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Significant Figure and Rounding Data Analysis Example: Data Analysis for free fall Plotting graph Writing Conclusion OUTLINE

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Significant Figure Rules The number of significant figures in a quantity is the number of trustworthy figures in it, the last significant digit in a measurement is somewhat uncertain (but still useful), because it is based upon an estimation All non-zero digits considered significant Zeros appearing anywhere between two non-zero digits are significant Number# of Significant Figure Significant Figure 12.34551,2,3,4,5 398.543,9,8,5 85675.268,5,6,7,5,2

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Zero is accepted as a significant figure if there is a significant figure before it. If zero comes before the non-zero integer If zero comes after the non-zero integer Number# of Significant FigureSignificant Figure 313 0.313 0.0000313 Number# of Significant FigureSignificant Figure 1.021,0 1.0000061,0,0,0,0,0 0.010031,0,0

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Number# of Significant FigureSignificant Figure 50.7045,0,7,0 0.12331,2,3 1.0000561,0,0,0,0,5 Examples: Number# of Significant FigureSignificant Figure 30013 300.33,0,0 300.0053,0,0,0,0

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Operations with Significant Figures Result 153.1+ 12.256165.356 153.1- 12.256140.844 153.1* 12.2561876.3936 153.1/ 12.25612.49184073 165.4 140.8 1876. 12.49 In addition or subtraction, the result can be as precise as the quantity with the lowest precision in the operation. Result may have a different number of significant figures than the inputs. When we do multiplication or division, the number of significant figures of the obtained result should be same as the one with the least significant figures in the operation. Result with correct SF

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Rounding A number is rounded off to the desired number of significant figures by dropping one or more digits to the right. When the first digit dropped is equal to or more than 5, we add 1 to the last digit retained. When it is less than 5, the last digit retained does not change Number Desired # of significant Figure Last Digit Last Digit smaller,equal or bigger than 5 Rounded number 6.57636bigger6.58 86.2535equal86.3 6.57333smaller6.57

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FREE FALL t 0 =0 t 1 =0.88 s t 2 =1.28 s t 3 =1.63 s t 4 =2.18 s t 5 =2.31 s Free fall formulas

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Free Fall: Experimental Data y(m)t(s)t 2 (s 2 ) 5.00.880.78 101.281.64 151.632.68 202.184.77 252.315.34

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Graph Paper Plotting the axes and Writing their Names & Units Scaling the Axes

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Plotting Data Best Fit

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Slope Slope Of The Graph

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Analysis Slope Percentage of Error Calculation

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Random Errors A random error, as the name suggests, is random in nature and very difficult to predict. It occurs because there are a very large number of parameters beyond the control of the experimenter that may interfere with the results of the experiment. Example:You measure the mass of a ring three times using the same balance and get slightly different values: 17.46 g, 17.42 g, 17.44 g

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How to minimize random errors Take more data. Random errors can be evaluated through statistical analysis and can be reduced by averaging over a large number of observations.

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Systematic Errors Systematic error is a type of error that deviates by a fixed amount from the true value of measurement. All measurements are prone to systematic errors, often of several different types. Sources of systematic error may be imperfect calibration of measurement instruments, changes in the environment which interfere with the measurement process and sometimes imperfect methods of observation The cloth tape measure that you use to measure the length of an object had been stretched out from years of use. (As a result, all of your previous length measurements would be smaller than the recent one.) The electronic scale you use reads 0.05 g too high for all your mass measurements (because it is improperly tared throughout your experiment).

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How to minimize systematic Errors? Systematic errors are difficult to detect and cannot be analyzed statistically, because all the data is off in the same direction (either high or low). You can not fix systematic error by repeating the experiment. Systematic error can be located and minimized with careful analysis and design of the test conditions and procedure; by comparing your results to other results obtained independently, using different equipment or technique Or by trying out an experimental procedure on a known reference value, and adjusting the procedure until the desired result is obtained (this is called calibration).

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Conclusion Part Conclusion is an important part of a laboratory report. The main purpose of the conclusion section is to comment on the results mentioned in the lab report so it requires most critical thinking. You should show whether your results are in agreement with the theoratical values. If not, then you should discuss the possible reasons for the observed deviation from the theoretical expectations.

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When writing your concluison; Firstly, restate the purpose of the experiment. Discuss the significance of the experiment, think about what you learned You can link the results to what you read in the literature, review or other sources mentioned in the introduction. Do not write procedure as your conclusion! Suggest biases that may have affected the experimental design;for instance, random and systematic errors. Discuss how they can be eliminated in the future. Suggest any changes that can be made to the experimental procedure and how these changes might affect the data received in the lab. Conclusion Part

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