Physics 201 Lab 2: Statistics and Data Analysis Dr. Timothy C. Black Summer I, 2018 I. THEORETICAL DISCUSSION Data Reduction Theory: Data reduction theory furnishes techniques for mathematically, and hence logically, transforming uncertainties in measured quantities into uncertainties in beliefs. This is necessary because 1) We cannot usually directly measure or observe the things about which we may wish to make assertions, and 2) We cannot make infinitely accurate measurements. In physics, at least, an asserted belief usually takes the form of a number, as in, The bound-coherent neutron-deuteron scattering length is 6.6649 fm. In general, we do not, and cannot, directly measure this quantity. Instead we measure other quantities that, either by definition, or according to some theory, are mathematically related to the quantity whose value we are asserting[1]. If we denote our quantity of interest by f, and our directly measured quantities by {α} = α 1, α 2,..., α N, then f is a function of the set {α}; i.e., f = f(α 1, α 2,..., α N ) The most elementary task in data reduction is to determine our uncertainty in f, denoted by δf, given uncertainties in the actually measured quantities, δα 1, δα 2,..., δα N. In other words, we would like to know by how much f will change if we change one of the α s, say the j th α, α j, by a small amount δα j. Because we have taken some calculus, we know that if f is an analytic function of α, meaning that f, and all orders of derivatives of f with respect to α j are continuous in some neighborhood of the central (measured) value α j0, then we can expand it in a Taylor series about the point α j0 : 1 n f f(α j0 δα j ) = n! α n (δα n=0 j j ) n αj0 = f(α j0 ) f α j (δα j ) 1 2 f (δα αj0 2 j ) 2 1 3 f (δα 6 j ) 3 αj0 αj0 We have used partial rather than full derivatives here, because we are concerned only with how f varies with changes in α j, assuming that all other variables are constant. This amounts to taking all of the measured quantities to be independent of one another. We will investigate at a later time how to deal with the dependence of one variable on another. Assuming that the uncertainty δα j0 is small enough, we can neglect all terms greater than those linear in δα j0, so that:[2] α 2 j f(α j0 δα j ) f(α j0 ) f α j δα j αj0 If we use the notation δf αj0 to symbolize our uncertainty in f due to our uncertainty in α j at the point α j0, then we can write so that f(α j0 ) δf αj0 = f(α j0 ) f α j δα j αj0 δf αj0 = f α j δα j αj0 α 3 j
Under the assumption that the uncertainties in each of the variables are independent, we can think of the uncertainty δf αj0 as the αj th component of an uncertainty vector whose total length is equal to the total uncertainty in f due to our uncertainties in all of the measured variables. If we denote this total uncertainty by δf, and use the generalized Pythagorean theorem to find the length of an N-dimensional vector, it follows that the total uncertainty δf in our determination of f(α 10, α 20,..., α N0 ), given our actually measured values α 10, α 20,..., α N0 and our presumptive uncertainties δα 1, δα 2,..., δα N in these measured quantities, is δf = = j ( δfαj0) 2 ( 2 ( 2 ( f α 1 (δα 1 ) α10) 2 f α 2 (δα 2 ) α20) 2 f α N αn0 ) 2 (δα N ) 2 This is the fundamental equation for propagating the uncertainty in a set of N measured variables {α j } to determine the resulting uncertainty in a quantity f({α j }) which is a function of those variables. One then typically quotes one s belief by stating, We have measured the quantity f 0 ± δf, which is a statement that you believe that this quantity f has the value f 0, and δf is a measure of your certainty in that belief. It is customarily assumed, though not necessarily the case, that one s uncertainties are distributed according to a Gaussian function. This means that the probability that you assign to any particular value of f being the correct value is given by P (f) = Ae 1 2( f f 0 σ ) 2 where σ = δf and A is a normalization constant chosen so that P (f)df = 1 An Example: Suppose that you have conducted an experiment in which you are attempting to determine the mass density ρ of a circular slot (i.e., a washer without a hole it makes the calculus easier to do it without a hole). The density is defined by the equation ρ = m V = where m is the mass of the slot, r is its radius, and t is its thickness. There are thus three directly measurable quantities; m, r, and t. You have measured these quantities and obtained the following results: m πr 2 t (1) m = 84.2 g r = 2.35 cm t = 1.4 mm You have assessed the uncertainties in these quantities to be, respectively, δm = 0.5 g δr = 0.4 mm δt = 0.4 mm According to our prescription for determining uncertainties, the uncertainty in the density is given by
δρ = ( ) 2 ( ) 2 ( ) 2 ρ ρ ρ m δm r δr t δt (2) The partial derivative of a function with respect to a variable is simply the derivative of that variable under the assumption that all other variables are constants. For our convenience in taking these derivatives, we can re-write equation 1 as Then the partial derivatives we need are given by ρ = 1 π mr 2 t 1 (3) ρ m = 1 π r 2 t 1 ρ r = 2 π mr 3 t 1 ρ t = 1 π mr 2 t 2 What values of m, r, and t do you put into the expressions for these partial derivatives? You put in your measured values!! Where do you get values for δm, δr, and δt? You must estimate them, by whatever means seems appropriate to you. In your first lab, you estimated these quantities by making multiple measurements and calculating the spread in values. You then insert your partial derivatives and your estimated uncertainties in the measurables into equation 2, to get δρ = ( ) ( ) ( ) 1 π 2 r 4 t 2 (δm) 2 4m 2 π 2 r 6 t 2 (δr) 2 m 2 π 2 r 4 t 4 (δt) 2 (4) It is often easier to work with the fractional uncertainties δm m, δr δt r, and t Thus, we can write than the absolute uncertainties. ( ) ( m 2 δm δρ = π 2 r 4 t 2 m (δm ) 2 = ρ 4 m ) 2 ( 4m 2 π 2 r 4 t 2 ( δr r ) 2 ( δt t ) 2 ) ( ) 2 ( δr m 2 r π 2 r 4 t 2 ) ( ) 2 δt t Inserting the measured values of m, r, and t, as well as the associated uncertainties δm, δr, and δt into this equation, we obtain ρ = 3.434 ± 0.9882 10 5 kg m 3 = 343.4 ± 98.8 g cm 3 Kaleidagraph: In this lab, we will be using Kaleidagraph to do all of our numerical and graphical analysis. Kaleidagraph is a very powerful graphical analysis program that is happily available to all students via the Tealware platform. Here is how you access this program:
1. Go to the Tealware site. It is located at Accessing Kaleidagraph on Tealware https://tealware.uncw.edu/citrix/xenapp/auth/login.aspx Then login using your UNCW username and password. 2. You should see a screen that looks like the image shown in Figure 1. (Without the red 7-pointed star, of course). FIG. 1: Tealware homepage 3. Click on the Kaleidagraph icon. After some time, perhaps, you will be rewarded with the image shown in Figure 2. If it asks you any questions, just answer in the most accomodating manner possible. 4. The raw data can be placed in the various columns. You can insert the labels for the different columns in the boxes marked A, B, C, etc. Each column has an alias, which is how the program recognizes which column you are talking about. Starting from the far left, the aliases for the columns are c0, c1, c2, etc. 5. I have entered some random phony data into the first column (column c0). For reasons unknown to any rational persion, I decided to let c1 = 100e c0 (5)
FIG. 2: Open Kaleidagraph Application The result is shown in Figure 3. 6. The values in column c1 were generated using formula entry. The easiest way to get to formula entry is to enter Crtl-f. This will bring up a dialog box that looks like Figure 4: To populate the column, just type into the box and hit Run. c1 = 100*exp(c0) 7. We now want to graph this data. To create a simple scatter plot of the data, go to the Gallery menu and select Linear-Scatter, as shown in Figure 5. This will bring you to a dialog box that looks like that shown in Figure 6. We select negative evens for the x axis and f(x) for the y axis, then hit New Plot, as shown in Figure 6. We should be rewarded with a graph that looks like that shown in Figure 7. This is OK, I guess, IF YOU DON T MIND THINGS THAT ARE UNBELIEVABLY UGLY!!. So let s pretty it up. The first thing we do is get rid of the legend. We don t need it because there is only one function plotted, which we can describe much better in the graph title. We do this by going to the Plot menu, and deselecting Display Legend. We can edit the x and y labels by double clicking on them. This opens dialog boxes that are rich with possibilities. The graph title can similarly be edited by double clicking on the title. It also has many and various options. I have taken full advantage of these options.
FIG. 3: Random Phony Data FIG. 4: The Formula Entry Dialog Box I can change the axes by hitting Ctrl-t, then making the appropriate changes in the dialog boxes. For the x axis, whose dialog box is shown in Figure 8, I have truncated the plot at x = 20 and reversed the axes. For the y axis, whose dialog box is shown in Figure 9, I have changed it to a log scale, so as to show off the variation for large values of x more clearly. 8. I can change the plotting symbols (as well as the curve fit template) by hitting Ctrl-m, which will bring up the dialog box shown in Figure 10, and making appropriate selections. After making sundry other changes, I obtain the graph shown in Figure 11 I will admit that it is quite garish, but I wanted to show you the range of what is possible, and also, it is late and I am getting very bored. II. SUMMARY PROCEDURE 1. Open a kaleidagraph window, and enter the data set shown in Figure 12. Try to figure out how I changed the format of the column entries.
FIG. 5: Making a Scatter Plot: Part 1 FIG. 6: Making a Scatter Plot: Part 2 2. Populate the f(x) column by letting f(x) = x ( x ) x 1.5 sinc 2 Make sure that the degree mode is set to radians in the formula entry box when you populate the
FIG. 7: An Ugly Scatter Plot FIG. 8: Changing the x Axis column. 3. Make your graph pretty, or at least interesting and informative. 4. Save it as both a kaleidagraph plot (qpc extension) and a gif file (gif extension). When saving the gif file, select Minimize White Space. 5. Write a report detailing what you have done. Insert your graph into your Word document. Make sure
FIG. 9: Changing the y Axis FIG. 10: Changing the Plotting Symbols your data table includes both your x data as well as your f(x) data. 6. Turn the report in. [1] To assert,in the modern practice of science, is synonymous with to publish. You cannot be criticized for things you have not published, but by the same token, unpublished beliefs do not take part in the formation of the scientific consensus and are therefore scientifically irrelevant. At least at the present time, beliefs put on the Web do not have the same status as beliefs published in a peer-reviewed journal, although this hierarchy may change. [2] In this approximation, we can relax our requirements on the analyticity of f(α j), requiring only that f be a smooth function of α j in the neighborhood of α j0. A smooth function is one that is continuous and has a continuous first derivative.
FIG. 11: My final plot
FIG. 12: x Axis Data