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Report Writing Tips You may be asking yourself, "Why do I have to write a ’Lab Report’ ?" The answer to that question is simple. As a budding Scientist, Engineer or Professional you need to learn how to communicate the results of your research and technical developments to your colleagues and other interested parties; and learning how to do this well, takes lots of practice. You will never perfect this skill, but you will be refining it throughout your professional life. The purpose of this handout is to provide you with a few tips that will help you get started. The instructors written comments, on the graded lab reports, draw your attention to deficiencies present in the write-up. Please review the instructors comments and seek out the instructor for clarifications. Try not to make the same mistakes in succeeding reports. A laboratory report should communicate, as clearly and concisely as possible, the rational basis for the experiment. It should also communicate what results were obtained and what significance they hold. While writing your reports, you must assume that the reader comes from outside your area of expertise, hence you must assume they are completely ignorant of what your experiment entails, even if this is known not to be the case. To this end, we have adopted a report format that is very similar to that of a scientific journal article. The lab report consists of: • A Title Page or Cover Sheet • An Abstract • An Introduction, consisting of a Theory subsection and Procedure subsection • A Discussion of Results • A Conclusion • A Calculation Section • and any Appendices that you believe will help clarify your presentation. Title Page On your title page, identify the experiment by name and indicate the date the data were collected. Also, put your name (first, middle initial, and last) and then that of your partner(s). Make sure to underline your name to indicate that the report is yours and not that of any other. Also indicate your lab section, either by day and time it meets or by the designation in the course offerings paper (ie. L50, L51, etc.). Abstract The Abstract provides a short synopsis of the purpose of the experiment and your principle results. It should also include what parameters were measured and it should list the percent difference between the theoretical and measured 1 principle results. Do not present raw data nor give details to how the measurements were obtained; those details will be discussed in the Procedure and Results sections. This can all be accomplished in three or four sentences. Introduction The Introduction is made up of two subsections; Theory and Procedure. The Theory provides the Physics behind the experiment. An easy way to organize this section can be found in the following; Start the section by stating the reason for the experiment. Follow this with the physical reasoning and essential ideas behind the experiment (this includes stating what features and numerical values are predicted). Use 2D diagrams to help clarify your discussion. Include only the equations that are important to the experiment and provide a derivation of these equations. Make sure you state what the variables in each equation represent. The Procedure tells the reader how you made the measurements. Start this section by describing the apparatus used. Use 2D diagrams to help describe how the apparatus is assembled and tell how the apparatus was manipulated to collect the data. Be very detailed and describe the measurements in the sequence performed. It is best not to refer to the lab manual when writing this section because this usually results in prose that tells the reader ’what to do’ instead of telling ’how the measurements were made’. In others words, do not present the procedure as a ’cookbook recipe’; it will result in a significant loss of points towards the final grade given for the report in question. To get an idea of how the prose should be crafted, read the sample report by ’Yutz and Yutz’ located in Appendix C. Discussion and Interpretation of Results In this section, you will present the data and results. These should be displayed in an organized form. Make use of tables, charts and graphs. Make sure that the SI units are stated and all numbers are quoted to the proper significant figures. Do not use the data sheets from your lab manual for this purpose. All data should be recopied and reformatted. Theoretical predictions and calculations should be compared to the data. If the experiment requires a graph and/or curve fitting (ie. Linear Regression Analysis) make sure you discuss the shape of the resultant curve, the slope of the line, the y-intercept and the Regression Coefficient. Also discuss the uncertainties in the measured quantities and any statistical quantities that may arise from a Regression Analysis. It is essential to note how the uncertainties in the Principle Results compare to the Theoretical Predictions. Conclusions The Conclusions are a summary of what you conclude from the results and whether they are what you expect. Restate the results with the Theoretical expectations and include the percent error when appropriate. Do not use terms such as "fairly close" and "pretty good" or "the experiment was a success;" give explicit quantitative differences from the expected result. State whether these differences fall within your expected errors and state possible explanations for unusual differences. If possible, note any sources of uncertainty that were not specifically determined in the measurements. Give plau2 sible evidence of the magnitude of the same and (if you are brave) quoted the calculable effects this unaccounted uncertainty may have on the interpretation of the results. If you do such calculations, these must be shown in the Calculations Section. Do not blame poor results on "human error". Human error is a mistake. If your poor results are possibly defective due to a mistake, then the experiment has to be redone. If you have any constructive criticism or ideas on how to improve the experiment, please include them here. Calculations Give one sample calculation for each different type of calculation (this includes uncertainty calculations) that you make in the course of your data processing. You need not show how a Linear Regression Analysis is done (that is why computers were invented), but you must show how any results from such a calculation were used. A sample calculation consists of three lines; put the equation used on the first line, the numbers substituted into the equation on the second, and the numerical result on the third line. Be sure to include the units for each result and use the proper significant figures. Do not submit scratch work! Regression Analysis - The Method of Least Squares Additional Example Suppose a student does an experiment to measure the local acceleration due to gravity, g, using a simple pendulum. The period of the pendulum is given by the equation, s l (eqn.3) T = 2π g where, • T : is the period • l: is the length of the pendulum • and g: is the local acceleration due to gravity. Squaring both sides we get: T2 = which is the same as T2 = (2π)2 l (eqn.4) g 4π 2 l (eqn.5) g which, if we plot T 2 on the vertical axis and l on the horizontal axis, implies 2 that the slope of the resulting line will be 4πg with a y-intercept equal to zero. Or, symbolically 3 m= 4π 2 (eqn.6) g Now, all that is left is to take the student’s data and obtain the slope using the Least Squares method. Once the slope has been found, we can calculate g. Using a spreadsheet, with a built-in utility to perform a Least Squares fit, we get the following: Period Squared (sec2 ) 1.35909 1.572516 1.755625 1.993744 2.226064 2.4336 2.621161 2.965284 3.305124 3.728761 4.227136 5.076009 5.597956 Pendulum Length (m) 0.335 0.396 0.440 0.505 0.562 0.609 0.658 0.740 0.830 0.940 1.049 1.262 1.394 Constant Std Err of Y Est R Squared No. of Observations Degrees of Freedom X Coefficient(s) Std Err of Coef. -0.002391 0.019168 0.999812 13 11 4.030133 0.016645 The first table contains the data the student obtained in the simple pendulum experiment; Period Squared and Length. The second table is the output from our spreadsheet regression. It contains values for, the y-intercept (ie. Constant), a value for the statistical error for the y-intercept, the R-Squared value, the number of data points (ie. No. of Observations), the slope (ie. X coefficient(s)), and a statistical error for the slope (ie. Std Err of Coef.). Since g has units of m/s2 , then the slope must have units of s2 /m [see (eqn.6)]. Now (eqn.6) implies: 4π 2 (eqn.7) m which, when we substitute the values into (eqn.7) gives: g= g= 4π 2 = 9.796 m/s2 4.030 4 Okay, we now have a value for g, but how reliable is it? Well, if we look at the graph of Period Squared versus Length, we see that it looks like a straight line and if we check the R-Squared value, 0.999812, we see that it is very close to 1. This implies that the student’s data fit a straight line which we expect according to (eqn.4), so at a first glance it appears the student has a reliable value for g; but this is not quite enough. We need to find an error for g before we can compare it to a standard value or another student’s experimental value. Statistical and Estimated Errors for Experimental Data What do we mean by error? This question is best answered in two parts. The first part describes what it is not; the term "error", as used by those working in Natural Science, does not mean mistake! Mistakes can be eliminated; measurement error cannot. If you forget to zero your ohmmeter before you make a resistance measurement - that is a mistake, not a scientific error. When we say a value has error, we mean that it has a specific uncertainty associated with it; it has a fuzziness. This "fuzziness" means the measurement has a range of values that it could be. By choosing the appropriate techniques and tools, we can reduce the size of this error, but we can never eliminate it. This is because measurement error, provided that it is not systematic, is random or statistical in nature. Going back to the table with the regression output, we see that the regression has calculated an error for the slope of the line. This means that the slope may be as small as 4.013 (if I keep only 3 decimal places) but no larger than 4.047; it can be any value within this range. We state this as: slope = m = 4.030 ± 0.0167 s2 /m This range has a very important place in Experimental Natural Science. It is called the ’Absolute Error’ or the ’Absolute Uncertainty’. If an experimenter makes a measurement directly, then the Absolute Error is an estimate of the uncertainty in the measurement. In other words, it is an educated guess. Another way to express this error is to state it as a percent. The ’Percent Error’ is defined as: P ercent Error = Absolute Error Of Quantity Quantity · 100 % So, the percent error for the slope is: P ercent Error = 0.0167 4.030 · 100 % = 0.414 % So we can specify the uncertainty in the slope as: slope = m = 4.030 ± 0.0167 s2 /m (1) slope = m = 4.030 s2 /m ± 0.414 % (2) or as: 5 This gives us an uncertainty for the slope, but we still need to determine the uncertainty in g. When an experimenter calculates a quantity using values that have uncertainty associated with them, there are very specific rules that govern how the error for the calculated quantity is found. The absolute error for the slope is a statistical error, but we treat it in just the same manner as we would for an estimated error, so we can treat the slope as if we had measured it and use the following rules to find the uncertainty in the calculated value of g. Rules For The Propagation Of Error When the computation involves Addition and/or Subtraction, we ADD the Absolute Errors of the measured quantities involved. When the computation involves Multiplication, Division and/or Raising To A Power we ADD the Percent Errors of the measured quantities, and if the quantity is Raised To A Power, the individual Percent Errors are first multiplied by their respective powers, BEFORE adding all the Percent Errors. Now since g is found by taking the inverse of the slope, m, [see (eqn.7)] we use the second rule to find its uncertainty, and since the slope, m, is the only quantity with uncertainty in (eqn.7), this says that the Percent Error in g is equal to the Percent Error in m. So, the student should report the value of g found in the Pendulum experiment as: g = 9.796 m/s2 ± 0.041 m/s2 (3) g = 9.796m/s2 ± 0.414% (4) or as: Now we have a figure of merit to judge how well the student’s value of g compares to another determination (experimental or theoretical). The Absolute Error just determined implies the value may be as small as 9.755 m/s2 , but not more than 9.837 m/s2 . Comparing the student’s laboratory standard value for g of 9.805 m/s2 , we see that this value falls within the range of uncertainty in the student’s experiment. This says that, within the error of this particular experiment, these two values, 9.796 m/s2 and 9.805 m/s2 , are the same. 6