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1. Dealing with Numbers A guide to Numerical & Graphical Methods

3. 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

4. 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 ?

6. 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

7. 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

8. 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

10. 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.

11. 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 .

12. Chapter 2 Mathematical Processes

13. 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

14. 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

16. 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%

18. 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.

19. Chapter 3 Graphical Methods

21. 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)

22. 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

25. 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

26. 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

27. 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

28. 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.

29. 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