Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement 1.2.1 Describe and give examples of random and systematic errors. 1.2.2 Distinguish between precision and accuracy. 1.2.3 Explain how the effects of random errors may be reduced. 1.2.4 Calculate quantities and results of calculations to the appropriate number of significant figures. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Error in measurement is expected because of the imperfect nature of us and our measuring devices. For example a typical meter stick has marks at every millimeter (10-3 m or 1/1000 m). Thus the best measurement you can get from a typical meter stick is to the nearest mm. EXAMPLE: Consider the following line whose length we wish to measure. How long is it? SOLUTION: It is closer to 1.2 cm than 1.1 cm, so we say it measures 1.2 cm (or 12 mm or 0.012 m). © 2006 By Timothy K. Lund 1 1 mm 1 cm

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Error in measurement is expected because of the imperfect nature of us and our measuring devices. We say the precision or uncertainty in our measurement is  1 mm. As a rule of thumb, use the smallest increment of your measuring device as your uncertainty. EXAMPLE: Consider the following line whose length we wish to measure. How long is it? SOLUTION: It is closer to 1.2 cm than 1.1 cm, so we say it measures 1.2 cm (or 12 mm or 0.012 m). © 2006 By Timothy K. Lund FYI We record L = 12 mm  1 mm. 1 1 mm 1 cm

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Random error is error due to the recorder, rather than the instrument used for the measurement. Different people may measure the same line slightly differently. You may in fact measure the same line differently on two different occasions. Suppose Bob measures the line at 11 mm  1 mm and Ann measures it at 12 mm  1 mm. Thus Bob guarantees that the line falls between 10 mm and 12 mm. Ann guarantees it is between 11 mm and 13 mm. Both are absolutely correct. © 2006 By Timothy K. Lund 1

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Random error is error due to the recorder, rather than the instrument used for the measurement. Different people may measure the same line slightly differently. You may in fact measure the same line differently on two different occasions. Perhaps the ruler wasn’t perfectly lined up. Perhaps your eye was viewing at an angle rather than head-on. This is called a parallax error. © 2006 By Timothy K. Lund FYI The only way to minimize random error is to take many readings of the same measurement and to average them all together.

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Systematic error is error due to the instrument being used for the measurement being “out of adjustment.” For example, a voltmeter might not be zeroed properly. A meter stick might be rounded on one end. Now Bob measures the same line at 13 mm  1 mm. Furthermore, every measurement Bob makes will be off by that same amount. © 2006 By Timothy K. Lund 1 Worn off end FYI Systematic errors are hardest to detect and remove.

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors FYI: RANDOM ERROR is where accuracy varies in a random manner. Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors FYI: SYSTEMATIC ERROR is where accuracy varies in a predictable manner. Uncertainty and error in measurement The following game where a catapult launches darts with the goal of hitting the bull’s eye illustrates the difference between precision and accuracy. First Trial Second Trial Third Trial Fourth Trial © 2006 By Timothy K. Lund Low Precision Low Precision High Precision High Precision Hits not grouped Hits not grouped Hits grouped Hits grouped Low Accuracy High Accuracy Low Accuracy High Accuracy Average well below bulls eye Average right at bulls eye Average well below bulls eye Average right at bulls eye

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Significant figures are the reasonable number of digits that a measurement or a calculation should have. For example, as illustrated before, a typical wooden meter stick has two significant figures. The number of significant figures in a calculation reflects the precision of the least precise of the measured values. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement Significant figure rules: Live with them! (1) All non-zero digits are significant 438 g 26.42 m 1.7 cm 0.653 L 3 4 2 (2) All zeros between non-zero digits are significant 225 dm 12060 m 900.43 cm 3 4 5 © 2006 By Timothy K. Lund (3) Filler zeros to the left of an understood decimal place are not significant 220 L 60 g 30. cm 2 1 (4) Filler zeros to the right of a decimal place are not significant. 0.006 L 0.8 g 1 (5) All non-filler zeros to the right of a decimal place are significant. 8.0 L 60.40 g 32.000 cm 2 4 5

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainty and error in measurement A calculation must be rounded to the same number of significant figures as in the measurement with the fewest significant figures. EXAMPLE CALCULATOR SIG. FIGS (1.2 cm)(2 cm) 2.4 cm2 2 cm2 π(2.75 cm)2 7.5625 cm2 7.56 cm2 5.350 m/2.752 s 1.944040698 m/s 1.944 m/s © 2006 By Timothy K. Lund (0.0075 n)(6 m) 0.045 nm 0.04 nm EXAMPLE CALCULATOR SIG. FIGS 1.2 cm + 2 cm 3.2 cm 3 cm 2×103 m + 2.1 m 2002.1 m 2000 m 5.30×10-3m – 2.10m -2.0947 m -2.09 m

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results 1.2.5 State uncertainties as absolute, fractional and percentage uncertainties. 1.2.6 Determine the uncertainties in results. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results Absolute, fractional and percentage uncertainties. Absolute error is the raw uncertainty or precision of your measurement. EXAMPLE: A student measures the length of a line with a wooden meter stick to be 11 mm  1 mm. What is the absolute error or uncertainty in her measurement? SOLUTION: The  number is the absolute error. Thus 1 mm is the absolute error. 1 mm is also the precision. 1 mm is also the raw uncertainty. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results Absolute, fractional and percentage uncertainties. Fractional error is given by fractional error Absolute Error Measured Value Fractional Error = EXAMPLE: A student measures the length of a line with a wooden meter stick to be 11 mm  1 mm. What is the fractional error or uncertainty in her measurement? SOLUTION: Fractional error = © 2006 By Timothy K. Lund 1 11 = .09

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results Absolute, fractional and percentage uncertainties. Percentage error is given by percentage error Absolute Error Measured Value Percentage Error = · 100% © 2006 By Timothy K. Lund EXAMPLE: A student measures the length of a line with a wooden meter stick to be 11 mm  1 mm. What is the percentage error or uncertainty in her measurement? SOLUTION: Percentage error = 1 11 ·100% = 9%

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results Absolute, fractional and percentage uncertainties. PRACTICE: A student measures the voltage of a calculator battery to be 1.6 V  0.1 V. What are the absolute, fractional and percentage uncertainties of his measurement? Find the precision and the raw uncertainty. Absolute uncertainty is 0.1 V. Fractional uncertainty is (0.1)/1.6 = 0.06. Percentage uncertainty is 0.06(100%) = 6%. Precision is 0.1 V. Raw uncertainty is 0.1 V. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results To find the uncertainty in a sum or difference you just add the uncertainties of all the ingredients. In formula form we have If y = a  b then ∆y = ∆a + ∆b uncertainty in sums and differences © 2006 By Timothy K. Lund FYI Note that whether or not the calculation has a + or a -, the uncertainties are ADDED. Uncertainties DO NOT EVER REDUCE ONE ANOTHER.

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results To find the uncertainty in a sum or difference you just add the uncertainties of all the ingredients. EXAMPLE: A 9.51  0.15 meter rope ladder is hung from a roof that is 12.56  0.07 meters above the ground. How far is the bottom of the ladder from the ground? SOLUTION: y = a – b = 12.56 - 9.51 = 3.05 m ∆y = ∆a + ∆b = 0.15 + 0.07 = 0.2 m Thus the bottom is 3.05  0.2 m from the ground. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results To find the uncertainty in a product or quotient you just add the percentage or fractional uncertainties of all the ingredients. In formula form we have If y = ab/c then ∆y/y = ∆a/a + ∆b/b + ∆c/c uncertainty in products and quotients © 2006 By Timothy K. Lund FYI Note that whether or not the calculation has a  or a /, the uncertainties are ADDED. Since you can’t add numbers not having the same units, we use fractional uncertainties for products and quotients.

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in calculated results To find the uncertainty in a product or quotient you just add the percentage or fractional uncertainties of all the ingredients. EXAMPLE: A car travels 64.7  0.5 meters in 8.65  0.05 seconds. What is its speed during this time interval? SOLUTION: r = d/t = 64.7/8.65 = 7.48 m s-1 ∆r/r = ∆d/d + ∆t/t = .5/64.7 + .05/8.65 = 0.0135 ∆r/7.48 = 0.0135 so that ∆r = 7.48(0.0135) = 0.1 m s-1. Finally we can state that the car is traveling at 7.48  0.1 m s-1. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in graphs 1.2.7 Identify uncertainties as error bars in graphs. 1.2.8 State random uncertainty as an uncertainty range (+/-) and represent it graphically as an ‘error bar.’ 1.2.9 Determine the uncertainties in the gradient and intercepts of a straight-line graph. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in graphs Identify uncertainties as error bars in graphs. IB has a requirement that when you conduct an experiment of your own design, you must have three trials for each variation in your independent variable. This means that for each independent variable you will gather three values for the dependent variable. The three values for each dependent variable will then be averaged. The following slide shows a sample of a well designed table containing all of the information required by IB. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in graphs Identify uncertainties as error bars in graphs. 3 trials Independent variable manipulated by you Sheets n / no units n = 0 Rebound Height hi / cm hi = 0.2 cm Average Rebound Height h / cm h = 2.0 cm 55 54.8 55.1 54.6 Trial 1 Trial 2 Trial 3 2 52 53.4 52.5 49.6 4 49 50.7 48.7 48.6 6 48 49.0 47.1 48.5 8 45 45.9 45.0 44.6 10 41 39.5 41.4 42.4 12 35 35.8 34.0 35.1 14 33 31.1 33.5 33.0 16 29 29.7 27.2 29.3 Dependent variable responding to your changes © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in graphs Identify uncertainties as error bars in graphs. In order to determine the uncertainty in the dependent variable we reproduce the first two rows of the previous table: The uncertainty in the height was taken to be half the largest range in the trial data, corresponding to the row for 2 sheets of paper: Sheets n / no units n = 0 Rebound Height hi / cm hi = 0.2 cm Average Rebound Height h / cm h = 2.0 cm 55 54.8 55.1 54.6 Trial 1 Trial 2 Trial 3 2 52 53.4 52.5 49.6 © 2006 By Timothy K. Lund 53.4 - 49.6 2 = 2

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in graphs Identify uncertainties as error bars in graphs. The size of the error bar in the graph is then up two and down two at each point in the graph on the next slide… © 2006 By Timothy K. Lund

Error bars go up 2 and down 2 at each point. © 2006 By Timothy K. Lund

Topic 1: Measurement and uncertainties 1.2 Uncertainties and errors Uncertainties in graphs Determine the uncertainties in the gradient and intercepts of a straight-line graph. Now to determine the uncertainty in the slope of a best fit line we look only at the first and last error bars, as illustrated here for a different set of data: A sample of a well-done graph for the Bounce-Height lab is shown on the next slide: mmax mbest mmin © 2006 By Timothy K. Lund uncertainty in slope mmax - mmin 2 mbest  m = mbest 

mmax - mmin m = 2 m = m = 0.25 m = -1.6 0.3 -1.375 - -1.875 2 © 2006 By Timothy K. Lund m = 0.25 m = -1.6 0.3