Lab_01 - Laboratory 1 Introduction to Statistics and Data...

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Unformatted text preview: Laboratory 1 Introduction to Statistics and Data Analysis Introduction Statistics is the science of obtaining meaningful information from data. The methodology of statistics allows scientists to “separate information from noise” and to quantify how certain they are about the results of their measurements. If scientists did not consistently analyze the results of their experiments within a statistical framework, they would frequently be misled by anecdotal evidence, or by the results of a few anomalous trials. In this laboratory, we will explore a few of the most important concepts in statistics, and we will also discuss what it means for measurements to be “accurate” and “precise”. We will then use these tools to see how well some real data fit to a simple model of uniform acceleration. The concepts we study here will provide a framework for our analysis of experimental data throughout the rest of the course. Theoretical Background: Systematic vs. Statistical Error In general, there are two types of error that creep into experimental data: systematic and statistical error. Systematic errors are biases in measurement that often lead to measured values’ being consistently too high or too low. These can be the result of poorly calibrated equipment or other consistent human errors that repeat themselves in every trial. Systematic errors are distinguished from other errors because the addition of more data does not reduce systematic error. Statistical errors are a very different type of error, and do not arise from any “mistake” on the experimenter’s part. A statistical error is any error that can be reduced by repeating the experiment a large number of times. Many statistical errors in experimental measurements are caused by unknown and unpredictable changes in the experiment or measurement apparatus that lead to “noise” in the measurement. Since the processes that generate noise tend to be random, it is expected that averaging over multiple trials will reduce the noise. Now consider a situation in which you have an experimental quantity that you are trying to measure, such as g, the gravitational acceleration at Earth’s surface. The “true” value of this quantity is out there, but you do not know what it is. So you devise an experiment to measure it, and you perform a series of trials. Precision refers to the degree of mutual agreement among your measurements. They may all be very far from the true value of whatever quantity you are trying to measure, but as long as the 1 measurements themselves are close together, your data are precise. In Figure 1, the bottom left target shows a set of measurements that are precise but not accurate. Accuracy is the degree of conformity of a measured quantity to its true value. For a set of measurements to be considered accurate, they must be clustered relatively randomly around the “true” value, but each individual measurement does not need to be close to the true value. In Figure 1, the top right target shows a set of measurements that are accurate but not precise. Figure 1: The difference between precision and accuracy. The true value of the measurement is at the center of the target. Problem 1.1 According to the definitions above, does systematic error have a greater impact on a measurement’s precision or its accuracy? What about statistical error? Problem 1.2 Can the impact of statistical error be reduced by simply repeating a measurement lots of times and taking the average over all of those trials? Why or why not? Could you reduce systematic error by repeating a measurement? Why or why not? Theoretical Background: Basic Statistical Concepts The mean is defined as the arithmetic average of a set of values obtained from a series of N observations. The formula for calculating the mean is x≡ ¯ 1N xi , N i∑ =1 2 (1) where the individual measurements are labeled xi , and x signifies the mean of x. The variance is ¯ related to the arithmetic average of the square of the difference between the value of each observation and the mean, written as 1N 2 σ≡ ¯ (2) ∑ (xi − x)2 . N − 1 i =1 It is not exactly equal to the arithmetic average, because the denominator includes N − 1 instead of N . There is a good reason for this (related to the concept of degrees of freedom), and a full explanation can be found in any introductory statistics text. In Equation 2, the differences are squared so that positive and negative differences will not cancel each other out. The variance is a measure of how closely the data cluster around the mean. If the variance is small, most data points are close to the mean. The standard deviation is the square root of the variance, usually written as σ , where σ is defined mathematically as ￿ 1N σ≡ ¯ (3) ∑ (xi − x)2. N − 1 i =1 Its units are the same as those of whatever it is you are measuring, which gives it an advantage over the variance. For example, if all of your xi have units of meters, their standard deviation will also have units of meters. Their variance, however, will have units of meters squared, which makes it harder to interpret. The probability associated with a particular outcome of a measurement is the likelihood that this outcome will occur. For example, if you toss a fair coin, the probability of landing on “heads” is 1/2, or 0.5. The probability will always be expressed as a number between 0 and 1. A probability distribution function for a measurement is the set of probabilities for every possible outcome of that measurement. For example, for a single coin toss, there are two outcomes (“heads” and “tails”), each of which has a probability 1/2 of occurring. Therefore, the probability distribution function for a single coin toss is {1/2,1/2}. • The sum of the elements of a probability distribution must always be 1, because something must happen. • Probability distributions fall into two categories: discrete and continuous. Discrete probability distributions occur when the outcome of a measurement takes on only certain values (typically integers). Continuous probability distributions occur when the outcome can be any decimal value. • One important type of probability distribution function is the Gaussian distribution function, which has the functional form ￿ ￿ (x − x)2 ¯ P(x) = C0 exp − . (4) 2σ 2 This formula actually represents a set of distributions, each of which has different values of 3 x and σ . The shape for these distributions look like those in Figure 2 - they follow the classic ¯ “bell curve”. In Equation (4), x is the mean of the distribution: the value of the variable x that ¯ is at the center of the distribution. The term σ is the standard deviation of the distribution. It √ is a measure of the distribution’s width. For C0 = 1/(σ 2π ), the area under the bell curve equals 1, and the distribution is a special type called a Normal distribution function. • The Central Limit Theorem states that the sum of a large number of probability distributions from independent random processes will be approximately normally distributed. This is the reason why so many random processes in nature (such as mechanical “noise”) follow normal distribution functions - they are the result of multiple independent processes acting together. Problem 1.3 In one experiment, you take a fair coin and flip it twice. You measure the number of heads that appeared in those two flips and call that number x. In this experiment, what is the probability distribution function for the number of heads that appear? What is the mean number of heads that appear? Problem 1.4 Look at Figure 2. Do the two distributions on the left have identical means, identical standard deviations, or both? How about the distributions on the right? Figure 2: Two normal distributions (see Question 1.4). Now, say that you are making measurements of a quantity we will call “x”. If you measured x an infinite number of times and made a histogram of your experimental values, your graph would look something like the one in Figure 3. Without worrying too much about the values on the vertical axis, think about what this graph means. It means that, for any given single measurement, you are very likely to get a result that is close to the true value of x (which is at the center of the distribution), and less and less likely to get results that are further and further from the true value. 4 Figure 3: The red curve shows the result of taking an infinitely large number of measurements of x, making a histogram of their values, and fitting a smooth curve to the histogram. It also represents the probability distribution function (P(x)) for the outcome of a single measurement of x. As you can see from Figure 3, your chances of obtaining a single measurement close to the true value of x are good, but they are not that good. There is still a sizeable probability that you will think x is something very far from its actual value. This problem can be solved, in part, by repeating your measurement several times and taking the mean of your experimental values. For example, assume you repeated your measurement of x, but this time you made six separate measurements and took their mean to be the measured value of x. What would happen? It is true that some of those values might still be far from the true x. But it is very unlikely that all of your values would be far from the true x, and even less likely that they would all be off in the same direction. More likely, some of your six measurements would be too low, and some would be too high. By taking their mean, you would cancel out some of the error. The chance that your reported value is close to the true value goes up as you take the mean of more and more measurements. We quantify this by defining a quantity known as the error of the mean for a set of N measurements. It is defined as σ σx ≡ √ , (5) ¯ N where σ is the standard deviation of the N measurements as defined in Equation 3. Given that σ is a measure of the uncertainty associated with taking a single measurement, σx is a measure of ¯ the uncertainty associated with taking the mean of N measurements. As you can see, this formula states that you can determine the mean to arbitrary accuracy by simply taking a larger and larger number of measurements: as N goes up, the error of the mean gets smaller and smaller. This is a very important result, because statisticians have shown that, provided certain assumptions are met, we can expect the true value of whatever we are trying to measure to be within the interval 5 x ± σx ¯ ¯ x ± 2 σx ¯ ¯ x ± 3 σx ¯ ¯ 68.2% of the time 95.4% of the time 99.7% of the time. Another way of looking at this is: if we repeat our measuring (and averaging) process an infinite number of times, the interval x ± 2σx will include the true value in 95.4% of our results. Alterna¯ ¯ tively, for a single set of N measurements, there is a 95.4% chance that the mean of our measurements is within ±2σx of the true value. In other words, the distribution of the means (x) of repeated ¯ ¯ √ measurements is a Gaussian with a standard deviation of σx = σ / N . Please note the difference ¯ between a single measurement without averaging and a set of N measurements with averaging. For a single measurement, we get a value x. The distribution of repeated single measurements is also a Gaussian, but with a standard deviation of σ as shown in Fig. 3. Perhaps you have heard the term confidence interval before; this is exactly what we are describing here. Most scientists report their results as a mean and a 95% confidence interval. (The 95% is just convention; it has no special significance.) To do this, they report their data as x ± 1.960 σx . ¯ ¯ Practically - and non-rigorously - speaking, this means that the scientists are “95% sure” that the interval they are reporting contains the unknown true value of the quantity of interest. Problem 1.5 If the standard deviation of four measurements is 0.1, what is the error of the mean? Problem 1.6 If the error of the mean for four measurements is 0.05, how many trials need to be performed to get the error of the mean below 0.01? Below 0.001? √ Problem 1.7 Notice again that σx depends inversely on N , where N is ¯ the number of independent measurements that go into the average value x. What happens to the width of a particular confidence interval as N in¯ creases? What does this tell you about your confidence in the accuracy of your measurement as N increases? Experimental Background: The Ultrasonic Sensor In this laboratory, we will take position measurements of a falling object using a device called an ultrasonic sensor. This device is capable of measuring the position of an object to a precision of a few millimeters. The sensor produces 1-millisecond bursts of 50 kHz sound waves. It works in a way similar to the locating system employed by bats: it measures the time delay of the echo to establish the distance from it to the object. 6 Figure 4: A picture of the ultrasonic ranger and vacuum release system used for measuring gravitational acceleration. The ultrasonic sensor is beneath a wire mesh housing in the bottom of the bucket. Experiments Experiment 1: Fitting Data to a Model To demonstrate some of the statistical concepts described above, we will begin by fitting several sets of data to a simple model. In this experiment, our model will be one of y-position as a function of time for an object in free-fall. We will assume that the object starts from rest at t = 0, such that 1 y = y0 − g t 2 . 2 (6) Our data will be a set of real measurements of g obtained by dropping a rubber ball several times and fitting Equation 6 to the y vs. t measurements provided by the ultrasonic sensor. Each of the data sets will look like that in Figure 5. Since the ball is small and heavy, we can safely assume that the effect of air drag will be negligible. 1. Use the vacuum release system to clamp the ball in place directly above the ultrasonic sensor. You can apply pressure to the ball by compressing the vacuum hand pump a few times while it is in place. 7 Figure 5: Raw position (y) vs. time (t ) data for one experimental trial. The region corresponding to the ball drop is highlighted in gray, and a best-fit quadratic curve is included. 2. Ensure that the ball falls directly onto the ultrasonic sensor by releasing it using the trigger on the hand pump. Adjust the position of the ultrasonic sensor as necessary. 3. Set up LoggerPro so that it takes 3 seconds of position vs. time data at 20 samples per second. You can do this by opening the template file free fall.cmbl from the “LoggerPro Files” folder. 4. Begin taking data by clicking the green Collect button on LoggerPro. As the data points begin to plot across the screen, pull the trigger to allow the ball to drop. You want to make sure you are recording a complete drop. Problem 1.8 How can you measure gravitational acceleration from the position vs. time graph of a falling object? 5. If you are satisfied with how your data look, fit an appropriate function (quadratic) to the position vs. time graph. (See Appendix A Section 1.4 for how to do this.) Paste the graph with fit lines included into your lab report. 6. Drop the ball three more times and record position vs. time data for its fall for each trial. Perform the same fit procedure and paste your graphs into your lab report. 7. Using your method from Problem 1.8, calculate g for each of your trials. 8. Record your values for |g| (magnitude of g) using the online data collection tool. Instructions for using the online tool can be found in Appendix A Section 3. 8 Experiment 2: Statistical Analysis We will now statistically analyze your data and the class data to determine which set provides the better measurement of g, whose true value (we know in advance) is 9.81 m/s2 . We will also analyze larger data sets, which represent data taken by hundreds of different groups of students. Figure 6: Screenshot of the statistical analysis page for Laboratory 1. 1. Open the LoggerPro template statistics.cmbl. You will see four histograms, two of which are blank (see Figure 6). 2. Record your group’s four values of g in the “Your Values” column, and record the class values of g (from the online data collection tool) in the “Class Values” column. Two histograms should appear. There are also two more histograms, representing data taken by 250 groups and 600 groups. 3. Add a statistics box to each histogram and paste all of the histograms into the appropriate places in your lab report. (See Appendix A Section 1.1.) In your lab report, calculate the 95% confidence interval for the mean of each set of data. You may want to consult Appendix A Section 2.1 for help inputting functions into Excel. 4. Perform a Gaussian fit to the distribution of 600 groups’ data (see Appendix A Section 1.4). Paste this histogram including the fit and fit constants into the corresponding designated area of your lab report. 5. Using only the fitting constants from the previous step, calculate the standard deviation (σ ) and the error of the mean resulting from this fit. Also, calculate the percent deviation of 9 this error of the mean from the error of the mean obtained with the information given by the statistics box. Problem 1.9 Look at the confidence intervals for your four graphs. Which confidence interval is the widest? Which is the narrowest? Why is this so? Your answer should show that you understand the two factors that affect the size of a confidence interval. Problem 1.10 Which of the four sets of data above is the most precise? Which is the least precise? Which is the most accurate? Which is the least accurate? Why? Explain your answers thoroughly for all four questions. Problem 1.11 What do you notice about the shapes of the histograms as more and more data points are added? What would the shape look like if you had an infinite number of data points? Problem 1.12 . In the last step of the procedures, you fit a Gaussian distribution to the largest data set. How does the error of the mean for this Gaussian fit compare to the error of the mean you estimated from the “Statistics Box”? (Refer to Equation 4) Is it consistent with what you expect from the Central Limit Theorem? Why or why not? √ Problem 1.13 The error of the mean is proportional to 1/ N . (See Equation 5.) Given this fact, what could possibly explain a situation where the error of the mean for the class data is larger than the error of the mean for your group’s data alone? Your explanation should refer to the concepts of systematic and statistical error. Problem 1.14 The distributions for 250 groups and 600 groups look very similar; they have approximately the same width and are centered at about the same mean. Given this fact, what advantage do we gain by having all that additional data? Remember: we are concerned with how close our mean is to the true (unknown) value. Problem 1.15 If you did not know that the real value of g is 9.8 m/s2 , could you determine how accurate your data are? What implications does this have for experimental situations where a researcher is measuring an unknown parameter? 10 ...
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Lab_01 - Laboratory 1 Introduction to Statistics and Data...

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