Introduction to Experiment: Part 1

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1 Introduction to Experiment: Part 1 Nate Saffold nas2173@columbia.edu Office Hours: Mondays 5-6PM Pupin 1216 INTRO TO EXPERIMENTAL PHYS-LAB 1493/1494/2699

2 General Announcements Labs will commence February 6th Lab room assignments change from week to week, will be posted on 5th floor bulletin board Waitlist Protocol First quiz on January 30th 2

3 General info At the beginning of each lecture I will give you a 10 minute long quiz about the topic covered in the previous week The quizzes are graded and they will contribute to 10% of the final grade These lectures have essentially three goals: 1. Introduce you to the basics of error analysis. The aim is to provide the tools to have an idea of how errors work and what they mean (useful in many different circumstances) 2. Review the main concepts of physics needed to perform the weekly experiments 3. Introduce the weekly experiments and describe the procedures Do not hesitate to stop me and ask questions! 3

4 Useful resources Some useful resources like the pdf version of the lab manual and the calendar of the weekly experiments can be found on the preceptors website: (you can also simply Google columbia physics preceptor ) I will post lecture slides and Mathematica tutorials on the course website: 4

5 What is an experiment? Essentially two types: (often both at the same time) You will mostly perform this kind Parameter determination Hypothesis testing Examples include: measurement of g, e/m, speed of light, etc... Examples include: Newton s gravity, quantum mechanics, Maxwell's equations (E&M), etc... Experiments must be reproducible! You cannot be the only one to ever make a particular observation The scientific description of the experiment must allow other scientists to check your results independently 5

6 Bad Science: Reproducible experiments 1989: Cold fusion experiment. Created lots of hope and excitement. Nobody has been able to reproduce the results again Good Science: 1798: Cavendish Experiment: Measured gravitational constant G. Confirmed Newton s Theory of Gravitation 1887: Michelson-Morley experiment. Attempted to find ether for E&M waves. Surprisingly, they found no evidence of the ether. Later explained by the Theory of Special Relativity 2012: LHC@CERN discovers the Higgs boson. This confirms the Standard Model of particle physics What can we do to avoid misinterpreting our results? CAREFUL ERROR ANALYSIS 6

7 Error Analysis Unfortunately, in most classes we are led to believe that we are done after obtaining a numerical value Equally important as your results are the errors of your results! Quantification of confidence in experiment Where U U = mgh = J U ± U = 19.3 ± 0.8 J is the uncertainty in the result (Also note significant figures in reporting the value!) An experimental value without an error does not make sense! Errors determine the boundaries of our ignorance on the measured quantity Much of the data analysis will be spent trying to quantify errors 7

Types of Errors Essentially fall into two categories: Statistical Errors: Due to random fluctuations from measurement to measurement They are unavoidable Can be reduced by taking

8 Types of Errors Essentially fall into two categories: Statistical Errors: Due to random fluctuations from measurement to measurement They are unavoidable Can be reduced by taking more measurements They can be quantified using statistical analysis Systematic Errors: Always bias the data in one direction Should be identified and corrected Hard to quantify 8

9 Statistical Errors A statistical (random) error defines a typical spread in a measurement caused by random effects Governed by probability and statistics Could have a real physical value e.g. Thermal fluctuations in the room temperature make rulers contract/expand a little bit at every measure It can also be a result of the measurement technique Timing experiment: random fluctuations in reaction time Use of a ruler: starting position is different at every trial because of hands shaking Statistical errors are quantifiable. Math allows us to go from fluctuations on the measured quantities to uncertainties on the final results 9

10 Systematic Errors They can be caused by many factors. This makes hard their identification Wrong calibration of instrument Examples: Observer bias: Faulty meter stick: Shorter or longer than one meter Scales: not calibrated to zero Parallax error in reading angle on analog display Timing: Jumping the gun Use of approximate equations: Neglecting some piece of math in the data analysis Ideally, systematic errors should be small correct them if possible If unable to remove, identify source and acknowledge it in lab report and state how this effects the measurement 10

11 Accuracy vs. Precision Attention: accuracy and precision of a measure are two different things! Accuracy: How close your measure is to the true value of the quantity of interest Precision: How close the measured values are to each other The smaller the spread, the more precise a measurement. Very often one has measurements that are: Accurate but imprecise or precise but not accurate The best experiment aims for both! 11

12 Accuracy vs. Precision To make things more concrete, let s look at an example. In the following case: Center of bullseye = True value Black points = Measurement (data) Red point = Average. Accurate but imprecise Precise but inaccurate Accurate and precise 12

13 A real example: measurement of G Accurate and Precise Accurate but imprecise Precise but inaccurate 13

14 Quantifying statistical errors How do we quantify statistical errors? Values from different measurements fluctuate. However, they tend to fluctuate around a certain value. The tools to help us in quantifying statistical errors (random fluctuations in measurement) are: Mean: Average of measurementsthe measurements will fluctuate around this value. Standard deviation: It is the average distance between a measure and the mean Standard error of the mean: Given N measurements, it is the expected distance between the mean and the true value 14

15 Statistical mean: Given N measures, the mean of the measured distribution is defined as: Quantities like this one are called estimators If we were able to perform infinite measures, the estimator would give the exact average value of the quantity of interest, The error in the accuracy of an experiment is then defined as: The smaller is this quantity the more accurate is the experiment 15

16 Standard deviation: s The sample standard deviation of the N measures is defined as: It quantifies how far from the statistical average each measure is (this is the reason for the square, no negative distances). As for the statistical average, the sample standard deviation is an estimator for the true spread of the population, 16

17 Standard error of the mean: The spread in each individual measurement is often less interesting than the uncertainty in the mean of x. Be careful of the distinction! We expect the precision in our measurement of the mean value of x to improve with larger sample size. So we define the spread of in the measurement of the mean as the standard error of the mean: In contrast to the sample standard deviation, the standard error of the mean gets smaller with more measurements. 17

18 Practice Given the timing measurements below Find the mean Find the sample standard deviation Find the standard error of the mean Timing measurements: s, s, s, 9.10 s The total number of measures is then N=4 18

19 Calculating the mean: Calculations 19

20 Calculating the mean: Calculations Sample Standard deviation: 20

21 Calculating the mean: Calculations Sample Standard deviation: Standard error of the mean: 21

22 Calculating the mean: Calculations Sample Standard deviation: Standard error of the mean: IMPORTANT: Note that standard deviation and standard error of the mean have units as well! 22

23 How to report an experimental value Significant figures: Since the error is the measure of our ignorance it does not make sense to add too many significant figures to it Only one or two significant figures should be assigned to your uncertainty. In the previous example: The number of decimal figures on the error dictates the number of decimal figures (not the significant ones!) on the value 23

24 Gaussian distribution and confidence intervals 24

25 Gaussian distribution For certain types of distributions of the measured quantities, the errors can be predicted before the experiment is perfomed Gaussian distribution (a.k.a. Normal distribution) For this semester: Unless otherwise noted, assume statistical uncertainty in parameter is Gaussian distributed. 25

26 Estimating Confidence Intervals If the measures follow a Gaussian distribution one can compute the probability for the true average,, to fall at a certain distance from the statistical mean, 26

27 Confidence Intervals 27

28 Confidence Intervals 28

29 Uncertainties and Experiment Using Gaussian distribution we can estimate whether a certain result is reasonable. Result is within of expected value: Experiment is in good agreement with expected value. Result is within of expected value: Experiment and expectation are consistent. (Probably need more measures) Result is within of expected value: Experiment and expectation may disagree. Result is or more from expected value: The difference between experiment and expectation is statistically significant. They do not agree. 29

30 Summary Defined What is an experiment? Accuracy vs. precision Statistical (random) vs. Systematic errors Quantifying measurements and uncertainties (mean, standard deviation, error of the mean) Reporting results (significant and decimal figures) Gaussian distribution and confidence intervals Agreement of your data with theory or another accepted value (use of the confidence levels) 30

Experiment 2: Projectile motion and conservation of energy

Experiment 2: Projectile motion and conservation of energy Nate Saffold nas2173@columbia.edu Office Hour: Mondays, 5:30PM-6:30PM @ Pupin 1216 INTRO TO EXPERIMENTAL PHYS-LAB 1494/2699 Overview The physics