Dealing with Numbers
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DESCRIPTIONDealing with Numbers. A guide to Numerical & Graphical Methods. 1.0 The Importance of Experiments. Scientists and engineers spend a lot of time performing experiments. Why ? They form the basis for scientific and technical advances. They allow theory to be put to the test. - PowerPoint PPT Presentation
Dealing with NumbersA guide to Numerical & Graphical Methods
1.0 The Importance of ExperimentsScientists and engineers spend a lot of time performing experiments. Why ?They form the basis for scientific and technical advances.They allow theory to be put to the test.They may reveal new, unexpected effects leading to new or modified theoretical models or explanations.In the case of students in a VCE class, experiments are unlikely to break new ground, but they do provide you with the opportunity to acquire:KnowledgeSkillsUnderstanding through investigating the real world
1.1 Experimental Results The DataOK youve done an experiment and collected some results.What are the important features of the data you have collected ?Measurements made or taken during an experiment generate raw data.This data must be recorded then presented and analysed.All data will have some uncertainty attached.It doesnt matter how good the experimenter, how well designed the experiment or how sophisticated the measuring device, ALL collected data has some uncertainty.(27.5 0.5)0CThis statement of temperature indicates both its measured value and the uncertainty.The temperature could be anywhere between 27.5 0.5 = 27.00 and 27.5 + 0.5 = 28.00
1.2 Uncertainty The uncertainty of the measurement is determined by the scale of the measuring device.The uncertainty quantifies (gives a number to) the amount of variation that has been found in a measured value.An alternative term to that of uncertainty is to use the term EXPERIMENTAL ERROR.This does NOT imply a mistake in your results, but simply the natural spread in the values of a repeatedly measured quantity.Uncertainty generally comes in three forms:Resolution Uncertainty how fine is the scale on the measuring device ?Calibration Uncertainty how well does the measuring device conform to the standard ?Reading Uncertainty - how well did the operator use the device ?
1.3 Systematic and Random UncertaintyEach form of uncertainty can have 2 categories: Systematic Uncertainty can exist without the experimenters knowledge. Can skew all readings or values one way. Mostly due to instruments rather than humans.
Random Uncertainty produces scatter in measurements. Environmental factors often cause this type of error. Mostly due to humans rather than instruments. Elimination of these experimental errors is the holy grail of experimental scientists and engineers.Systematic uncertainties can be reduced or eliminated from the measuring device by calibrating (comparing to a known standard) known to a high degree of both accuracy and precision.Random uncertainties can be controlled (but not eliminated) by taking multiple readings and using statistical analysis on the collected results.
1.4 PrecisionPrecision is a measure of how closely a group of measurements agree with one another.Close agreement translates to a small uncertainty.However, precision DOES NOT mean that the measurements are close to the true value. An example here should explain:The true value on a dart board is the bullseye.This player is precise - all darts fall within a small area (small uncertainty) but he is certainly not accurateA player throws 5 darts
1.5 AccuracyAccuracy is how closely the measurements agree with the true value.Again using the darts analogy: This player is BOTH accurate AND precise.What can you say about the following measurements ? Each dot represents one persons attempt to measure the length of a piece of stringInaccurateImprecisePreciseImprecisePreciseInaccurateAccurateAccurate
1.6 Significant FiguresSignificant Figures can be regarded as another method of indicating the uncertainty in a measured quantity.Significant Figures THE RULES:1. All NON ZERO integers are significant.
2. Zeros(a) Captive Zeros they fall between two non zero numbers they always count as significant figures.(b) Decimal Point Zeros Zeros used to place a decimal point are NOT significant.(c) Trailing Zeros any zeros following a decimal point are significant.
1.7 Significant Figure ManipulationADDITION & SUBTRACTION:When adding and subtracting numbers, round the result of the calculation to the same number of decimal places as the number with the fewest decimal places used in the calculation. 46.379Rounding to the least number of decimal places of those numbers added (21.1 with 1 decimal place). 2. MULTIPLICATION AND DIVISION: Identify the number in the calculation with the least number of significant figures. Give your answer to the same number of sig figs. = 152.729383With 56.4 having 3 sig figs, the answer should have 3 sig figsAnswer = 153Answer = 46.4
1.8 Scientific NotationIt is not always clear how many figures in a number are significant. By changing the unit in which a number is expressed it can appear that the amount of significant numbers changes.For example a time measurement could be 125 sec. Writing this time in milliseconds would give 125,000 ms.Both numbers have 3 significant figures.However, say somebody asks for the time measurement in ms and assumes (incorrectly) that our measuring device is accurate to within 1 ms, then the time would be seen as a 6 significant number. To get around this problem, Scientific Notation can be used. This has all numbers expressed as a number between 1 and 10, multiplied by a power of 10 The time 125,000 ms becomes 1.25 x 105 sec and now only the numbers to the left of the multiply (x) sign are significant.
1.9 Orders of MagnitudeWhen performing experiments, such as measuring the distance to the stars, determining the strength of gravity or measuring the speed of cars passing the college, you expect to gets answers within a certain range.If your measurements and subsequent calculations gave answers for g of 99 Nkg-1 or speeds of 400 kmh-1 hopefully you would suspect your calculations or measurements.The ability to make an estimate of an expected answer at least to within a factor of 10 can often save an embarrassing and career threatening mistake.This ability is called knowing an answer to within an order of magnitude.For gravity you would expect to get an answer in the range 9.7 to 9.9 Nkg-1For the speeds of the cars maybe a range between 40 to 80 kmh-1.
Chapter 2Mathematical Processes
2.0 RoundingWhen the result of a calculation has too many figures, which normally happens when using a calculator, you may need to reduce the number of figures that appear in the answer, so that it is becomes both meaningful and acceptable.For example, you are asked to measure the length of a thigh bone (femur) from a skeleton and put that measurement into a formula to calculate the height of the person before death.Since the original measurement had 2 significant figures, the answer you quote should be no more that 2 sig figs.Thus the height of the person was 2.1 m. The process of reducing the number of significant figures is called ROUNDING the number.When a calculation has a number of steps dont round until you get to your final answer, as rounding during the calculation could lead to large errors in the final answer.You do this and the calculator gives you an answer of 2.064655089. Your original measurement for the femur was 0.33 m
2.1 The MeanA team of students collected the following data in an experiment aimed at finding the Speed of Sound. To determine the average or MEAN (usually labelled as x) of these values: add them and divide by the number of measurements: How many Significant Figures should the Mean be quoted to ?The data has 4, so the mean should also have 4, right ? So, in this case the Mean or Average speed for sound on this day was 341.8 ms-1Is there an uncertainty in the Mean ? If so, how is it calculated ?
2.2 Uncertainty in the MeanWhat is the uncertainty associated with the calculated speed of sound of 342.8 ms-1 ?To calculate the uncertainty in the mean:Calculate the range of values (largest smallest)Divide the range by the number of termsThus uncertainty = 345.5 338.58 = 0.875 ms-1Since uncertainties are about determining the probable range of a measured or calculated quantity, there is little use in quoting them to any more than 1 Significant Figure. So the uncertainty here becomes 0.9 ms-1 (NOTE: if the first number of the uncertainty is a 1, then quote to two sig figs., so an uncertainty of 1.425 becomes 1.4)Thus, the speed of sound, and its associated uncertainty, as determined by the students is (342.8 0.9) ms-1
2.3 Fractional and Percentage UncertaintyThe function of uncertainties is to quantify the probable range of the values of the measured quantity.Thus it is usual to quote uncertainty to, at the most, 2 significant figures and often only 1 significant figure.For the speed of sound - (342.8 0.9) ms-1 FRACTIONAL UNCERTAINTY = Uncertainty in Quantity Value of Quantity = 0.9 342.8 = 0.0026NOTE: Fractional Uncertainty has NO unitsPERCENTAGE UNCERTAINTY = Fractional Uncertainty x 100 = 0.0026 x 100 = 0.26%
2.4 Combining UncertaintiesIn experiments often you will collect two or more sets of data which need to be used in an equation to calculate a final result. The uncertainties in each of the pieces of data will affect the final result in a process called error propagation.Uncertainties in measured or calculated quantities are quoted in a number of ways:If the quantity measured is a Volume (V), its uncertainty could be quoted as V or V or VMathematical Operations:Sums and Differences If V = a + b or V = a - bthen V = a + bIn words, the un