Physics 201 Lab 10: Data Weighting Dr. Timothy C. Black Summer I, 2018

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1 Physics 20 Lab 0: Data Weighting Dr. Timothy C. Black Summer I, 208 I. THEORETICAL DISCUSSION In experimental physics, we often combine different measurements to obtain a composite result. For instance, in the lab on Centripetal Force, you found the period of the orbital revolution of a spring loaded mass by averaging together the values of a number of different trials in which you measured the total period for 20 revolutions. In this case, you took the unweighted mean of the 20 revolution periods for all of the trials you conducted. Simply put, let us say that you had 6 different measurements of the time it took for the mass to complete 20 orbits. Then you added them together and divided by 6. This constituted an unweighted mean of your 6 trials. It was unweighted in as much as each of the six trials was included in the average with equal weightings of unit apiece. You then normalized the mean by dividing by 6. Another way of putting it is that each of the 6 trials was accorded a relative weight of ( 6) th. A. Weighted Means Whenever you take an unweighted mean of several quantities, you are implicitly assuming that each of the quantities in the mean is equally probable. There are many other occasions in which you may wish to combine data where you have reason to believe that the results that you are averaging are not equally probable. In this case, you would find the weighted mean of your data. Let us say that you assign each of your data a relative weight of w j. These weights should be normalized, so that the sum of all of the weight factors is equal to one. Thus, w j = j. This is of course exactly what you do with unweighted means also. In the case of taking the unweighted average of 6 20-orbit periods, each weight factor was equal to ( th. 6) The sum over all 6 of these weight factors is. But in taking a weighted average, it is not necessarily the case that all of the weight factors are equal. Each weight factor should represent your belief in the relative probability of each piece of data. In the case in which one is combining data that have different uncertainties σ j associated with them, it is typical to take the weight factors associated with the j th datum as being equal to w j = j () Therefore, the weighted mean of N different data f j, each with uncertainties σ j, is equal to f = N j= N f j = j (2) B. Weighted Data In Curve Fitting When performing a curve fit, it is often advantageous to weight the dependent variable when conducting the fit. In an unweighted curve fit, the aim is to find the set of M parameters α, α 2,..., α M that makes the

2 function of the N independent variables x, x 2,..., x N as close as possible to the set of dependent variables y, y 2,..., y N. As an equation, we require the set {α, α 2,..., α M } f (x, x 2,..., x N, α, α 2,..., α M ) y, y 2,..., y N However, it may be the case that not all of the dependent variable data y, y 2,..., y N are equally reliable. In this case, we can specify weights w j for each of the N values of y j that govern how much each contributes to the determination of the parameter set α, α 2,..., α M. The method of determining these weights depends on the curve-fitting algorithm used. Kaleidagraph uses an unnormalized version of the weight factors we described earlier in the section on weighted means. In other words, the weight factors in Kaleidagraph are taken to be equal to w (unnorm) j = (3) When performing a curve fit without data weighting, the weights on each data point are simply equal to one. Recall from the hot wheels lab the relationship between x and V, namely that x = v j 2h g (4) Consider Austin s data from the previous lab. If we fit this function to the form shown in figure, we obtain x exp = mv j + b (5) m = ± b = ± If we weight the curve fit, shown in figure 2, we obtain instead m = ± b = ± Note that both the expected slope and its uncertainties are different for the two fits. C. Weighted Data Averaging in Kaleidagraph Suppose that you have a set of N data points y, y 2,..., y N in column c0, and a set of associated uncertainties δy, δy 2,..., δy N in column c, shown in figure 3. Note that the values and uncertainties do not necessarily need to be in these specific columns, but since I am showing you an example, I have to choose something. In this example, my data points represent the fractional fat content of a bunch of hamsters that I have rendered. I can obtain a number of statistical parameters from this data by using the Statistics function. If I select column c0, then click on Functions and then on Statistics, I obtain a clipboard showing the statistical data, depicted in figure 4. In this data set, the unweighted mean is given as f = , and the uncertainty σ f, which is called the standard deviation in this table, is σ f = Thus, my unweighted hamster fat content is equal to

3 FIG. : An unweighted curve fit of x exp vs. v j FIG. 2: A weighted curve fit of x pred vs. v j f = ± The variance is simply the square of the standard deviation. In order to take the weighted average of this data set, I must first create a third column, c2, which I will call weight factors. The unnormalized weight factors are found from equation 3. I populate this column by using the Formula Entry and typing c2 = /c^2

4 FIG. 3: Some weighted hamster fat data FIG. 4: Unweighted statistics on hamster fractional fat content Using the Statistics function on this new column, I find that the sum of the column entries is equal to S = j = (6) I divide the entries in column c2 by this sum, using the Formula Entry with the command c2 = c2/.507e5 This gives me the normalized weight factors, represented in equation. I now form a fourth column, c3, from the product of the fractional fat (column c0) and the normalized weight factors (column c2), and call this column weighted fracfat. c3 = c0*c2 Using the Statistics function, I can find the statistical parameters associated with this column, shown in figure 5. The weighted average of my fractional fats is just the Sum of this column, which is equal to

5 f weighted = Note that it is not equal to the mean of this column, because these weight factors are already normalized. The uncertainty in this weighted average is found from our definition of the weighted uncertainty, namely, FIG. 5: Weighted statistics on hamster fractional fat content σ = j (7) Putting in the numbers from our previous result in equation 6, we have σ = j = = Thus, the weighted average of the fractional fat content of my hamster collection is equal to f weighted = ± Note that this is very different from the unweighted average. This is because the uncertainty associated with the data point in the fifth row is vastly smaller than the uncertainties associated with the other data points. This significantly weights the average towards this particular value. Sometimes this is appropriate, and sometimes it is not. You must use your judgement to determine which is the case for your specific situation. D. Weighted Curve Fitting in Kaleidagraph In order to weight the curve fit, we must use the General form of curve fitting. When defining the fitting function, which in this case corresponds to that shown in equation 5, we must check the Weight Data box in the curve fit dialogue box, shown in figure 6. A new dialogue box will appear, which asks us to select the column in which our uncertainties are located, as shown in figure 7. After making the appropriate choice of column, and hitting OK, our weighted curve fit will then be generated.

6 FIG. 6: General Curve fit dialogue box FIG. 7: General uncertainty column selection dialogue box II. SUMMARY I have sent you all the class results for the experimental values of the local gravitational constant (in units of cm/s 2 ).. Using the procedure of section I C, find the weighted mean value g weighted of the class determination of the local gravitational acceleration constant. 2. Using the procedure of section I C and equation 7, determine the overall uncertainty σ g in the class determination of the local gravitational acceleration constant. 3. Report your values of g weighted and σ g in the form g ± σ g