PHYS 342L NOTES ON ANALYZING DATA Spring Semester 2002 Department of Physics Purdue UniversityA major aspect of experimental physics (and science in general) is measurement of some quantities and analysis of experimentally obtained data. While there are a lot of books devoted to this problem, in the next paragraphs we will summarize some of the important ideas that will be needed to successfully analyze data acquired in PHYS 342L. Students are advised to consult with [1] for more detailed discussion on the topic. 1. The importance of estimating errors. Suppose you are asked to measure the length of a piece of notebook paper. You grab a ruler and proceed with a measurement. The ruler shows 276 mm. Does it mean the length is 276.0000 mm? Most probably not. Why? Because the distance between the neighboring marks on your ruler is 1 mm, by saying 276 mm you cannot exclude, for example, length 276.2 or 259.9 mm. Thus you assume certain precision (or error) in your measurement, in this case it is probably ~0.5 mm, as the distance between the closest marks is 1 mm. The result of the meausurement is not just the length of the paper but also the error of this measurement: (276.0±0.5) mm. In a scientific experiment, both parts of measurement are important. Suppose you measure the length of the next sheet of paper to be (275.5±0.5) mm. Within the error of your measurement these two sheets of paper have the same length. 2. Precision (or Accuracy) of a Measurement. Distinguish between absolute uncertainty and relative uncertainty: absolute uncertainty relative uncertainty 27.6 ± 0.1 003623188.06.271.0±=± All these numbers don’t mean much when calculating the relative uncertainty, so round off to ± 0.004, or, expressed as a percent, ± 0.4%. 3. Combining Uncertainties. Suppose that you measure two quantities A and B. Suppose you measure A to an accuracy of ±δA and B to an accuracy of ±δB. How do you algebraically combine these uncertainties? a) When adding: (A ± δA) + (B ± δB) = ? there are four possibilities: (A + δA) + (B + δB) = (A + B) + (δA + δB) (A + δA) + (B - δB) = (A + B) + (δA - δB) (A - δA) + (B + δB) = (A + B) - (δA - δB) (A - δA) + (B - δB) = (A + B) - (δA + δB) clearly, the worst case will be (A+B)±(δA+δB) (1) b) When subtracting: (A±δA) - (B±δB) =?Again consider four cases. From above, it should be obvious that the worst case will be given by (A-B)±(δA+δB) (2) c) When multiplying +±≈+±≈±±+±+±+=±×±AABBABABBAABBAABBAABBBAAneglectsmallδδδδδδδδδδ1)())(()()()()(, (3) d) When dividing ?=±±BBAAδδ After some algebra, you find that +±≈±±BBAABABBAAδδδδ1 (4) Remember: • relative uncertainties add when multiplying or dividing. • absolute uncertainties add when adding or subtracting 4. Systematic and random errors. X1 X2 X3 X4 X6 X5 X8 X7 Xtrue X X1 X2 X3 X4 X6 X5 X8 X7 Xtrue X Figure 1: Spread in the measurement of some quantity x in the absence of systematic error (left) and in its presence (right). If error in your measurements is random, then the average value should be close to the actual value. In the case of systematic error, that is not true. This situation may occur when, for example, using a clock which is running slow to measure some time period. Random errors are inevitable, while systematic errors can be taken into account or eliminated. 5. Average value and standard deviation. In order to decrease the influence of random error multiple measurements xi are taken and averaged: ∑==niixnx11 (5) How close this average value x is to the actual value X? If we have a set ofmeasurements we can find an average error for a single measurement. The commonly accepted value to characterize error is called standard deviation σ, or root mean square (rms): ( )∑=−=niiXxn1221σ (6) Since the actual value X is usually unkown, we must use x instead. It can be shown [1] that in this case: ( )∑=−−≈niixxn12211σ (7) The value σ characterizes error in a single measurement of value X. If we take several measurements of the same value x and average them, the resulting value x must in average be closer to actual value x as a single measurement. It can be shown [1] that standard deviation σn for the average value of n measurements is: nnσσ= (8) 6. Distribution of measurements A series of measurements may be represented as a histogram (Fig. 2). Figure 2. A simple histogram after taking just five data points (n=5). There was only one data point falling into range of x marked as A, B and D, and two measutrements where in region C. It is difficult to see any trends after taking just a few data points. Make more measurements and use smaller bins and you’ll eventually get a histogram that might look like this.Figure 3. An histogram after taking hundreds of measurements. In a limit of large n the distribution is given by continuous distributin function f(x), so that f(x)dx is the probability that a single measurement taken at random will lie in the interval x to x+dx. The average value can be then found as: ∫∞∞−⋅= dxxfxx )( (9) And standard deviation: ( )∫∫∞∞−∞∞−−≈−== dxxfxxdxxfXxx )()()(2222σ (10) In many cases error distribution function is well described by Gaussian (also called normal distribution (Fig. 4): ( )22221)(σπσXxexf−−= (11) σ fwhm X Figure 4. Gaussian distribution function.The standard deviation σ for Gaussian distribution can be also expressed as: fwhmfwhm×≈= 425.0)2ln(22σ (12) where fwhm if full width at half maximum, which can be estimated graphically. Suppose now that we performeed a single measurement which resulted in value x and we also know the standard deviation of this measurement σ. Since the Gaussian distribution is a continuous function which becomes zero only in infinity, the measured value x may lay anywhere from -∞ to +∞. What is the probability that the actual value X which we are trying to measure is within distance σ from this measured value? Since f(x)dx is the