Dealing with Numbers A guide to Numerical & Graphical Methods
1.0 The Importance of Experiments <ul><li>Scientists and engineers spend a lot of time performing experiments. Why ? </li></ul><ul><li>They form the basis for scientific and technical advances. </li></ul><ul><li>They allow theory to be “put to the test”. </li></ul><ul><li>They may reveal new, unexpected effects leading to new or modified theoretical models or explanations. </li></ul><ul><li>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: </li></ul><ul><ul><li>Knowledge </li></ul></ul><ul><ul><li>Skills </li></ul></ul><ul><ul><li>Understanding </li></ul></ul><ul><li> through investigating the “real world” </li></ul>
1.1 Experimental Results – The Data OK you’ve 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 doesn’t 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) 0 C This statement of temperature indicates both its measured value and the uncertainty. The temperature could be anywhere between 27.5 – 0.5 = 27.0 0 and 27.5 + 0.5 = 28.0 0
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 Uncertainty <ul><li>Each form of uncertainty can have 2 categories: </li></ul><ul><li>Systematic Uncertainty – can exist without the experimenters knowledge. </li></ul><ul><li>Can skew all readings or values one way. </li></ul><ul><li>Mostly due to instruments rather than humans. </li></ul><ul><li>Random Uncertainty – produces scatter in measurements. </li></ul><ul><li>Environmental factors often cause this type of error. </li></ul><ul><li>Mostly due to humans rather than instruments. </li></ul><ul><li> </li></ul>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 Precision Precision 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 accurate A player throws 5 darts
1.5 Accuracy Accuracy 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 person’s attempt to measure the length of a piece of string Inaccurate Imprecise Precise Imprecise Precise Inaccurate Accurate Accurate True Value
1.6 Significant Figures Significant 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. Number 12.5 0.003002 49,000 0.000234 123.00 Significant Figures 3 4 2 3 5
1.7 Significant Figure Manipulation <ul><li>ADDITION & SUBTRACTION: </li></ul><ul><li>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. </li></ul>46.379 Rounding to the least number of decimal places of those numbers added (21.1 with 1 decimal place). <ul><ul><li>2. MULTIPLICATION AND DIVISION: </li></ul></ul><ul><ul><li>Identify the number in the calculation with the least number of significant figures. Give your answer to the same number of sig figs. </li></ul></ul>= 152.729383 With 56.4 having 3 sig figs, the answer should have 3 sig figs Answer = 153 Answer = 46.4 16.54 8.269 0.47 +21.1 65.64 (32.787 + 98.443) 56.4
1.8 Scientific Notation It 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 10 5 sec and now only the numbers to the left of the multiply (x) sign are significant.
1.9 Orders of Magnitude When 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 -1 For the speeds of the cars maybe a range between 40 to 80 kmh -1 .
2.0 Rounding When 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 don’t 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 Mean A 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 -1 Is there an uncertainty in the Mean ? If so, how is it calculated ? Speed of Sound (ms -1 ) 341.5 342.4 342.2 345.5 341.1 338.5 340.3 342.7 x = 2734.2 8 = 341.775 ms -1
2.2 Uncertainty in the Mean What is the uncertainty associated with the calculated speed of sound of 342.8 ms -1 ? To calculate the uncertainty in the mean: <ul><li>Calculate the range of values (largest – smallest) </li></ul><ul><li>Divide the range by the number of terms </li></ul>Thus uncertainty = 345.5 – 338.5 8 = 0.875 ms -1 Since 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 Uncertainty The 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 = 0.0026 NOTE: Fractional Uncertainty has NO units PERCENTAGE UNCERTAINTY = Fractional Uncertainty x 100 = 0.0026 x 100 = 0.26%
2.4 Combining Uncertainties In 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 σ V <ul><li>Mathematical Operations : </li></ul><ul><li>Sums and Differences </li></ul><ul><li> If V = a + b </li></ul><ul><li>or V = a - b </li></ul><ul><li>then ∂V = ∂a + ∂b </li></ul>In words, the uncertainty in V is the sum of the uncertainties of a and b Mathematical Operations: Products and Quotients If V = a x b or V = a / b In words, the fractional uncertainty in V equals the sum of the fractional uncertainties in a and b then ∂V = ∂a + ∂b V a b
2.5 Data Selection A vital question for all experimental scientists and engineers is: Are ALL my data equal ? For many investigators ALL data is valid and NONE can ever be rejected. While others can simply look at a set of data and label it as spurious and reject the lot. And there are yet others who can look at individual data points and reject them whilst keeping the rest. Confidence in the “correctness” of experimental data really comes when you are satisfied that the experiment is repeatable. If you do have a suspect data point the best thing to do is to repeat the experiment. Of course this is not always possible, especially when testing to destruction, as in breaking a wire or bursting a balloon. Statistical tests which help eliminate “spurious” data do exist, but their rigid and unquestioning application to all data may mask a trend that you should know about. There are situations where a data point may be neglected or rejected. For example, during a series of events being hand timed, the operator lost concentration during one of the events.
3.0 Why Graphs ? A picture is worth a thousand words. Humans generally find it easier to understand information when presented as a picture rather than as a table of figures. <ul><li>A graph will indicate: </li></ul><ul><li>The range of the measurements taken </li></ul>(d) The existence of “outlying data” (c) The existence or otherwise of trends (b) The uncertainty in each measurement Temperature 0 C Time (sec) 0 80 60 40 20 0 20 40 60 80 100 120 Data point showing error bars for both Temperature (vertical) and Time (horizontal) outlying data point
3.1 Graphs – The Basics The most used graph in science is the Cartesian Coordinate Graph, better known as the x – y graph. The y axis is known as the ordinate and the x axis as the abscissa. The quantity that is controlled or deliberately varied throughout the experiment is the INDEPENDENT Variable and is plotted on the x axis The quantity that varies in response to changes in the independent variable is called the DEPENDENT Variable and is plotted on the y axis Temperature versus Time ALL graphs require a TITLE, and AXIS labels and UNITS X axis Abscissa Y axis Ordinate Independent Variable Dependent Variable Temp ( 0 C) Time (Sec)
3.2 Graphs – Origins, Scales & Symbols On most graphs the numbering of both the axes begins at zero, so the bottom left hand corner of the graph is the point (0,0) and is called the ORIGIN. The scale should be chosen so as to allow the graph to fill the whole page, while leaving enough space for labels units and a title However there is no law that states that an origin must be included in a graph. Sometimes including an origin will produce too coarse a scale which may hide important information. Data points (with or without error bars) should be too big rather than too small so as they cannot be mistaken for a smudge on the page Temperature 0 C Time (sec) 0 80 60 40 20 0 20 40 60 80 100 120 ORIGIN Good data points . Bad data point
3.3 Error Bars & Line Drawing Uncertainties in the quantities being graphed are indicated by attaching “error bars” to each of the data points. They can be vertical, horizontal or both. <ul><li>Error bars may: </li></ul><ul><li>remain the same size for all data points or, </li></ul><ul><li>vary in size from data point to data point. </li></ul>Where error bars are very small, due to the scales used, it is advisable to omit them from the graph. Their length indicates the size of the uncertainty associated with that data point. When connecting data points it is difficult to draw freehand “smooth curve” A rubberised flexible ruler called a “flexi - curve” is probably the best way to draw curves through data. As long as the curve fits within the error bars, the data has been joined together in a valid way. Temperature 0 C Time (sec) 0 80 60 40 20 0 20 40 60 80 100 120
3.4 Linear Graphs It is hard to determine exact mathematical relationships from curved graphs. Converting the graph to a linear or “straight line” graph allows quantitative relationships to be determined. <ul><li>Linear graphs are important in the analysis of experimental data because: </li></ul><ul><li>The slope or gradient and y intercept can be calculated </li></ul><ul><li>Departure from linearity can be easily seen </li></ul><ul><li>Outliers are easily identified </li></ul><ul><li>A mathematical relationship between “y” and “x” is easily determined </li></ul>Recognising that the temperature – time graph shown previously indicates an inverse relationship (Temp α 1/Time) and manipulating the data will give: Temperature 0 C 1/Time x 10 -2 (sec) 0 80 60 40 20 0 5.0 2.5
3.5 Line of Best Fit Is the red line the only line that can be drawn to join the data points ? Obviously not, other lines can be drawn. Is the red line the “best” line to join the data ? Yes, because it meets the criteria for a “line of best fit”. It passes through all the error bars. It has as many data points above the line as below and the distances above and below total about the same. Rules for drawing a Line of Best Fit: 1. Place a clear plastic ruler over the data points. 2. Move the ruler until the data points are equally placed above and below the straight edge. 3. Generally the origin is not a special point, don’t force the line through it. 4. Use a pencil to draw a fine line along the straight edge. Temperature 0 C 1/Time x 10 -2 (sec) 0 80 60 40 20 0 5.0 2.5
3.6 Determining Relationships Linearising the relationship between variables allows you to use the general equation for a straight line (y = mx + c) to determine the mathematical law which relates the variables. In this case: y = Temperature ( o C) m = Slope of Graph x = 1/Time (sec) c = Temperature axis intercept Slope = Rise/Run = (75 – 5)/(5 x 10 -2 - 0) = 1400 = 1.4 x 10 3 Thus: Temp = 1.4 x 10 3 (1/Time) + 5 Temperature 0 C 1/Time x 10 -2 (sec) 0 80 60 40 20 0 5.0 2.5 Run Rise c = +5
3.7 Interpolation & Extrapolation When the “y” or “x” value falls within the range of known data points INTERPOLATION is occurring. Determining a value of a variable (y and/or x) outside the range of those already known, EXTRAPOLATION is occurring. Once a line of best fit has been drawn for the available data, it becomes quite easy to determine a “y” value from a given “x” value or visa versa. Of the two processes, interpolation is inherently more reliable than extrapolation. Interpolation Region Extrapolation Regions Temperature 0 C 1/Time x 10 -2 (sec) 0 80 60 40 20 0 5.0 2.5
4.0 In Summary It is very easy to enter data incorrectly into a calculator or computer which will ultimately lead to ridiculous values for gradients and intercepts. This can go unnoticed unless you have an approximate value obtained from a hand drawn graph for comparison. Computers and calculators are excellent for fast and repetitive calculations. But they cannot match the eye/brain combination when it comes to spotting patterns or anomalies.
Information Sources: 1. Experimental Methods – An Introduction to the Analysis and Presentation of Data Les Kirkup – Jacaranda Wiley Ltd. ISBN 0 471 33579 7 2. Dr. Fred Omega Garces - Chemistry 100 – Powerpoint - Miramar College