Download Experiment 2: Random Error & Basic Stats - Measuring Time Intervals & Analyzing Errors and more Assignments Physics in PDF only on Docsity!
Experiment 2
Random Error and Basic Statistics
Homework 2: Turn in at start of experiment.
Readings: Taylor chapter 4: introduction, sections 4.1, 4.6 can be read together; then read the rest of chapter 4; then read chapter 5 through section 5.2. Also the experiment will refer to the Reference Guide, which has a summary of results from error analysis. Keep this Reference Guide, as you will refer to it throughout the term.
Do Taylor problems 4.2, 4.10, 4.16, and 4.23. For problem 4.2, do the calculation laid out in table style initially so you see exactly how it works; the entries in the table you can calculate either by a spreadsheet or with your calculator. But if you use a spreadsheet, you should spot-check results with your calculator! For 4.2 and 4.10, the checking calculations requested can be done with either your calculator, or (easier) Excel —but you should really do them (the purpose is to be sure you know how, and the check is that you get the right answer). You can use either explicit formulas or the built-in functions.
1. Introduction
Contrary to the naïve expectation, the experiments in physics typically involve not only the measurements of various quantitative parameters of nature. In almost all the situations the experimentalist has also to present an argument showing how confident she is about the numeric values obtained. Among other things, this confidence in the validity of the presented numeric data strongly depends on the accuracy of the measurement procedure. As a simple example, it is impractical to measure a mass of a feather using the scale from the truck weighing station, which is hardly sensitive to the weight less than a few pounds.
Another challenge has to be met when the scientist tries to compare the results of her experiment with the data from another experiments, or with the theoretical predictions. Since the conditions of the measurement almost always vary from an experiment to an experiment, and since they are also different from the idealized situation of the theoretical model, the compared values most likely will not match each other exactly. The task is then to figure out how important the factors creating this discrepancy are. If these factors are stable ( do not change from measurement to measurement ) and noticeable, they are called systematic errors. If, on the contrary, these factors are variable, they are called random, or statistical, errors. An important fact is that the uncertainty due to the random errors can be reduced by increasing the number of measurements to average over the random variations.
The main purpose of this experiment is to introduce you to methods of dealing with the uncertainties of the experiment. The basic procedures to correctly estimate the
uncertainty in the knowledge of the measured value ( the error of the measurement ) include:
- correct treatment of the random errors and systematic errors of the experiment ( Taylor Chapters 4 and 5 );
- rounding off the insignificant digits in the directly measured and calculated quantities (Chapter 2 Taylor and the Reference Guide).
2. Goals
- Understand basic statistical measures of uncertainty.
- Learn when standard deviation vs. standard deviation of the mean is appropriate.
- Distinguish between systematic and random errors.
- Learn one method for estimating the random errors.
- Learn how one way to estimate systematic errors.
- Test for a statistically significant difference from an expected value.
- To measure a time with accuracy better than half a percent.
3. Preliminary discussion (10-15 minutes).
Before the lab, you are asked to read and understand the theoretical material for this lab (Exp 2 and Taylor ). Before the experiment starts, your group needs to decide which information will be relevant to your experiment. Discuss what you will do in the lab and what preliminary knowledge is required for successful completion of each step. Warning: this lab is a bit short, and the next is a bit long. Don’t leave early: but start Exp3.
Think hard about organizing your work in an efficient way. What measurements will you need to make? Go through your lab manual with a highlighter, then make checklist of the needed measurements. What tables or spreadsheets will you need to make to organize the calculations data? How should you use Kgraph to expedite your calculations and unit conversions (when necessary)? What tables will you need to summarize your analysis and conclusions from the data? This lab will have more explicit reminders about tables than future labs, but you should be thinking about this organization of data taking, data reduction, and summarization in every lab.
Questions for the preliminary discussion
You should write your own answer to each question in your lab book, but leave space to change it after discussion. If you do change your answer, say why.
3.1 Calculators and computers typically return results with as many digits as possible, including digits well beyond our measurement uncertainty. What procedure will you follow to systematically get rid of these insignificant digits? 3.2 What tables will you need to record your input data? What tables will you need to summarize your results? Scan the lab, write down your tables, then see Appendix.
look at the data sheet until all 25 measurements have been recorded. This is essential; otherwise, you will introduce a bias into your measuring procedure! Make a few practice runs before taking data. Exchange places with your partner, and time 25 more counting intervals. Each person will have a data sheet with 25 timings recorded on it. Assuming you have prepared well for the lab, you should both be able to analyze your own data. In any case, you should provide your own answers to the questions at the end.
5. Data Analysis
5.1 Plot your data first! Then if it is wildly non-Gaussian, make another trial before sinking a lot of analysis time. Use Kgraph to make a histogram from your twenty-five measurements. The x axis represents the time measured T, and the y axis the number of measurements falling in the k th^ time bin. The data should be mainly peaked at a center value, and roughly symmetrical. You can consult with your instructor to see whether to take more data.
To analyze your data, you will calculate 3 quantities, the mean, the standard deviation, and the standard deviation of the mean value. If the formulas don’t make sense to you,
check Taylor chapter 4 again. The average or "mean" time per interval, T (pronounced “T bar”), is
=
N
i 1
Ti N
T
where the Σ stands for a summation, Ti represent the i -th single measurement of the time per interval, and N = the number of measurements. This formula directs you to add up
the N values of T, and divide by N. Next you can use this value of T to compute the standard deviation, σ, of the values of T, defined as:
∑^ (^ )
=
σ=
N
i 1
2 N 1 Ti^ T
Finally, the standard deviation of the mean value you arrived at is related to the standard deviation of the individual values:
N
m
σ σ = (3) The standard deviation of the mean σm , is the best estimate of the uncertainty in the measurement of the mean. Note that, unlike the standard deviation, this uncertainty can be made arbitrarily small by taking a sufficiently large number of measurements.
5.2 To demonstrate that you understand them, write out the calculations of these three quantities explicitly (say with Excel but not using Excel statistical formulas) for the first 3 measurements (N=3), as if these were the only data you had taken. You may then use the computer for the full 25 values. Write in your notebook which item from the output of Kgraph Functions | Statistics does this calculation for you (How could you check that you are looking at the right entry? Hint : try N=3 first before N=25). You will need the values of σ and σm in what follows.
5.3 Now predict how T, σ and σm vary with N, the number of measurements involved in their calculation. Then use Kgraph to calculate them for the first N=3, 5, 10, and 25 (all) of your measurements, again imagining that you had only the first 3, 5, 10, or 25 data points. Then comment whether on your predictions for the changes as N gets larger are approximately correct. In particular, why do σ and σm behave differently?
5.4 Compare your N=25 value of σ with your previous guess of the variability of your reaction time.
5.5 Now use Kgraph to make your final histogram from your twenty-five measurements. Adjust settings so the bin width is about w ≈ 0.4σ Show the calculation in your notebook. How can you find the bin width Kgraph is using?
5.6 Clearly mark the points of T and T^ ± σ for your measurements. These quantities can be shown to be the best estimate of your measurement. The region included in the range ±σ should contain about 68% of your data points if your errors are random and consequently the distribution of your measurements is normal or Gaussian (Taylor, chapter 5). What fraction of your data lies within this range?
5.7 Extra Credit: (Taylor Chapter 5.2 – 5.3 ) Draw on your histogram an appropriate Gaussian distribution given by a curve (function) of the form
2 2
1
( )
⎟⎟⎠
⎞ ⎜⎜⎝
⎛ (^) − − =
σ
T T
g T Ae (4)
To do this, you need values for the constants in the function g(T). For T^ and σ, use your best estimates for the mean and standard deviation of your time measurements. Choose
A to match your histogram. Hint : what is the value of the function g(T) at T= T?
Calculate g(T) at five points. ( Hint : Kgraph and Excel both use exp for the exponential function). Draw the points by hand on your histogram plot. Then connect these points with a smooth curve, which should resemble the Gaussian curves in Taylor. With a finite number of measurements such as 25, your histogram may not resemble the expected "bell" shape curve to a great degree.
6. Drawing Conclusions from the Timing Measurements
6.1 Check to see is there a (statistically) significant discrepancy in the time measured by the large clock. We want to check the hypothesis that the large digital clock is running correctly. Statistical significance is tested for not by just looking at the size of the difference and saying “that seems small” or “looks big to me”. Rather we compare the relative size of the difference with the uncertainty of our measurement. If the difference isn’t substantially larger than our uncertainty, then we say that our statistical analysis left
its systematic error? Explain how you would give the best estimate (in seconds) of a time interval of 120 counts. What uncertainty would you report for that time interval?
6.6 Suppose you had recorded only your first 5 measurements. What would you have concluded about the existence of a significant discrepancy? Were the remaining 20 measurements necessary in your opinion? Give a quantitative justification.
6.7 Suppose you measured with the hand timer a different counter with a known period, and found your measured period was (statistically significantly) too low. What kind of flaw in your measurement procedures might have caused this bias?
6.8 What was the muddiest point of this experiment? Where, specifically, should the write-up be improved?
Appendix: Summary Table for Experiment 2:
all times in seconds N= mean std dev std dev mean 3 5 10 25
