Uncertainty & Errors in Measurement

1. Uncertainty & Errors in Measurement

2. Waterfall by M.C. Escher

3. Keywords • Uncertainty • Precision • Accuracy • Systematic errors • Random errors • Repeatable • Reproducible • Outliers

4. 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.

5. 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

6. 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?

7. Significant Figures • ____ significant figures in 62cm3 • ____ significant figures in 100.00 g. The 0s are significant in (2) What is the uncertainty range?

8. Exercise • Express in scientific notation. • 0.04g • 222 cm3 • 0.030g

9. Exercise 2. Express in scientific notation. • 15.00 cm3 • 150 s • 0.0123 g • 150.0 g

10. Random (Precision) Errors • An error that can based on • individual inter__________. • 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.

11. 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.

12. 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.

13. 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 • What is the mean temperature? The temperature is reported as as it has a range of Read example in the notes.

14. 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.

15. 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 ____________. • 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

16. 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 ______ .

17. Minimizing Systematic Errors • Control the variables in your lab. • Design a “perfect” procedure ( not ever realistic)

19. How trustworthy is your reading?

20. If all the temperature reading is 200C but the true reading is 190C . This means, is 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

21. systematic error accuracy random error precision

22. Exercise

23. Putting it together Example The accurate pH for pure water is 7.00 at 250C. Scenario I You consistently obtain a pH reading of 6.45 +/- 0.05 Accuracy: Precision:

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

25. What can be done realistically to improve the investigation? Scenario 2 Choose a better instrument or method of obtaining the pH so that your random error is lower. The more precise your reading , the more likely you can reproduce the same results among your partners. Scenario 1 Recalibrate pH prole. Do not use a better probe because the instrument is very precise

26. Absolute & Percentage Uncertainty Consider measuring 25.0cm3 with a pipette that measures to +/- 0.1 cm3. We write Absolute Uncertainty Percentage Uncertainty

27. Percentage Uncertainty & Percentage Error

28. Presenting Uncertainties in your reading When adding and subtracting, the number of decimal places is important. Example: Suppose we need to find the total mass of two pieces of zinc of mass 1.21g and 0,56g. The total mass is

29. Example: Report the total mass of solution prepared by adding 50g of water to 1.00g of sugar. Would the use of a more precise balance for the mass of sugar result in a more precise total mass? The precision of the total is limited by the precision of the mass of the water. Using a more precise balance for the mass of sugar would not improve the precision.

30. When +/- measurement , add the absolute uncertainties Example (15.5 +/- 0.5 mL) + (12.2 +/- 0.2 mL)

31. Example: (13.5 +/- 0.2 g) + (14.55 +/- 0.05 g)

32. Example: When using a burette ( +/- 0.02 cm3) , you subtract the initial volume from the final volume. The volume delivered is Final vol = 38.46 +/- 0.02 cm3 Initial vol = 12.15 +/- 0.02 cm3 Total volume delivered =

33. Multiplying & Dividing When multiplying or dividing, it is the number of significant figures that is important. The number with the fewest significant figures used in the calculation determines how many significant figures should be used when quoting the answer. Example When the temperature of 0.125kg of water is increased by 7.20C. 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 ____________

34. With the absolute uncertainties included, the percentage uncertainties should be used and then convert it back to absolute uncertainties when the final results is presented. Example Consider a sample of sodium chloride with a mass of 5.00+/- 0.01g and a volume of 2.3 +/-cm3.What is the density?

35. Example

36. Example The concentration of a solution of hydrochloric acid = moldm3 and the volume = cm3 . Calculate the number of moles and give the absolute uncertainty.

38. Factory made thermometers Assume 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.

39. Instrument s 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.

40. Graphical technique • 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.

41. 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.

42. Line of Best Equation