Topic 11: Measurement and data processing 11.1 Uncertainty and error in measurement 11.2 Uncertainties in calculated results 11.3 Graphical techniques -later!
From the syllabus
Precision v. Accuracy The accuracy of a measurement is an expression of how close the measured value is to the correct or true value. Expressed as a Percentage Difference The precision of a set of measurements refers to how closely the individual measurements agree with one another. Precision is a measure of the reproducibility or consistency of a result.
Imagine a dartboard!
Significant Figures What is 22.1 x 13.7? It s NOT 302.77! It s 303 Your answer cannot be more certain than the data you started with!
Remember! All non-zero digits count as significant figures: 4 Leading zeros are those that precede the first nonzero digit. Leading zeros are not counted as significant figures. 0.003 Captive zeros are zeros between non-zero integers. Captive zeros always count as significant figures. 406 Trailing zeros are zeros to the right of a number. Trailing zeros are significant if the number contains a decimal point.if there is no decimal point, the number of significant figures may be unclear. 1000
Scientific Notation 400 vs. 400. To avoid this confusion the number should be written in standard form or scientific notation. Rewrite as 4 x10 2 Rewrite as 4.00 x10 2
A Caveat Sig figs are only applied to measurements and calculations involving measurements. They do not apply to quantities that are inherently integers or fractions a stoichiometric ratio such as 2 or ½ mole, defined quantities (for example, one metre equals 100 centimetres), or conversion factors (multiplying by 100 to get a percentage or adding 273 to convert C to K).
Multiplication and division The result should have the same number of significant figures as the factor with the least number of significant figures. 2.54 2.6 = 6.604.
Addition and Subtraction The result should have the same number of decimal places as the number used with the fewest decimal places. Note that when adding and subtracting we are interested in decimal places, not significant figures. 3.467 + 4.5 + 3.66 = 11.627
Exercise 2.568 x 5.8 4.186 5.41 0.398 3.38 3.01 4.18 58.16 x (3.38 3.01)
Uncertainty of a Digital Reading +/- the last measurable digit If a measured mass is read on a digital scale as 18.34 g, then it s expressed as: 18.34+/- 0.01 g If a measure mass is read on a digital scale as 18.3g, then it s expressed as 18.3 +/- 0.1 g Imagine when you re on the scale and the last digit flickers between two...
Uncertainty of an Analog Reading +/- 0.5 of the last measurable digit. The volume of solution in this burette is 48.80 cm 3. The burette reading can be recorded as 48.80 ± 0.05 cm 3. The burette reading is between 48.75 cm 3 and 48.85 cm 3.
Reading a Meniscus When reading a meniscus you should always have your eyes level with the meniscus, and for aqueous solutions the volume is read from the bottom of the meniscus.
MISTAKES Sometimes, during an experimental investigation, a student may make a mistake: misreading a scale or a digital reading using different balances for a number of related measurements wrongly transferring raw data to a table of results pressing the wrong buttons on a calculator or making arithmetical errors in mental calculations failing to carry out a procedure as described in the method. MISTAKES CAN BE AVOIDED!!! ERRORS CANNOT
Systematic Error Poor accuracy in measurements is usually associated with an error in the system a systematic error. Using a balance that has been incorrectly zeroed (for example, so that the zero reading is in fact a mass of 0.1 g) will produce measurements that are inaccurate (below their true value). Systematic errors are always biased in the same direction. The incorrectly zeroed balance will always produce measurements that are below their true value. Repeating the measurements will not improve the accuracy of the result.
Systematic errors can be the result of: poorly calibrated instruments instrument parallax error (reading a scale from a position that is not directly in front of the scale) badly made instruments poorly timed actions (such as the reaction time involved in clicking a stopwatch).
Random uncertainty Poor precision in measurements is associated with random uncertainty. These are the minor uncertainties inherent in any measurement. Error associated with estimating the last digit of a reading is a random uncertainty. How can we reduce random uncertainty???
Repeat, repeat, repeat! Repeating the measurement a number of times and averaging the results reduces the effect of random uncertainty.
Percent Uncertainty The absolute uncertainty is the size of an uncertainty, including its units. 20.00 ± 0.05 cm 3 The percentage uncertainty changes with the amount of material that you are measuring. Percentage uncertainty is found by dividing the absolute uncertainty by the measurement that is being made. Percentage uncertainty = absolute uncertainty x 100 m measurement 0.05/20 x100 =
Processing Data Determine the uncertainties in results When adding or subtracting absolute uncertainties can be added together For multiplication/division, percentage uncertainties can be added together
Example a 12.5 ± 0.16 cm 3 of HCl is added from a burette to 4.0 ± 0.5 cm 3 of water that had been measured in a measuring cylinder. Calculate the absolute uncertainty and hence the percentage uncertainty in the final solution. b Solution a Total volume of solution = 12.5 + 4.0 = 16.5 cm 3 Absolute uncertainty = ±(0.16 + 0.5) = ±0.66 cm 3 Percentage uncertainty = 066. 16. 5 = = 100 = ±4% 1
Example b A sample of copper has been produced during a multi-stage experiment. The 250 cm 3 beaker was weighed at the start of the experiment and found to have a mass of 180.15 ± 0.02 g. Several days later the beaker and copper was found to have a mass of 183.58 ± 0.02 g. Calculate the absolute uncertainty and hence the percentage uncertainty in the final mass of copper. c Calculate the percentage uncertainty, and hence the absolute uncertainty, b Mass of copper = 183.58 180.15 = 3.43 g Absolute uncertainty = ±(0.02 + 0.02) = ±0.04 g Percentage uncertainty = 004. 343. m(na CO ) 100 = ±1% 1
Homework Complete sig fig worksheet