Topic 11: Measurement and Data Processing and Analysis Topic 11.1- Uncertainties and Errors in Measurement and Results
Key Terms Random Error- above or below true value, usually due to limitations of equipment Systematic Error- in one direction, usually due to instrument or method (procedure) error Precision- a measure of the certainty (±) Accuracy- How close the value is to the accepted value
Systematic vs Random Error A result is said to be accurate if it is relatively free from systematic error A results is said to be precise if the random error is small
Some Causes of Systematic Error Physical Errors in Measuring Devices Thermometer was dropped and has small air bubbles in it Improper or sloppy use of measuring device Measured values in instead of Not selecting correct size/range for/ of the instrument Instrument wasn t calibrated or cleaned Parallax Error Chemical Splashes Ambient Conditions Temperature, pressure, or air currents changed during the experiment
What to do if Systematic Error Occurs? Best Answer Start Over- Change your method / procedure to reduce the error There will ALWAYS be systematic error anyway, BUT we can do our best to reduce it if we notice it during an experiment Decent Answer Keep consistent! Example- You always measured the volume of the liquid from above (parallax error) - keep doing that so your results are precise and discuss how this systematic error would affect the accuracy of your results in the conclusion
How to Reduce Random Error It will ALWAYS occur if measurements are being taken Reduce by repeating the experiment (multiple trials and average the results) Repetition of at least 3 Range of 5 to determine a relationship
In a Conclusion ALWAYS discuss Systematic Error- get specific! WHAT were the systematic errors? How would they affect the accuracy of your results (mass too high? Too low? How would it affect final result?) Random Error- get specific! Where in your experiment would random error inherently occur? (any measurement) What did you do to reduce random error? (repetition)
Determining Uncertainty Unless the instrument tells you, the uncertainty is measured in one of two ways: Analog: For glassware and similar instruments Digital: The uncertainty (±) is half the smallest increment of the instrument The uncertainty is the smallest digit
Analog Uncertainty Example 1: The volume is between 36cm 3 and 37cm 3 The uncertainty is half the smallest digit = 0.5 You always estimate 1 place value past the smallest increment (estimate the 0.1 place) when measuring Answer: 36.7 ± 0.5 cm 3
Analog Uncertainty Example 2: A B Don t forget to estimate 1 digit of your measurement Ruler A: Uncertainty = ½ of smallest increment of 10cm = ± 5cm Measurement A = 23 ± 5cm Ruler B: Uncertainty = ½ of smallest increment of 1cm = ± 0.5cm Measurement A = 22.5 ± 0.5cm
Digital Uncertainty Example 1: Uncertainty = Smallest digit 25.04 ± 0.01g
Significant Figures Rule 1- Zeros in the middle of a number are significant Ex. 94.072 = 5 sig figs Ex. 94072 = 5 sig figs Rule 2- Zeros at the beginning of a number are not significant Ex. 0.0834 = 3 sig figs Can also be written as 8.34 x 10-2 = 3 sig figs)
Significant Figures Rule 3- Zeros at the end of a number and after the decimal point are significant Ex. 138.200 = 6 sig figs Can be written as 1.38200 x 10 2 = 6 sig figs) Rule 4- Zeros at the end of a number and before the implied decimal point may or may not be significant (use scientific notation to distinguish!) Ex. 138200 = could be 4 sig figs or 6 sig figs (I say 4!) Can write as 1.38200 x 10 5 = 6 sig figs Can write as 1.382 x 10 5 = 4 sig figs
Rules for Sig Figs (Multiplication and Division) Rule 1- Answer cannot have more sig figs than any original number Round answer to least number of sig figs
Practice: Multiplication and Division 1) 32.27 x 1.54 = 49.6958 = 2) 3.68 0.07925 = 46.4353312 =
Rules for Sig Figs (Addition and Subtraction) Rule 2- Answer should only have the same number of unit placings as the most imprecise number Round answer to fewest number of decimal places Example: Example: 0.011 + 0.01= 0.021 0.02 (round to 2 decimal places) 8.251 + 921 = 929.251 930 or 9.30 x 10 2 (round to the 1 s place)
Practice: Addition and Subtraction 25.5 + 34.270 59.770 59.8 320 + 12.5 332.5 330 0r 3.30 x 10 2
Key Terms Absolute Uncertainty- Shows the value of the uncertainty in a reading 1.0 ± 0.5 g Percentage Uncertainty: Provides meaningful context to the uncertainty measurement - Example: 56.6 ± 0.5 g (0.5/ 56.6) x 100 = 0.883 % Round to 1 S.F. = 56.6 ± 1% uncertainty - Example: 1.1± 0.5 g (0.5/ 1.1) x 100 = 45.45% Round to 1 S.F. = 1.1 ± 50% Uncertainty
Propagation of Uncertainty- Woohoo! Each measured value you record (Raw Data) contains uncertainty When you used your Raw Data and perform calculations with you, you now have Processed Data Processed Data must always take into account the uncertainty values from the raw data ---- Here s where the Propagation of Uncertainty rules come in!
Propagation of Uncertainty Provides understanding of Random Error Addition and Subtraction Rule: add the uncertainties together Table 1: Determining the Density of Water Mass of Beaker (± 0.01g) Mass of Beaker + Water (± 0.01g) Volume (± 0.5 cm 3 ) 20.01 30.03 10.5 Mass of Water = (Mass of Beaker + Water) - (Mass of Beaker) Example: Mass of Water = (30.03) - (20.01)= 10.02g Uncertainty = 0.01g + 0.01g = 0.02g Answer = 10.02 ± 0.02 g
Propagation of Uncertainty Mult. and Division Multiplication and Division Rule: Step 1: Turn absolute uncertainty into percent uncertainty % Uncertainty = 1 S.F. (or whatever makes sense- generally 1 decimal place) Step 2: Add percent uncertainties together Step 3: Round the final answer to a similar number of places as the final absolute uncertainty Example: Molarity= 3.7 ± 0.2 mol dm -3 Volume = 0.65 ± 0.05 dm 3 Step 1: Moles = 3.7 x 0.65 = 2.4 moles Molarity % Uncertainty = (0.2 / 3.7) 100 = 5% Volume % Uncertainty = (0.05 / 0.65) 100 = 8%
Propagation of Uncertainty- Mult. and Division Step 2: Add the percent uncertainties together Percent Uncertainty of Final Answer = 5% + 8 % = 13% Step 3: Round the final answer to a similar number of places as the final absolute uncertainty 13% of 2.4 = 0.312 mol, therefore we can continue rounding to 2.4 moles in our answer Final Answer = 2.4 moles ± 13 % or 2.4 ± 0.3 moles Percent Uncertainty Absolute Uncertainty
Key Terms Percentage Error- How close the experiment/measured value is to the literature/ accepted value (should provide insight into systematic error was involved) Example: A student measures the density of Aluminum to be 2.81 g cm -3. The accepted value is 2.70 g cm -3. [ (2.81-2.70) / (2.70) ] x 100 = 4.07% Error
Conclusion Reminders Always use the following words in your conclusion! Random Error Systematic Error Accuracy Precision
Measuring with Glassware- How to reduce error
Using Glassware- Graduated and Volumetric Pipette
Using Glassware- Burette
Using Glassware- Volumetric Flask