"= fol;'"' Introduction: vel\e*\ In this lab, standard deviation was determined to be consistent with statistical theory in

Transcription

1 D. Constantino 1 Introduction: vel\e*\ In this lab, standard deviation was determined to be consistent with statistical theory in hopes ofev.aluating the uncertainty in experimental our data. In.all.experiments, rneasurements of data can never be free of error, despite the carefulness of your procedure. Thus, experimental uncertainty exists; by using data analysis, we are able to evaluate and interpret the uncertainty of a measurement. In this lab, we organized and analyzed quantitative data in rolled values ofdice. When rolling dice, a theoretical value is known (or rather - a best estimate), but due to random error, this is not the case. In order to better estimate the uncertainty of a roll, we repeatedly \ rolled the dice and gathered data to calculate its standard deviation. Figure I shows the equation used to calculate the standard deviation (o). Nis the number of measured quantities gathered in the data. The term ds in Figure 2 refers to a deviation, which is the difference between the it measurement & and the mean value 7. CIhe "= fol;'"' mean value.is the bes{ estimot cf dr tflre v*}e - it is cd dated by adding all the data and dividing that with the N numba of measurements). Figure 3 is a combination offigure I and Figure 2 and it is tlte eouation we used to calculate the standard deviation. 1 s-r ;j. The standard deviation reveals the best estimate of the uncertainty in the set ofvalues. Statistically, ifwe were to make another roll, there is a 68% chance the value is between one standard deviation. Furthermore, there is a 95% chance it will fall within two standard deviations. The standard deviation is a reliable measurement of uncertainty because it.determines llrhere exactly tlre unoe$ai*y tries. Our results revealed that this exporiment is consistent because our theoretical value falls between one standard deviation. (mean-c) < theoretical value < (mean + o)

2 Eouinment: o Assorted Dice (4/618/12 sided): Froccduret l.l r." I Pan 1: Step RigM ap and RoU the Dice l. Determine/cornpute a Theoretical Value for each ofthe four dice. 2. Determine/compute a theoretical value for the average of the fow dice rolled together. 3. Roll the set of 4 dice together and record the total in a data table. Repeathis a total of 5 time. Find the average or mean value ofyour 5 roll. 4. Repeat step 3 a total of 15 times (Remember, each experiment is the average of 5 rolls). Record the average values for all 15 experiments in EXCEL. 5. Restate your experimental results in the following form: "Our obsemation is that the average value of the 4 dffirent dice rolled together is _ +/-." 6. Go to whiteboard and add your results to the group chart. 7. Make a histogam chart to display the results of 15 rolls 8. Using excel, calculate the standard deviation (o). 9. On the histogram, draw a bell curve that approximately fits yow histogram and mark the 1o and 2o "confidence intervals". Part 2: Compare Your Theoretlcal Preiliction to Your bperimcntal Values 10. Using this "test for consistency", answer the following question'. "Would you conclude that your theoreticat vahre is cor8'istent with your experimental measuremerttt? Explain" Part 3: Compare Your Values with ostalistical Theory" I 1. Analyze the first standard deviation. 12. Alalyze the second standard deviation.

3 D. constantino 3 Tablel. Theoretical Values Each Die 4-sided 6-sided 8-sided l2-sided VALIIE ALL dice VALUE all 17.0 Table 2. Exnerinrental data Roll 1 Roll 2 Roll 3 Roll 4 Roll 5 Average Deviation Deviation^2 Exp Exp.2 zo Exo.3 17 to 't4.t( 4E e IJ Extr4 2n 1a Exp Exo z Exo z.z 4.U Exo Exp.9 18 ' u.o N?A Exo.10,o I U 9.00 Exp ' ExD o.2 0.M Exp.13 zc Exp aa 1.44 Exp i = 17.2 Std.Dev sum

4 D. Constantino 4 Graph 1. Histograph 6 Experimental Uncertainty - t*,i a-.-+>-- f,o 5 4.c R3 2 L n No15 No16 No17 No18 Dice number Sample Calcalations: (Theoretical Value 6.sided die) (Theoretical Value: ALL dice),.eochside ( ) 27 vaiue: -;t;: 60 :;;= r.) Uu1o. :! all the averages of each ilie : ( ) : 17.0 (Exp. I - Average) Avg. : Eggl r9g - (1e;0+1e's+1s'o+22'o+1o;0) # RoUs S (Mean of all Avgs.) - t rr (16, , , ,2+1A,4+17,O+70,6+ 18, ) rv (Exp.l - Deviation) d1 - i) = ( ) = -l.l (Standard

5 Enor Analysis (mean-o) < theoretical value < (mean + o) : (t ) = 17.o < (r ) : 15.5 < 17.0 < 18.9 b vu"u;; ht.fi Jy'U Nip. H'l*' vv Since our theoretical value falls between the mean/deviation of , the results of this experiments are consistent. ln part 1, we determined the theoretical value for rolling all four dice to be a value of 1?-0- br our first experiment of rolling the dice 5 timeg the average did not equal the. theoretical average value. The discrepancy in this experiment was a value of 1.0 (the percent discrepancy being 5.9%). Even though the experimental value did not match the theoretical value, it does not mean our theory was inconsistent. The theoretical value is an 'average' of the rolled dice, therefore, multiple rolls and calculations are need to determine the consistency. I do not think tle discrepancy is due to 'human error' because human enor is something tlat could be fixed., \, \\ These differences occurandomly I art V After repeating the experiment a totat of 15 times, we were pretty confident that the "k resutts willfalt between 14 and 20 (a confidence range of /- 3). After adding our results to the class chart, we see that "most of our results fall within the range 15 to 19", The interval of the class confidence range is smaller than our individual groups'. Although we got the sarne results as the class, the confidence range is nrrorruer in tie class chart. '

6 D. Constantino 6 After recording our own histogram chaxt in the 1ab, the next step was to calculate the Standard Deviation. Our experiment resulted in a 1.67 standard deviation. When marking the lo and 2o "confidence intervals' on our histogram chart, it was clear that the theoretical value fell within the 1"'deviation. Also, the peak ofthe bell curve is very close (ifnot located) on the mean value. When part 2 ofthe lab required us to test for consistency, we concluded that our theoretical value was in fact consistent with our experimental measurements because we have (mean-o) < theoretical value < (mean + o) :( ) s I7.0 < (17.2+ r.67) : 15.5 < 17.0 < 18.9 proven both graphically and analytically that the theoretical value falls within 1 Standard Deviafion. 1 standard deviation was orr test for consistency. Lastly, in part tlree, we compared our values with 'Statistical Theory'. It is statistically predicted that 68% of the values of any measrnsment falls between one standard deviations. Our results showed that 67% ofour values fell between our standard deviation - a 2:i-::-:::::::: Y.# medstrements between 10 --'t^ - ^^ oz x LOO% = 67To very consistent result. Furthermore, approximately 95% ofthe value of any measurement should statisticallv fall within 2 standard deviations. Our results revealed that I 007o of our values L# measurements between ^^ru. 15 xtoayo =!OO% fell between 2 o. When detsrmining experimental cursisteflcy t rough standard deviation, our results have shown quite a bit of consistency. An expected number ofvalues had rolled between I and 2 standard deviations, making our experiment consistent with statistical theory as well. Thus, experimental mcertainty is not so uncertain with the calculation powers of standard deviation.