Systematic error

• Example:
• The mass of a piece of magnesium ribbon is measured several times and the following results obtained (however, the balance was not zeroed correctly):
• 0.1236 g 0.1234 g 0.1235 g 0.1236 g 0.1237 g 0.1238 g
• *All values too high by 0.0002 g
• Average mass Mg: [0.1236 + 0.1234 + 0.1235 + 0.1237 + 0.1238]/5 = 0.1236 g
• Average mass Mg: 0.1236 ± 0.0002 g
• Examples of systematic errors:
• Measuring liquids from the top instead of the bottom of the meniscus
• Consistently overshooting the volume of a liquid delivered in a titration
• Instruments not calibrated correctly
• How do you know if you have a systematic error?

• Affects the accuracy of an experiment.

Slide 13: Accuracy vs. Precision

• Smaller __________________ error = greater accuracy
• Smaller random error = greater precision (____________________)

• How close a ___________________ is to the accepted value.

• How close ____________________ are to each other

Slide 14: Percentage uncertainties and errors

• ___________________ can be expressed using absolute, fractional, or percentage values.
• Absolute ___________________: actual uncertainty in raw data
• Fractional ___________________: absolute uncertainty
• (aka Relative ___________________) measured value
• Percent ___________________: absolute uncertainty × 100%
• ________________ value
• percent ___________________ ≠ percent error
• Percent error: |accepted value – ____________________ value| × 100%
• ________________ value

• _______________: Raw data: 23.4 mol dm⁻³ ± 0.2 mol dm⁻³

• 0.2 mol dm⁻³ = 0.0085
• 23.4 mol dm⁻³

• = 0.85% ___________________

• *Note: ____________________ values in calculations should not be rounded in order to avoid imprecision
• * If calculated ___________________ is ≥ 2%, report to 1 sig fig. If calculated uncertainty is < 2%, report to 2 sig figs.

Slide 15: Propagation of uncertainties in calculated results

• _____________________ in raw data must be propagated in a consistent manner during data processing.
• Addition and ___________________:
• Two burette ________________:
• _______________ reading: 15.05 ± 0.05 cm³
• Final _______________: 37.20 ± 0.05 cm³
• Reported value for volume _________________: 22.15 ± 0.10 cm³

• }

• Add absolute _____________________

Slide 16: Propagation of uncertainties in calculated results

• ______________________ and Division:
• ___________________ uncertainty when determining density:

• _______________: 12 g cm⁻³ ± 0.8 g cm⁻³

• *You can add the individual % _____________________ to get the total % uncertainty, then use the result to calculate the absolute uncertainty

Slide 17: Propagation of uncertainties: Summary

• Adding/Subtracting ____________________:
• Total absolute uncertainty is the sum of the individual absolute _____________________
• Multiplying/Dividing ____________________:
• Total % uncertainty is the sum of the individual percent _____________________
• Use the total % ___________________ to calculate the total absolute uncertainty

Slide 18: Sig Figs: Addition/Subtraction

• When adding/subtracting ____________________, the number of decimal places determines the precision of the calculated value.
• _______________ 1: adding the masses of 2 pieces of zinc
• Mass 1: 1.21 g ± 0.01 g
• Mass 2: 0.56 g ± 0.01 g
• Total mass: 1.77 g ± 0.02 g

• Example 2: ___________________ of a temperature increase
• Temp. 1: 25.2 °C ± 0.1 °C
• Temp. 2: 34.2 °C ± 0.1 °C
• ___________________ change: 34.2 – 25.2 = 9.0 ± 0.2 °C

• When adding/___________________: You cannot report a measurement with more precision than the measurement with the smallest number of decimal places (least precise measurement).

• Suppose a ________________ was made by adding 50 g of water to 1.00 g of sugar. 50 g + 1.00 g = 51 g
• The precision of the total mass is limited by the least precise ___________________ (the water).

Slide 19: Sig Figs: Multiplication/Division

• % ___________________:
• Mass: 0.01/5.00 = 0.002 = 0.2%
• Volume: 0.1/2.3.= 0.0435 = 4%
• Add the _____________________ to get 4.2%
• Absolute ___________________: 0.0455(2.1739) = 0.0989 ≈ 0.1 g cm-3

• When multiplying/dividing ____________________, the number of significant figures determines the precision of the calculated value.
• Example: ___________________ the density of sodium chloride
• Mass: 5.00 g ± 0.01 g
• Volume: 2.3 cm³ ± 0.1 cm³
• _______________: 5.00/2.3 =2.173913043 g cm⁻³ ≈ 2.2 g cm⁻³ ± 0.1 g cm⁻³

• When ___________________/dividing: You cannot report a measurement with more precision than the measurement with the smallest number of sig figs (least precise measurement).

Slide 20: Let’s Practice!

• The lengths of the sides of a wooden block are measured and their uncertainties are shown on the diagram below.
• What is the percentage and absolute uncertainty in the calculated area of the block?

• 45.0 ± 0.5 mm

• 25.0 ± 0.5 mm

Slide 21: How can errors/uncertainties help your experimental design?

• One measurement with a much higher uncertainty will have the major effect on the uncertainty of the final result
• Thermometers often produce the most uncertain results (may require more repetitions of experiment)
• Enthalpy changes in exothermic reactions that are lower than literature values can be due to heat loss to surroundings (suggests systematic error and an alteration in experimental design may be necessary).

Slide 22: Lord Kelvin (1824-1907): “When you can measure what you are speaking about and express it in numbers, you know something about it, but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind.”

• Brown, Catrin, and Mike Ford. Higher Level Chemistry. 2nd ed. N.p.: Pearson Baccalaureate, 2014. Print.

• How would you rewrite this quote using your own words?