6. Distribution of measurementsReferencePHYS 342LNOTES ON ANALYZING DATASpring Semester 2002Department of PhysicsPurdue UniversityA major aspect of experimental physics (and science in general) is measurement ofsome quantities and analysis of experimentally obtained data. While there are a lot ofbooks devoted to this problem, in the next paragraphs we will summarize some of theimportant 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 aruler and proceed with a measurement. The ruler shows 276 mm. Does it mean the lengthis 276.0000 mm? Most probably not. Why? Because the distance between the neighboringmarks on your ruler is 1 mm, by saying 276 mm you cannot exclude, for example, length276.2 or 259.9 mm. Thus you assume certain precision (or error) in your measurement, inthis case it is probably ~0.5 mm, as the distance between the closest marks is 1 mm. Theresult of the meausurement is not just the length of the paper but also the error of thismeasurement: (276.00.5) mm. In a scientific experiment, both parts of measurement areimportant. 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 samelength.2. Precision (or Accuracy) of a Measurement.Distinguish between absolute uncertainty and relative uncertainty: absolute uncertainty relative uncertainty 27.6  0.1003623188.06.271.0 All these numbers don’t mean much when calculating the relative uncertainty, soround 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 anaccuracy of A and B to an accuracy of B.How do you algebraically combine these uncertainties? a) When adding: (A  A) + (B  B) = ?there are four possibilities: (A + A) + (B + B) = (A + B) + (A + B)(A + A) + (B - B) = (A + B) + (A - B)(A - A) + (B + B) = (A + B) - (A - B)(A - A) + (B - B) = (A + B) - (A + B)clearly, the worst case will be (A+B)(A+B) (1)b) When subtracting: (AA) - (BB) =? Again consider four cases. From above, it should be obvious that the worst case willbe given by(A-B)(A+B) (2)c) When multiplying AABBABABBAABBAABBAABBBAAneglectsmall1)())(()()()()(,  (3)d) When dividing ?BBAAAfter some algebra, you find that BBAABABBAA1 (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 systematicerror (left) and in its presence (right).If error in your measurements is random, then the average value should be close to theactual value. In the case of systematic error, that is not true. This situation may occurwhen, 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 oreliminated.5. Average value and standard deviation.In order to decrease the influence of random error multiple measurements xi are takenand 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 commonlyaccepted value to characterize error is called standard deviation , or root mean square(rms): niiXxn1221(6)Since the actual value X is usually unkown, we must use x instead. It can be shown[1] that in this case: niixxn12211(7)The value  characterizes error in a single measurement of value X. If we take severalmeasurements of the same value x and average them, the resulting value x must inaverage be closer to actual value x as a single measurement. It can be shown [1] thatstandard deviation n for the average value of n measurements is:nn(8)6. Distribution of measurementsA 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 onedata point falling into range of x marked as A, B and D, and two measutrements where inregion C.It is difficult to see any trends after taking just a few data points. Make moremeasurements and use smaller bins and you’ll eventually get a histogram that might looklike 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 thatf(x)dx is the probability that a single measurement taken at random will lie in the intervalx to x+dx. The average value can be then found as: dxxfxx )((9)And standard deviation:  dxxfxxdxxfXxx )()()(2222(10)In many cases error distribution function is well described by Gaussian (also callednormal distribution (Fig. 4): 22221)(Xxexf(11) fwhm X Figure 4. Gaussian distribution function.The standard deviation  for Gaussian distribution can be also expressedas:fwhmfwhm 425.0)2ln(22(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 andwe also know the standard deviation of this measurement . Since the Gaussiandistribution is a continuous function which becomes zero only in infinity, the measuredvalue x may lay anywhere from - to +. What is the probability that the actual value Xwhich we are trying to measure is within distance  from this measured value? Sincef(x)dx is the probability of measuring value between x and x+dx, the probability ofmeasuring x between X- to X+ is given