ࡱ> 5 bjbjT~T~ .668iE)SS87D{LG[t("~~~ Y Y Y Y Y Y Y\]_ Yk\"~kk YSSK[3#3#3#kLSl Y3#k Y3#3#2S,X<vU.X[0G[U`y 2`\X`XT~^3#`L~~~ Y Y"~~~G[kkkk`~~~~~~~~~ :  Introduction In science, the goal is truthful understanding of our world based on careful observations and measurements of natural phenomena. To build our understanding, we struggle to develop conceptual models. These models (theories) must be tested and/or validated to really be worthwhile. Often, observations and/or experimental results cause us to realize the limitations of our models (theories) and so we must work to develop better (or, at least, more complete) ones. To see if our models are valid, we must carefully compare their predictions to what is actually seen in nature. Imprecise or sloppy comparisons of our theoretical models with nature do not improve anyones understanding of either and, in fact, often just lead to greater confusion. Consequently, true understanding will ONLY come from extreme care with each and every measurement and theoretical calculation. So, in this spirit, a healthy self-critical approach and skepticism about the accuracy of any result should be developed and nurtured. To this end, we should endeavor to make each and every measured value as accurate and precise as possible. We must work to minimize the error or uncertainty of each and every measurement. To be specific about expressing these qualities of a quantitative measurement, we use the rules for so-called significant figures. These and other related terms are defined as follows: Significant Figures: the number of digits that are known (or specifically given) for a value. This may differ from the total number of digits actually used in writing out the value. A standard set of rules determines the actual number of significant figures (SF) used in an expression of a value (see below). Measurement Accuracy: how close a measured value is to the true value (if it is known). If the true value is not known, then the accuracy of measurement can only be estimated (typically, this must be done with extreme care). Measurement Precision: an indication of the reliability and/or repeatability of a measurement, as reflected by the number of significant figures used to represent the measured value. Measurement Uncertainty/Error: the estimated deviation of a measured value from the true value. The true value may or may not be known. There are three types (sources) of error: measurement mistakes, random errors, and systematic errors. Measurement mistakes are illegitimate errors since they are due to sloppiness and/or lack of care in the measurement process and are avoidable. Mistakes errors should always be completely eliminated. Random errors result from (hopefully small) uncontrolled variability of the environment, equipment, and/or other subtle aspects of the measurement. The individual measured values randomly deviate high or low of an average value. Systematic errors result in the consistent deviation of a measurement (on average, either high or low as compared to the true value) due to equipment problems or neglect (or ignorance) of some other important factor in the measurement process. The following pages present a detailed discussion of measurement errors and the interpretation of measurements. This discussion is organized into a set of sections: What is meant by an Uncertainty (or an error). Rules for working with Significant Figures. How to compute Percentage Error & Percentage Difference Ideas for Preparing a Scientific Table or Graph Rules for calculation of the Propagation of Errors, and Mean Value, Standard Deviation, and Standard Error: A set of repeated measurements subject to random errors (normally distributed) can be treated statistically. The mean value is the best approximation to the true value. The precision of the measurement can be estimated by calculating the standard error from the standard deviation. The procedure for Fitting a Straight Line to Data (Linear Regression) Uncertainty The best estimate of a measured value A is often given with an explicit uncertainty A. This means that one should realize and remember that the written value is likely to be wrong by a bit; our uncertainty in its value is related to A. So, in science and elsewhere, values are often written in the form A A. For example, an accurate measurement of the acceleration due to Earth s gravity  g in our TCU physics lab location might turn out to be equal to 9.80 0.01 m/s2. What do these numbers really mean? According to our statistical definitions of uncertainties, this value for g has a 68% likelihood (chance) of being within 0.01 m/s2 of the true value. Said another way, there is a 32% chance of the measurement being further than 0.01 m/s2 from the true value for g. Also, from the same statistics, the measured value has only a 5 % chance of being off from the actual value by more than 0.02 m/s2 ( 2) and a much smaller 0.2 % chance of being wrong by more than 0.03 m/s2 ( 3). Based on this information and the measured value, we can deduce that (at the location of our TCU lab) it is unlikely (a chance of about 1:20 or 5%) that the actual value of g is greater than 9.82 m/s2 or less than 9.78 m/s2, and it is very unlikely (a chance of about 1:500 or 0.2%) that the actual value of g is, for example, greater than 9.83 m/s2 or less than 9.77 m/s2. Note that when a value is written in the form A A, such as 9.80 0.01 m/s2, that the value is rounded off to the same digit as determined by the uncertainty A. It is pretty meaningless, for example, to write such a value as 9.8013521 0.01 m/s2, even if this resulted from a detailed experimental analysis. Given the size of the uncertainty, the extra digits are pretty useless and the value should be rounded-off to the appropriate digit. This kind of reasoning has led to a convention called using significant figures. Significant Figures When writing numbers down it is important to communicate the actual significance of the digits and numbers used. If a fellow says I have five hundred dollars how many does he really have? Did he round-off the number? Perhaps he really has 499 and rounded up by one, or maybe he has 523 and he simply reported it to the nearest hundred. To communicate concretely, we must avoid this ambiguity. We need rules for explicitly and consistently reporting the significance of digits used to represent values. The conventional rules for determining the number of significant figures (SF) in a quantity are as follows: The most significant digit is the leftmost nonzero digit. Zeros at the left are never significant. For example, the value 0.0023 has two SF (while it has 5 digits) it could also be written as 2.3e-3, for example (still with 2 SF). If no decimal point is explicitly given, the rightmost nonzero digit is the least significant digit. Ex.: the value 300 has one SF, while 3.00e2 has three SF and the value 300.0 has four SF. If a decimal point is explicitly given, the rightmost digit is the least significant digit, regardless of whether it is zero or nonzero. See the above example. The number of SF is determined by counting significant digits from most significant to the least significant. So, in the example in the previous section, the value of g is determined and reported with three significant figures. If the uncertainty was actually 0.1 m/s2 then it should be reported as 9.8 0.1 m/s2, with only two significant figures. Significant Figures and Calculations When adding, subtracting, multiplying or dividing values, it is necessary to pay attention to significant figures. For example, if a value 2.1 (apparently precise to 0.1) is multiplied by 4.555 (apparently precise to 0.001) then the calculated number is equal to the value 9.5655. The actual number of significant figures is only two (due to the precision of the 2.1 value) so the result (if it is a final one) should be expressed as 9.6, after rounding off at the second significant figure. This is easily understood by repeating the multiplication using 2.2 and/or 2.0 the reasonable range of values given the implied precision of 0.1). Similar reasoning can be applied to other calculations to determine their effect on the actual significant figures in the result, for example 100.0 added to 0.08 is written 100.1 due to significant figures and appropriate round-off. Percentage Error If the true value of a quantity is known, the percentage error of a measurement is simply the difference between the measurement M and the true value T, divided by the true value, and then multiplied by 100%. This can be written: %Error = 100%(MT)/T. Percentage Difference Sometimes the difference between two measurements (or other values) is represented as a percentage difference. This is typically done when the two values are similar to each other (they must have exactly the same units). To calculate the percentage difference between values M1 and M2 you do a similar calculation as for percentage error, but instead you divide by the average of the two values. This can be written: %Difference = 200%(M1M2)/(M1+M2). Preparing a Scientific Table or Graph Measurements can be displayed and represented in Tables or Graphs. Tables are multi-dimensional lists that show the relationships between different sets of measurements. For example, the location of a moving car might be represented by a table giving the time of the measurements and the measured X and Y coordinates. To be meaningful, the units of each listed value must be included. Suppose the cars positional coordinates are measured at various times, to a precision of one meter and 0.01 second. Here is the data shown in a Table: Time (seconds)X coordinate (km)Y coordinate (km)0.002.3510.14423.502.8800.04055.213.235-0.05573.563.511-0.147 Likewise, the same data can be displayed graphically:  EMBED Excel.Chart.8 \s  Note how the axis scales are chosen and labeled. The point of a graph is to communicate the values and the trends in the data. So, every attempt should be made to ensure that the graph title, axis scales and labels give complete and accurate information. In the above case, the data points are large to help them show up. The values are known very accurately, more accurately than can be determined from the graph (compare with the accuracy shown in the Table). In contrast, if the measurements were, in fact, only certain to within 0.25 km and 5 sec, then we should graphically display this measurement uncertainty as we show the data. This is done with so-called graphical error-bars attached to the data points. This is demonstrated as follows:  EMBED Excel.Chart.8 \s  Mean Value, Standard Deviation, and Standard Error If measurements were perfect, and so therefore perfectly exact, we would know a desired answer from a single measurement of a quantity. Since measurements are not perfect, we work for improved accuracy by repeating the measurement process. How can we obtain our best estimate of the desired value from a repeated set of measurements? How can we determine our confidence (or our uncertainty) in our resulting estimate? A measurement is repeated N times, resulting in a set (or list) of values Ai where the index i identifies each specific measurement (i taking values between 1 and N). If the variations of the values in this set are entirely due to random errors, we may obtain an improved estimate of the true value by applying a simple and standard statistical treatment to summarize our results. This treatment becomes increasingly meaningful for larger numbers of measurements. For example, suppose a mass is measured five times and the following set of values is obtained: {9.15, 9.13, 8.96, 9.18, and 9.09 grams}. Suppose each measurement was done in the same fashion and so we have no reason to suspect (or respect) any one value over the others. To analyze this set of measurements (our sample) and to determine the precision of a resulting best measurement value, the mean value, sample standard deviation, and standard error may be calculated. These quantities (and other related ones) are defined as follows: The Mean Value  is the average of the measured values. The mean is calculated from the sum all Ai (from i = 1 to i = N) and then division of this sum by N. Our above example of five mass measurements has a mean value of 9.102 g. This is our best estimate of the correct true value of the mass we are measuring. The RMS Deviation is obtained by taking the square root of the mean of the squared deviations (hence, the RMS-deviation). This is calculated in the following steps: First, for each measured value a squared-deviation is calculated using the expression (Ai  )2. The set of these for our example is {0.002304, 0.000784, 0.020164, 0.006084, and 0.000144 grams2}. Second, a sum-of-the-squared-deviations (Ai  )2 is calculated and this sum is divided by N, the number of measurements (here, the symbol  means the sum of quantities for i =1 to N). In our example the sum is 0.02948 g2 and after division by 5, we obtain a mean-squared-deviation of 0.005896 g2 (keeping extra digits to avoid round-off error). Third, the rms-deviation is found from the square root of the previous result. For our example, the sample standard deviation is approximately equal to 0.0768 grams (extra digits kept to avoid round-off error). The Standard Deviation  is similar to the rms-deviation, except the  mean-squared-deviation is calculated by dividing the  sum-of-the-squared-deviations by the so-called  number of degrees of freedom (DOF), instead of N. Given this we can write a formula for the square of the Standard Deviation (the so-called  variance ): 2 = [ (Ai  )2]/DOF. For a set of N measurements of the same quantity (like our mass), the DOF is equal to N-1. Consequently, for our example above, the Standard Deviation is approximately equal to 0.0858 grams (keeping extra digits to avoid round-off error). The Standard Error represents an estimate of our uncertainty for the measured mean value (as determined by the number of measurements and the variations in our set of values). The Standard Error is an estimate of the standard deviation of the distribution of mean values expected if the same set of measurements was repeated many times. The Standard Error S is calculated by dividing the Sample Standard Deviation  by the square root of the number of measurements N. As a formula: S =  /"N. In our example, the standard error is 0.04 grams (generally, it is only necessary to keep one or perhaps two significant figures for the standard error). The Result of a Set of Measurements: The best estimate of the measured value (and its estimated uncertainty) is given by  S. For our example, the result of our measurements is 9.10 0.04 grams (note that mean value is rounded off to the same digit as the uncertainty). Propagation of Error Final results should always reflect the appropriate precision. So, final results (typically involving a calculated value based on measurements) should include appropriate estimates of uncertainty. Of course, intermediate calculations should be done with extra digits (to avoid the accumulation of round-off errors). So, how do we determine the uncertainty of a quantity that depends on several measured values? Values calculated from several measured quantities will certainly inherit their uncertainties in a way that depends on the form of the calculation. A few examples of how errors propagate (from the measurements to the result) are shown below for measured quantities X and Y (and a constant C that has little or no uncertainty): If A = CX,then A = CX If A = C+Xthen A = X If A = X+Ythen, (A)2 = (X)2 + (Y)2If A = XY, then (A)2 = Y2(X)2 + X2(Y)2. If A = X/Y, then, (A)2 = A2[(X /X)2 + (Y /Y)2 ]If A = XC, then A = AC(X )/X.  These formulae and those for other cases are derived from probability theory. The above relations are valid for the propagation of errors (or in deviations or variances, as shown above). The general rule for the propagation of error in a calculated quantity A that depends on measured quantities X, Y, and Z is given by the relationship (A)2 = (A/X)2(X)2 + (A/Y)2(Y)2 + (A/Z)2(Z)2 , where A/X means  the partial derivative of the quantity A with respect to variable X (from calculus). If necessary, it is simple enough (if you know calculus) to verify and/or extend the above list of formulae [dont worry about this if you have not had calculus]. Obtaining a Linear Fit to Data (Linear Regression) Often measurements are taken in pairs: by changing (and measuring) one variable X and then subsequently measuring a second variable Y. A set of such data pairs helps to determine the relationship of the two variables (quantities). In some cases, Y will depend on X with a linear relationship given by Y = MX + B, where M and B are constants. In a plot of Y vs. X for a straight line, the constant M is the slope of the plotted line and B is the Y-intercept (the value of Y when X = 0). Suppose we have a reason to think that some pair of quantities X and Y have a linear relationship. To verify this relationship, a set of N measurements of these variables will produce a set of X and Y pairs which we may denote as (Xi, Yi). These may be plotted on a graph and the trend of the data examined. Statistical theory states that there is a best-fitting line to the data. We may define the best-fitting line as the line that deviates as little as possible from the data. Consequently, we need a way to define and calculate the  quality of the fit of the line to the data.  Quality-of-Fit 2 Various formal definitions (or approaches) exist for determining the best fitting line. For the purposes of our course we will define a quantity 2 ( chi-squared ) as the sum of the squared deviations, comparing the data to the line. For a given choice of parameters M and B, this quantity is given by the following sum (over all data points, i = 1 to N): 2 =  [Yi  (MXi + B)]2. The values of B and M that produce a minimum 2 can be found (again from the use of calculus). From such an analysis best-fitting values can be calculated from the following formuale: B = (SxxSy - SxSxy) / , and M = (NSxy  SxSy) / , using the various quantities defined by: Sx: the sum of all Xi values, written as Xi Sy: the sum of all Yi values, written as Yi Sxx: the sum of all (Xi)2 values, written as Xi2 Syy: the sum of all (Yi)2 values, written as Yi2 Sxy: the sum of all (XiYi) values, written as XiYi , and : the quantity [N(Xi2 )-(Xi)2]. The uncertainties for the slope and intercept can be obtained by evaluating the propagation of errors from the measured values (Xi and Yi) to the calculated slope and intercept (see the previous section on error propagation). If the Yi errors dominate and all equal to a value Y, it turns out that the slope and intercept uncertainties are simply given by: B2 = (Sxx / )(Y)2 M2 = (N / )(Y)2 SPEED TRAP Example Here is a specific example: Traffic police use speed measuring devices that measure the round-trip travel time for reflected pulses of radio-waves (RADAR) or infra-red light (LIDAR). These devices actually measure the distance between the source and the reflector of the waves (the police device and the suspect car). Suppose a car is moving with constant velocity down a road and its position is measured at five different times. A graph of the measured position vs. measured time explicitly shows the motion of the car. In such a graph, the slope a line fitted to the plotted data points is our best estimate of the cars true velocity. In this case, N = 5 and the car positions are Yi and the measured times are Xi, as listed in the following table (with their estimated uncertainties): MeasurementTime (Xi)XiPosition (Yi)Yi10.150 sec0.001 sec120 m2 m20.300 sec0.001 sec115 m2 m30.450 sec0.001 sec111 m2 m40.600 sec0.001 sec107 m2 m50.750 sec0.001 sec101 m2 m To determine if the car is speeding or traveling at a legal speed (less than or equal to 65 miles per hour = 29 m/s), the device must calculate an estimated value and uncertainty for the cars speed. Clearly, the dominant uncertainties are for the Y position values, so the linear regression method described above will be sufficient. Using these measured values and the formulae given above we can obtain values (shown with extra digits to avoid subsequent round-o U ] b i 8 K XvU]! 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So, M = 4.22 m/sec Consequently, the measured quantity of interest being the car s speed, we should write it with a reasonable estimated uncertainty (converted into a standard error S =  /"N): Car s Speed = 31 2 m/s We can conclude: that it is LIKELY that the true speed of the car is between 29 m/s and 33 m/s, and that it is UNLIKELY that the true speed of the car is much less than 29 m/s or much more than 33 m/s. In this borderline case, where the difference is comparable to the uncertainty, a conservative or driver-friendly approach would be to conclude that the cars true speed is at least close enough to the limit to be considered legal. If, however, the measured value was instead found to be 35 2 m/s, then a speeding ticket would be justified (since it is very unlikely that the cars true speed was 29 m/s or less. LAB 1 ASSIGNMENT (100 pts) Student_Name________________________ 1. INTRODUCTION: Draw a line between the matching concepts/statements (12 pts) High AccuracyLarge number of significant figuresHigh PrecisionSubtle unavoidable variability Illegitimate ErrorConsistent error or deviationRandom ErrorEstimated deviation from true valueSystematic ErrorMeasurement mistakes due to sloppinessUncertaintySmall estimated uncertainty or error 2. UNCERTAINTY (15 pts) Several weights are measured and reported with their estimated statistical uncertainties as: A = 25 3 Newtons B = 30 1 Newtons Based on these values fill in the blanks and select the appropriate statements: There is a 95% probability that the true value of weight A is between __________ Newtons and ___________ Newtons. There is a 32% probability that the true value of weight A is less than ____________ Newtons or more than ___________Newtons It is ________________that the true values of weights A and B are equal. certain very likely very unlikely impossible 3. SIGNIFICANT FIGURES, PERCENTAGE ERROR/DIFFERENCE (18 pts) Give the number of significant figures for the following numbers: 300 __________ 20000.0 __________ 0.0000034 __________ 4.9 ( 103 __________ 0.123*13 __________ 0.123 + 13 __________ A mass measurement is repeated twice resulting in a first value of 1.235 g and a second value of 1.242 g. A third VERY ACCURATE and more reliable measurement (using a different mass scale) produces a value of 1.256 g. What is the percentage difference between the first and second values?_____________% What is the percentage error between the first and third values?______________% 4. MEAN VALUE, STANDARD DEVIATION, AND STANDARD ERROR (20 pts) Using a meter-stick, carefully measure the length of the hallway in front of the lab room (say, the distance between the doorways at opposite ends of the hallway). Record your measurements and complete the calculations listed below (including the units). MeasurementDistance Ai (m)Ai- (m)(Ai-)2 (m2)12345 What is the MEAN distance  ?_________________ ________. What is the RMS DEVIATION of the measurements?_____________ ______. What is the STANDARD DEVIATION A of the measured values?___________ _____. What is the RESULT (the best estimate of the measured value) given as the MEAN STANDARD ERROR? ____________ __________ __________. 5. LINEAR REGRESSION (35 pts) Here is a repeat of the example discussed in the section on data fitting (linear regression): Suppose a car is moving with constant velocity down a road where the posted speed limit is 35 mph. The cars position is measured at five different times by a police officer who uses a radar speed detector. A graph of the measured position vs. measured time explicitly shows the motion of the car. In such a graph, the slope of a line fitted to the plotted data is our best estimate for the cars true velocity. 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M&#$/Ifa$kdW!$$Iflr s 6! M064 lavx|N<-<<$! M&#$/If$$! M&#$/Ifa$kd!"$$Iflr s 6! M064 la<-$! M&#$/Ifkd"$$Iflr s 6! M064 la$$! M&#$/Ifa$ $8<kd#$$Iflr s 6! M064 la$$! M&#$/Ifa$8HTVZn<kd$$$Iflr s 6! M064 la$$! M&#$/Ifa$<:::kdI%$$Iflr s 6! M064 la$$! M&#$/Ifa$5 d,6Bsdhdh` hhdh^h`h`raveling at a legal speed (less than or equal to 35.00 miles per hour = 15.65 m/s), the device must calculate an estimated value and uncertainty for the cars speed. Clearly, the dominant uncertainties are for the Y (position) values, so the linear regression method described previously will be sufficient. Using these measured values and the formulae given previously we can obtain values (recorded with extra digits to avoid subsequent round-off errors): Sx = _______________ __________ Sy = _______________ __________ Sxx = _______________ __________ Syy = _______________ __________ Sxy = _______________ __________  = _______________ __________ So, by linear regression, the best values for the slope M and intercept B are: B = (SxxSy - SxSxy) /  = _______________ __________ M = (NSxy  SxSy) /  = _______________ __________ with squared uncertainties: B2 = (Sxx / )(Y)2 = _______________ __________ M2 = (N / )(Y)2 = _______________ __________ Since the measured quantity of interest is the cars speed, we should write it with a reasonable estimated statistical uncertainty (using its standard error): Cars Speed = _______________ __________ Is it LIKELY or UNLIKELY that the car was speeding? ________________________ In one or two complete sentences, support the above answer by comparing the estimated uncertainty with the difference between the estimated speed and the posted speed limit. ________________________________________________&`#$dh________________________________________________________________________________________________________________________________________________________________________     PAGE  PAGE 1 Lab 1 Accuracy & Error LAB 1: Measurement Accuracy & Error hHhHB*CJ$ph *hHB*CJ$phh=hlIXdh,1h/ =!"#$% $$If !vh5>5>5>#v>:V l065>4a $$If !vh5>5>5>#v>:V l065>4a $$If !vh5>5>5>#v>:V l065>4a $$If !vh5>5>5>#v>:V l065>4a $$If !vh5>5>5>#v>:V l065>4a iDd ZJ  C A? 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"     ! ! ! ! !    *8     8 x  < "< "|   ( # #  " x <  < "< "|    8@@ 8@ 8  @ 8@ 8 8 8 @ 8 8 " ! "   !X ! `~V Example Chart_Linear Regression(Chart1HSheet1wSheet2`i~v(   0e0e     Ad @  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S"@  Measurement Time (Xi) Xi Position (Yi) Yi1.5 sec0.1 sec12.0 cm0.1 cm3.0 sec14.4 cm4.5 sec16.7 cm6.0 sec19.2 cm7.5 sec21.5 cm Xi = 22.5 sec ! and (Xi)2 = (Xi)2 = 0.05 sec2     Yi = 83.8 cm Xi2 = 123.75 sec2 % and (Xi2)2 = 2(XiXi)2 = 2.48 sec4   $ Yi2 = 1461.14 cm2$ and (Yi2)2 = 2(YiYi)2 = 29.2 cm4   # XiYi = 412.8 cmsec and (XiYi)2 = 15.9 cm2sec2   D = 112.5 sec2 E = 178.5 cmsec and E2 = 159 cm2sec2F = 1082.25 cmsec2 and F2 = 34689 cm2sec4seccmTime Time ErrorPositionPosition ErrorD =F =cm secN = sec cm cm secSignificant Figures The most significant digit is the leftmost nonzero digit. Zeros at the left are never significant. Ex.: the value 0.0023 has two SF (while it has 5 digits)  it could also be written as 2.3e-3, for example (still with 2 SF). If no decimal point is explicitly given, the rightmost nonzero digit is the least significant digit. Ex.: the value 300 has one SF, while 3.00e2 has three SF and the value 300.0 has four SF. L cIf a decimal point is explicitly given, the rightmost digit is the least significant digit, regardless of whether it is zero or nonzero. See the above example.mThe number of SF is determined by counting significant digits from most significant to the least significant.Percentage ErrorPercentage Difference4Mean Value, Standard Deviation, and Standard Error 2The Sample Standard Deviation is obtained by taking the square root of the mean of the squared deviations (the rms-deviation). This is calculated in the following steps:? @K LW X First, for each measured value a squared deviation is calculated using the expression (Ai  )2. The set of these for our example is {0.002304, 0.000784, 0.020164, 0.006084, and 0.000144 grams2}. XY^_Second, a sum of the squared deviations is calculated and divided by N-1. In our example the sum is 0.02948 g2 and this is divided by 5-1 = 4, to obtain an average-squared-deviation of 0.00737 g2.noThird, the sample standard deviation SD is found from the square root of the previous result. For our example, the sample standard deviation is equal to 0.08585 grams.Propagation of ErrorValues calculated from several measured quantities will inherit their uncertainties. Final results should reflect the appropriate precision. Intermediate calculations should be done with extra digits (to avoid the accumulation of round-off errors).2Obtaining a Linear Fit to Data (Linear Regression)Often measurements are taken by changing one variable X, measuring its new value, and then subsequently measuring a second variable Y to determine its dependence on the first variable. In many cases, Y will depend on X with a linear relationship given by Y = MX + B,. the sum of all Xi values, denoted below as Xi-. the sum of all Yi values, denoted below as Yi-2 the sum of all (Xi)2 values, denoted below as Xi2012 the sum of all (Yi)2 values, denoted below as Yi2015 the sum of all (XiYi) values, denoted below as XiYi2343 the quantity [N(Xi2 )-(Xi)2], denoted below as D.z the quantity [N(XiYi)-(Xi)(Yi)], denoted below as E. The ratio E/D is the value of the optimally fitting line slope M.  the quantity [(Yi )(Xi2 )-(Xi)( XiYi)], denoted below as F. The ratio F/D is the value of the optimally fitting line intercept B. %&'( The uncertainties for the slope and intercept can be obtained by evaluating the propagation of errors from the measured values (Xi and Yi ) to the calculated slope and intercept (see the section on error propagation). Using this approach:m the square of the uncertainty for Xi is obtained from the sum of the squared uncertainties for each Xi value$%fg (Xi)2 = (Xi)2, h the square of the uncertainty for Yi is equal to the sum of the squared uncertainties for each Yi value$%ab (Yi)2 = (Yi)2,  the square of the uncertainty for Xi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv (Xi2)2 = 2(XiXi)2, the square of the uncertainty for Yi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv (Yi2)2 = 2(YiYi)2, the square of the uncertainty for the quantity XiYi is equal to the sum of the squared products of each Yi with the corresponding uncertainty Xi plus the sum of the squared products of each Xi with the corresponding uncertainty Yi1234jk" (XiYi)2 = (YiXi)2 + (XiYi)2,  ! the square of the uncertainty for the quantity D (defined as [N(Xi2 )-(Xi)2]) is equal to N multiplied by the uncertainty for Xi2 (equal to 2N(XiXi)2) added to the twice the squared prod< uct of Xi with its uncertainty Xi BCDJKLM$ D2 = 2N(XiXi)2 + 2(Xi2)(Xi)2, !"#A E2 = N[(YiXi)2 + (XiYi)2] + (Xi2)(Yi)2 + (Yi2)(Xi)2  %&'+-./567;=>?+ F2 = (Xi2)2(Yi)2 + (Yi)2[2(XiXi)2] +     "$%&'(1 (Xi)2[(YiXi)2 + (XiYi)2] + (XiYi)2[(Xi)2]   #$%&'(,./0Consequently, we can determine the uncertainties for the slope M. The square of the uncertainty for the slope is related to the above uncertainties for quantities E and D. Since M = E/D we have M2 = E2/D2 + D2(E2/D4).    Likewise, we can obtain the uncertainty for the Y-intercept B from the uncertainties for quantities F and D. Since B = F/D we have B2 = F2/D2 + D2(F2/D4).    Using these measured values and the formulae given above we can obtain values (shown with extra digits  to avoid subsequent round-off errors):M = E/D = 1.5867 cm/sec. B = F/D = 9.6200 cmHAlso we will obtain the squared uncertainties (shown with three digits):> M2 = E2/D2 + D2(E2/D4) = 0.0175 cm2/sec2 = (0.132 cm/sec)2   &'+,=2 B2 = F2/D2 + D2(F2/D4) = 2.92 cm2 = (1.71 cm)2   $%1Consequently, by taking the square roots of these values we can express our final best fitting results with uncertainties (rounding-off appropriately):*The car s velocity was 1.59 0.13 cm/sec.-The car s original position was 9.6 1.7 cm..MThe rules for determine the number of significant figures (SF) in a quantity:E =_ FOR A DIFFERENT SET OF MEASUREMENTS, BE SURE TO CHANGE THE UNITS LISTED THROUGHOUT THIS PAGE.u INSERT MORE ROWS ABOVE THIS ROW FOR ADDITIONAL DATA. THEN ADJUST THE N VALUE & SUMS TO OBTAIN CORRECT CALCULATION.Time2 Time Error2 Position Error2Xi XiYi YiXi2 (Xi)2Yi2 (Yi)2XiYi (XiXi)2 (YiYi)2 (XiYi)2 (YiXi)2sec2cm2 (cm sec)2 Xi =  (Xi)2 = (Xi)2=  sec2 Yi = (Yi)2 = (Yi)2=  cm2 Xi2 =  (Xi2)2 = sec4 Yi2 = (Yi2)2 = cm4 XiYi = (XiYi)2 =  cm2sec2 (XiXi)2=   (YiYi)2=   (XiYi)2=   (YiXi)2=   D2 = E2 = cm sec2 F2 = cm2sec4 cm/sec cm Optimal Slope (M) =Optimal Intercept (B) ='MechLab 1: Measurement Accuracy & ErrorPurpose^ In this activity, torque and rotational equilibrium are explored using an equal arm balance.Equipment Providedg Required: (1) one equal-arm balance, (2) a set of masses and 2 mass hangers, (3) thread, (4) a 5 g 7mass hanger and mass set, (5) a large sheet protractor.TheoryTime (seconds)X coordinate (km)Y coordinate (km)NOTE: The calculated values shown here have extra digits (to avoid subsequent round-off errors). The only numbers shown on this page with meaningful significant figures are the measured values and their estimated uncertainties. To control the way the va A measurement is repeated N times, resulting in a set (or list) of values Ai (where the index  i identifies each specific measurement -  i takes values between 1 and N). If the variations of the values in this set are entirely due to random errors, we KLY ce ~  The Mean Value  is the average of the measured values. The mean is calculated from the sum all Ai (from i = 1 to i = N) and then division of the sum by N. Our above example of five mass measurements has a mean value of 9.102 g. This is our best estimabcThe Standard Error represents an estimate of our uncertainty for the measured mean value (as determined by the number of measurements and the variations in our set of values). The standard error S is calculated by dividing the sample standard deviation  The Result of a Set of Measurements: The best estimate of the measured value (and its estimated uncertainty) is given by  S. For our example, the result of our measurements is 9.10 0.04 grams (note that mean value is rounded off to the same digit as$where M and B are constants. The constant M is the slope of the line and B is the Y-intercept (the value of Y when X = 0). To verify this relationship, a set of N measurements of the variables will produce a set of X and Y pairs which we may denote as ( the square of the uncertainty for the quantity E, with E = [N(XiYi)-(Xi)( Yi)], is equal to N multiplied by the squared uncertainty for XiYi added to the squared product of Xi with the uncertainty Yi plus the squared product of Yi with the uncert@ABCHINO !the square of the uncertainty for value F, with F = (Yi)(Xi2 )-(Xi)( XiYi), is equal to the sum of the squared product of Xi2 with the uncertainty for Yi added to the squared product of Yi with the uncertainty of Xi2 plus the sum of the squared p78<=>DEJKLM<O  '()*MNOPkHere is a specific example: Suppose a car is moving with constant velocity down a road and its position is accurately measured at five different times. A graph of the measured position vs. measured time explicitly shows the motion of the car. In such aFrom these values we may determine the best estimates for the car s average velocity and its original position. The optimally fitted slope and Y-intercept (shown with five digits  the uncertainties will determine the actual number of significant figures7 rhV+' ")07j; = p>! c? 4@ A_BChDM  @" caFing ??lo3` 4# ` 4# ` 4#` 4#` 4#` 4#` 4#hPH 0(   `h=:3d 3QQ ;Q ;Q3_ M NM  4E4 3QQQQ3_ O NM  MM<4JK4D$% MP+3O& Q4$% MP+3O& Q4FA57 3O 3*#M43*#M! M4% oM3O#&Q  Time (seconds)'4% &MZ3O#&P  Q  Position (cm)'4523  NM43d" 3_ M NM  MM<444%  OM3O/&Q &Position Vs. Time'4%  pb@M*?3OmA&PQ'44e?@@@@e(@,@333330@3333333@5@e>  @  dMbP?_*+%MLexmark 3200 Color Jetprinter@C odXXLetterDINU"0 },KMXL"dXX??U} } }  }  3$;$$,$,$,$,$,$,$ 7 7 ,$ ,$ J$J$J$J$J$J$J$J$J$Y$ J$J$,$,$,$J ;;; #! #" ## #$ #g #h ## #i$$$$$%% &j &k &l &m &n &o &p &q &r  &s  &t  &u  &v  %% & & & & 'w 'w 'x 'x ''  'w  'x  'y  'y  %%(?($@((@($@!)@ DD!)|Gz? DD!)b@ DD!)|Gz? DD!*2@ DD& * ףp= ? DD& * ףp= ? DD& + ףp= ? DD& + ףp= ? 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M4% ;M3OU&Q  Time (Seconds)'4% 0MZ3Or&Q ,Coordinates (meters)'4523  NM43d" < 3O9 % Mp73O&Q443_ M NM  MM<444% iUvM3OO&Q  Motion of Car'44e7@7@{GK@{GK@p= cR@p= cR@e+@;On? ףp= @{Gz?zG @)\(J + @7A`¿e> 7A` @  dMbP?_*+%"w??0U;;;;   ~ +@;On?7@r@@@8t@~ @J + @7A`¿*P*2>@7 @  dMbP?_*+%"??0U} }   h ,    , 7 ,7 7 7 , ; ; ; ; ;@ ;@ ;@ ;@ ;@ ;@ ;@ ;@ ;@ ;@ ; ; ; ; ; ; EEEEEEEE FFFFFFFF BBBBBBBBBBBBBBBBCCCCCCCC FFFFFFFFAAAAAAAA7 BBBBBBBBB B BBBBBBB7 CCCCCCCC7 F FFFFFFF  ,  c D-DDDDDDDDDDDDDDDDDDDDDDD" D.DDDDDDDDDDDDDDD" D/DDDDDDDDDDDDDDD" D0DDDDDDDDDDDDDDD"""""""" 1 2BX(((((((((0000(( ;! ;" w# w$ ;% & w' ;( ;) ;* ;+ ;, ;- ;. ;/ ;0 ;1 ;2 ;3 w4 w5 w6 7 8 w9 : w; < w= w> ? w 3! " # $4 %5 &6 '7 ( )* +8, -9. /: 0 1;1 2<2 334 4=5 5>6 6?7 7@8 8A9 9B :C: ;D; <E< =F= >G> ?H?Dl    @ A B C D E wF G H I wJ K L M N ;O ;P Q ;R ;S T ;U ;V 'W vX ;Y ;Z ;[ ;\ ;] ;^ ;_ ; @ I AJ B K CL D M EN F O GP H Q I JR K LS M!TN OU PVPQ RWR SXST UU VV~ W? W W W W~ X@ X X X X~ Y@ Y Y Y Y~ Z@ Z Z Z Z~ [@ [ [ [ [\ \ \ \ \]^_Dl    FFFFFB` ;a ;b ;c ;d ;e ;f wg h i j k l ;m n o ;p ;q ;r ;t u ;v ;w ;x ;y ;z ;{ ;| ;} ;~ ; ` a b c dY d ef ffg g gh hhi i ij j jk k kl llm m m n n o o pZp q[q r\r t^ t] u_u v`v waw x yby z { | } ~@8D"&"&&&"& ;40 (  v2  < @{=h]`"~v2  < @=]`P"~v2  < @=]`@#~v2  < @=]`#~v2  < @{43h4]`$~v2  < @{53h5]`$~v2  < @{6/h6]`%~v2  < @{7/h7]`&~v2   < @{83h8] `'~v2   < @{9/h9] `P'>@r   7 9^-:t7< ;! ;" w# w$ ;% & w' ;( ;) ;* ;+ ;, ;- ;. ;/ ;0 ;1 ;2 ;3 w4 w5 w6 7 8 w9 : w; < w= w> ? wC 4Mean Value, Standard Deviation, and Standard Error 2!)"A measurement is repeated N times, resulting in a set (or list) of values Ai (where the index  i identifies each specific measurement -  i takes values between 1 and N). If the variations of the values in this set are entirely due to random errors, we KLY ce ~ #The Mean Value  is the average of the measured values. The mean is calculated from the sum all Ai (from i = 1 to i = N) and then division of the sum by N. Our above example of five mass measurements has a mean value of 9.102 g. This is our best estimabc$The Sample Standard Deviation is obtained by taking the square root of the mean of the squared deviations (the rms-deviation). This is calculated in the following steps:? @K LW X%First, for each measured value a squared deviation is calculated using the expression (Ai  )2. The set of these for our example is {0.002304, 0.000784, 0.020164, 0.006084, and 0.000144 grams2}. XY^_&Second, a sum of the squared deviations is calculated and divided by N-1. In our example the sum is 0.02948 g2 and this is divided by 5-1 = 4, to obtain an average-squared-deviation of 0.00737 g2.no'Third, the sample standard deviation SD is found from the square root of the previous result. For our example, the sample standard deviation is equal to 0.08585 grams.(The Standard Error represents an estimate of our uncertainty for the measured mean value (as determined by the number of measurements and the variations in our set of values). The standard error S is calculated by dividing the sample standard deviation  )The Result of a Set of Measurements: The best estimate of the measured value (and its estimated uncertainty) is given by  S. For our example, the result of our measurements is 9.10 0.04 grams (note that mean value is rounded off to the same digit as$*+Propagation of Error,-Values calculated from several measured quantities will inherit their uncertainties. Final results should reflect the appropriate precision. Intermediate calculations should be done with extra digits (to avoid the accumulation of round-off errors)..;/2Obtaining a Linear Fit to Data (Linear Regression) 01Often measurements are taken by changing one variable X, measuring its new value, and then subsequently measuring a second variable Y to determine its dependence on the first variable. In many cases, Y will depend on X with a linear relationship given by12 Y = MX + B,23where M and B are constants. The constant M is the slope of the line and B is the Y-intercept (the value of Y when X = 0). To verify this relationship, a set of N measurements of the variables will produce a set of X and Y pairs which we may denote as (34s4.the sum of all Xi values, denoted below as Xi-5s5.the sum of all Yi values, denoted below as Yi-662the sum of all (Xi)2 values, denoted below as Xi201772the sum of all (Yi)2 values, denoted below as Yi201885the sum of all (XiYi) values, denoted below as XiYi234993the quantity [N(Xi2 )-(Xi)2], denoted below as D.:zthe quantity [N(XiYi)-(Xi)(Yi)], denoted below as E. The ratio E/D is the value of the optimally fitting line slope M. :C;the quantity [(Yi )(Xi2 )-(Xi)( XiYi)], denoted below as F. The ratio F/D is the value of the optimally fitting line intercept B.  %&'(;<The uncertainties for the slope and intercept can be obtained by evaluating the propagation of errors from the measured values (Xi and Yi ) to the calculated slope and intercept (see the section on error propagation). Using this approach:<=mthe square of the uncertainty for Xi is obtained from the sum of the squared uncertainties for each Xi value$%fg=O>(Xi)2 = (Xi)2, >?hthe square of the uncertainty for Yi is equal to the sum of the squared uncertainties for each Yi value$%ab?DlG - !  ?"(-Q ]@ A B C D E wF G H I wJ K L M N ;O ;P Q ;R ;S T ;U ;V 'W vX ;Y ;Z ;[ ;\ ;] ;^ ;_ ;O@ (Yi)2 = (Yi)2, ;Athe square of the uncertainty for Xi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv_B (Xi2)2 = 2(XiXi)2, ;Cthe square of the uncertainty for Yi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv_D (Yi2)2 = 2(YiYi)2,  Ethe square of the uncertainty for the quantity XiYi is equal to the sum of the squared products of each Yi with the corresponding uncertainty Xi plus the sum of the squared products of each Xi with the corresponding uncertainty Yi 1234jkF "(XiYi)2 = (YiXi)2 + (XiYi)2,  !Gthe square of the uncertainty for the quantity D (defined as [N(Xi2 )-(Xi)2]) is equal to N multiplied by the uncertainty for Xi2 (equal to 2N(XiXi)2) added to the twice the squared product of Xi with its uncertainty Xi BCDJKLMH $D2 = 2N(XiXi)2 + 2(Xi2)(Xi)2, !"#UIthe square of the uncertainty for the quantity E, with E = [N(XiYi)-(Xi)( Yi)], is equal to N multiplied by the squared uncertainty for XiYi added to the squared product of Xi with the uncertainty Yi plus the squared product of Yi with the uncert@ABCHINOJAE2 = N[(YiXi)2 + (XiYi)2] + (Xi2)(Yi)2 + (Yi2)(Xi)2  %&'+-./567;=>?Kthe square of the uncertainty for value F, with F = (Yi)(Xi2 )-(Xi)( XiYi), is equal to the sum of the squared product of Xi2 with the uncertainty for Yi added to the squared product of Yi with the uncertainty of Xi2 plus the sum of the squared p!78<=>DEJKLM  '()*MNOPkL+F2 = (Xi2)2(Yi)2 + (Yi)2[2(XiXi)2] +     "$%&'(M!1(Xi)2[(YiXi)2 + (XiYi)2] + (XiYi)2[(Xi)2]   #$%&'(,./0NOConsequently, we can determine the uncertainties for the slope M. The square of the uncertainty for the slope is related to the above uncertainties for quantities E and D. Since M = E/D we havePM2 = E2/D2 + D2(E2/D4).    PQRLikewise, we can obtain the uncertainty for the Y-intercept B from the uncertainties for quantities F and D. Since B = F/D we haveRSB2 = F2/D2 + D2(F2/D4).    STUHere is a specific example: Suppose a car is moving with constant velocity down a road and its position is accurately measured at five different times. A graph of the measured position vs. measured time explicitly shows the motion of the car. In such aUV MeasurementV~ W?W Time (Xi)WXiW Position (Yi)WYi~ X@X1.5 secX0.1 secX12.0 cmX0.1 cm~ Y@Y3.0 secY0.1 secY14.4 cmY0.1 cm~ Z@Z4.5 secZ0.1 secZ16.7 cmZ0.1 cm~ [@[6.0 sec[0.1 sec[19.2 cm[0.1 cm\\7.5 sec\0.1 sec\21.5 cm\0.1 cm]^_DlS?c?c #Y *p]]]]Y` ;a ;b ;c ;d ;e ;f wg h i j k l ;m n o ;p ;q ;r ;t u ;v ;w ;x ;y ;z ;{ ;| ;} ;~ ; ` a b c'dUsing these measured values and the formulae given above we can obtain values (shown with extra digits  to avoid subsequent round-off errors): d ef1fXi = 22.5 sec fg-g Yi = 83.8 cmqg!and (Xi)2 = (Xi)2 = 0.05 sec2     hEhXi2 = 123.75 sec2 hi?iYi2 = 1461.14 cm2i%and (Xi2)2 = 2(XiXi)2 = 2.48 sec4    $jCjXiYi = 412.8 cmsecj$and (Yi2)2 = 2(YiYi)2 = 29.2 cm4    #kkD = 112.5 sec2 _kand (XiYi)2 = 15.9 cm2sec2   llE = 178.5 cmseclm"mF = 1082.25 cmsec2Mmand E2 = 159 cm2sec2 nQnand F2 = 34689 cm2sec4oFrom these values we may determine the best estimates for the car s average velocity and its original position. The optimally fitted slope and Y-intercept (shown with five digits  the uncertainties will determine the actual number of significant figures o#pM = E/D = 1.5867 cm/sec. pqB = F/D = 9.6200 cmqQrHAlso we will obtain the squared uncertainties (shown with three digits):rt2B2 = F2/D2 + D2(F2/D4) = 2.92 cm2 = (1.71 cm)2   $%1t>M2 = E2/D2 + D2(E2/D4) = 0.0175 cm2/sec2 = (0.132 cm/sec)2   &'+,=uConsequently, by taking the square roots of these values we can express our final best fitting results with uncertainties (rounding-off appropriately):u]v*The car s velocity was 1.59 0.13 cm/sec.vcw-The car s original position was 9.6 1.7 cm.w x y.y z { | } ~@D9I]1c1*_kq ;40 (  v2  < @{=h]` ~v2  < @=]`h ~v2  < @=]`X~v2  < @=]`~v2  < @{43h4]`~v2  < @{53h5]`~v2  < @{6/h6]`~v2  < @{7/h7]`(~v2   < @{83h8] `~v2   < @{9/h9] `h>@r   7 ՜.+,0$ PXp x Fort Worth, TX Linear RegressionSheet1Sheet2Example ChartChart1  WorksheetsCharts_1092816018!Fn<n<Ole PRINT  \%CompObjb-Q      ''   Arialw@ bwbw0- Arialw@ bwbw0--- Arialw@ bwbw0- Arialw@ bwbw0-----"System 0-'-  -'-  --  $R R R--- -'---  N t t   ii   ]]   RR -- -'---  -- -'---  R R{ {RMMR  RRRRjjR<<R  R  -- -'---   - R R  R-- -'---   - R -ttii]]RR {{MM  jj<<    -- -'---  !!-- -'---  =-- -'---  =--x{ii-x{ xci0nnnx{xci0x{K{KfKiiT}}r}x{{fiFiFTFr-{ii%%5-{i*%%nn;n55 K{i*%%;55 K{{fiiT%5r{.{.f.i@i@T@%QQQ5bbrbx{- -  cf-- -'-- -   =i /~T-- -'-- -   = -- -'-- -   = r-- -'-- -   ={-  e-{f{{f{-- -'-- -  =i- ,S-iTi)~i)Ti~-- -'-- -  =%- =-%%:%:%-- -'-- -  =5- Mq-5 r5J5Jr5 -- -'-- -  =-- -'-- -  =-- -'-- -  ----- -'-- - o>h 2 Ss Motion of CarA////8+--- -'-- -  -- -'-- -  --- -'-- -    2 I-1 2 P-0.5  2 \01 2 ,0.55  2 E\11 2 ,1.55  2 \21 2 9,2.55  2 \31 2 ,3.55  2 .\41 -- -'-- -  -- -'-- -    2 -105 2 k01 2 -10 2 20 2 30 2 40 2 w50 2 J60 2 70 2 80 2  90 -- -'-- -  ----- -'-- - JW> 2 iGTime (s)(:$$--- -'-- -  ---- -'-- - $  Arialw@ bwbw0- 2 Coordinates (km)- --- -'-- -  -- -    -- -'---   -- -'---   - - -  2   X Coordinate+/$$$$$$-- -'-- -  -- -'-- -  -```- - wK- `L`t`t`L2 9  Y Coordinate*/$$$$$$-- -'---   -- -'---   -- -'---   - - '   '  ' !FMicrosoft Excel ChartBiff8Excel.Chart.89qOh+'0HPp Information ServicesInformation ServicesMicrosoft Excel@G;P@1پ?@DUObjInfoWorkbook6SummaryInformation(DocumentSummaryInformation8p @\pInformation Services Ba= D= <X@"1Arial1Arial1Arial1Arial1.Times New Roman1.Times New Roman1.Times New Roman1Arial1Arial1Arial1.Times New Roman1.Times New Roman1.Times New Roman1Arial16Arial16Arial16Arial1Arial1Arial1Arial1Arial1Arial1 Arial1Arial1Arial1Arial1Arial1dArial1dArial1Arial1Arial1dArial1Arial1Arial1KArial1KArial1Arial1Arial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)"Yes";"Yes";"No""True";"True";"False""On";"On";"Off" 000000.0 0.000                + ) , *   8@ @  8 @    8@  8 ! "     ! ! ! ! !    *8     8 x  < "< "|   ( # #  " x <  < "< "|    8@@ 8@ 8  @ 8@ 8 8 8 @ 8 8 " ! "   !X ! `W Example ChartN`Linear RegressiondCar Motion I (2) Car Motion ISheet1Sheet2`ih0   0e0e     Ad @  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S"@  Measurement Time (Xi) Xi Position (Yi) Yi1.5 sec0.1 sec12.0 cm0.1 cm3.0 sec14.4 cm4.5 sec16.7 cm6.0 sec19.2 cm7.5 sec21.5 cm Xi = 22.5 sec ! and (Xi)2 = (Xi)2 = 0.05 sec2     Yi = 83.8 cm Xi2 = 123.75 sec2 % and (Xi2)2 = 2(XiXi)2 = 2.48 sec4   $ Yi2 = 1461.14 cm2$ and (Yi2)2 = 2(YiYi)2 = 29.2 cm4   # XiYi = 412.8 cmsec and (XiYi)2 = 15.9 cm2sec2   D = 112.5 sec2 E = 178.5 cmsec and E2 = 159 cm2sec2F = 1082.25 cmsec2 and F2 = 34689 cm2sec4seccmTime Time ErrorPositionPosition ErrorD =F =cm secN = sec cm cm secSignificant Figures The most significant digit is the leftmost nonzero digit. Zeros at the left are never significant. Ex.: the value 0.0023 has two SF (while it has 5 digits)  it could also be written as 2.3e-3, for example (still with 2 SF). If no decimal point is explicitly given, the rightmost nonzero digit is the least significant digit. Ex.: the value 300 has one SF, while 3.00e2 has three SF and the value 300.0 has four SF. L cIf a decimal point is explicitly given, the rightmost digit is the least significant digit, regardless of whether it is zero or nonzero. See the above example.mThe number of SF is determined by counting significant digits from most significant to the least significant.Percentage ErrorPercentage Difference4Mean Value, Standard Deviation, and Standard Error 2The Sample Standard Deviation is obtained by taking the square root of the mean of the squared deviations (the rms-deviation). This is calculated in the following steps:? @K LW X First, for each measured value a squared deviation is calculated using the expression (Ai  )2. The set of these for our example is {0.002304, 0.000784, 0.020164, 0.006084, and 0.000144 grams2}. XY^_Second, a sum of the squared deviations is calculated and divided by N-1. In our example the sum is 0.02948 g2 and this is divided by 5-1 = 4, to obtain an average-squared-deviation of 0.00737 g2.noThird, the sample standard deviation SD is found from the square root of the previous result. For our example, the sample standard deviation is equal to 0.08585 grams.Propagation of ErrorValues calculated from several measured quantities will inherit their uncertainties. Final results should reflect the appropriate precision. Intermediate calculations should be done with extra digits (to avoid the accumulation of round-off errors).2Obtaining a Linear Fit to Data (Linear Regression)Often measurements are taken by changing one variable X, measuring its new value, and then subsequently measuring a second variable Y to determine its dependence on the first variable. In many cases, Y will depend on X with a linear relationship given by Y = MX + B,. the sum of all Xi values, denoted below as Xi-. the sum of all Yi values, denoted below as Yi-2 the sum of all (Xi)2 values, denoted below as Xi2012 the sum of all (Yi)2 values, denoted below as Yi2015 the sum of all (XiYi) values, denoted below as XiYi2343 the quantity [N(Xi2 )-(Xi)2], denoted below as D.z the quantity [N(XiYi)-(Xi)(Yi)], denoted below as E. The ratio E/D is the value of the optimally fitting line slope M.  the quantity [(Yi )(Xi2 )-(Xi)( XiYi)], denoted below as F. The ratio F/D is the value of the optimally fitting line intercept B. %&'( The uncertainties for the slope and intercept can be obtained by evaluating the propagation of errors from the measured values (Xi and Yi ) to the calculated slope and intercept (see the section on error propagation). Using this approach:m the square of the uncertainty for Xi is obtained from the sum of the squared uncertainties for each Xi value$%fg (Xi)2 = (Xi)2, h the square of the uncertainty for Yi is equal to the sum of the squared uncertainties for each Yi value$%ab (Yi)2 = (Yi)2,  the square of the uncertainty for Xi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv (Xi2)2 = 2(XiXi)2, the square of the uncertainty for Yi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv (Yi2)2 = 2(YiYi)2, the square of the uncertainty for the quantity XiYi is equal to the sum of the squared products of each Yi with the corresponding uncertainty Xi plus the sum of the squared products of each Xi with the corresponding uncertainty Yi1234jk" (XiYi)2 = (YiXi)2 + (XiYi)2,  ! the square of the uncertainty for the quantity D (defined as [N(Xi2 )-(Xi)2]) is equal to N multiplied by the uncertainty for Xi2 (equal to 2N(XiXi)2) added to the twice the squared prod< uct of Xi with its uncertainty Xi BCDJKLM$ D2 = 2N(XiXi)2 + 2(Xi2)(Xi)2, !"#A E2 = N[(YiXi)2 + (XiYi)2] + (Xi2)(Yi)2 + (Yi2)(Xi)2  %&'+-./567;=>?+ F2 = (Xi2)2(Yi)2 + (Yi)2[2(XiXi)2] +     "$%&'(1 (Xi)2[(YiXi)2 + (XiYi)2] + (XiYi)2[(Xi)2]   #$%&'(,./0Consequently, we can determine the uncertainties for the slope M. The square of the uncertainty for the slope is related to the above uncertainties for quantities E and D. Since M = E/D we have M2 = E2/D2 + D2(E2/D4).    Likewise, we can obtain the uncertainty for the Y-intercept B from the uncertainties for quantities F and D. Since B = F/D we have B2 = F2/D2 + D2(F2/D4).    Using these measured values and the formulae given above we can obtain values (shown with extra digits  to avoid subsequent round-off errors):M = E/D = 1.5867 cm/sec. B = F/D = 9.6200 cmHAlso we will obtain the squared uncertainties (shown with three digits):> M2 = E2/D2 + D2(E2/D4) = 0.0175 cm2/sec2 = (0.132 cm/sec)2   &'+,=2 B2 = F2/D2 + D2(F2/D4) = 2.92 cm2 = (1.71 cm)2   $%1Consequently, by taking the square roots of these values we can express our final best fitting results with uncertainties (rounding-off appropriately):*The car s velocity was 1.59 0.13 cm/sec.-The car s original position was 9.6 1.7 cm..MThe rules for determine the number of significant figures (SF) in a quantity:E =_ FOR A DIFFERENT SET OF MEASUREMENTS, BE SURE TO CHANGE THE UNITS LISTED THROUGHOUT THIS PAGE.u INSERT MORE ROWS ABOVE THIS ROW FOR ADDITIONAL DATA. THEN ADJUST THE N VALUE & SUMS TO OBTAIN CORRECT CALCULATION.Time2 Time Error2 Position Error2Xi XiYi YiXi2 (Xi)2Yi2 (Yi)2XiYi (XiXi)2 (YiYi)2 (XiYi)2 (YiXi)2sec2cm2 (cm sec)2 Xi =  (Xi)2 = (Xi)2=  sec2 Yi = (Yi)2 = (Yi)2=  cm2 Xi2 =  (Xi2)2 = sec4 Yi2 = (Yi2)2 = cm4 XiYi = (XiYi)2 =  cm2sec2 (XiXi)2=   (YiYi)2=   (XiYi)2=   (YiXi)2=   D2 = E2 = cm sec2 F2 = cm2sec4 cm/sec cm Optimal Slope (M) =Optimal Intercept (B) ='MechLab 1: Measurement Accuracy & ErrorPurpose^ In this activity, torque and rotational equilibrium are explored using an equal arm balance.Equipment Providedg Required: (1) one equal-arm balance, (2) a set of masses and 2 mass hangers, (3) thread, (4) a 5 g 7mass hanger and mass set, (5) a large sheet protractor.TheoryTime (seconds)X coordinate (km)Y coordinate (km)NOTE: The calculated values shown here have extra digits (to avoid subsequent round-off errors). The only numbers shown on this page with meaningful significant figures are the measured values and their estimated uncertainties. To control the way the va A measurement is repeated N times, resulting in a set (or list) of values Ai (where the index  i identifies each specific measurement -  i takes values between 1 and N). If the variations of the values in this set are entirely due to random errors, we KLY ce ~  The Mean Value  is the average of the measured values. The mean is calculated from the sum all Ai (from i = 1 to i = N) and then division of the sum by N. Our above example of five mass measurements has a mean value of 9.102 g. This is our best estimabcThe Standard Error represents an estimate of our uncertainty for the measured mean value (as determined by the number of measurements and the variations in our set of values). The standard error S is calculated by dividing the sample standard deviation  The Result of a Set of Measurements: The best estimate of the measured value (and its estimated uncertainty) is given by  S. For our example, the result of our measurements is 9.10 0.04 grams (note that mean value is rounded off to the same digit as$where M and B are constants. The constant M is the slope of the line and B is the Y-intercept (the value of Y when X = 0). To verify this relationship, a set of N measurements of the variables will produce a set of X and Y pairs which we may denote as ( the square of the uncertainty for the quantity E, with E = [N(XiYi)-(Xi)( Yi)], is equal to N multiplied by the squared uncertainty for XiYi added to the squared product of Xi with the uncertainty Yi plus the squared product of Yi with the uncert@ABCHINO !the square of the uncertainty for value F, with F = (Yi)(Xi2 )-(Xi)( XiYi), is equal to the sum of the squared product of Xi2 with the uncertainty for Yi added to the squared product of Yi with the uncertainty of Xi2 plus the sum of the squared p78<=>DEJKLM<O  '()*MNOPkHere is a specific example: Suppose a car is moving with constant velocity down a road and its position is accurately measured at five different times. A graph of the measured position vs. measured time explicitly shows the motion of the car. 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M4% aM3O/&Q Time (s)'4% @MZ3O^&Q $Coordinates (km)'4523  NM43d" ) 3O% % Mp73O&Q443_ M NM  MM<444% iUvM3OO&Q  Motion of Car'44e7@7@{GK@{GK@p= cR@p= cR@e+@;On? ףp= @{Gz?zG @)\(J + @7A`¿e> 7A` @  dMbP?_*+%"??U;;;;   ~ +@;On?7@r@@@8t@~ @J + @7A`¿*P*2>@7 @  dMbP?_*+%"??U} }   h ,    , 7 ,7 7 7 , ; ; ; ; ;@ ;@ ;@ ;@ ;@ ;@ ;@ ;@ ;@ ;@ ; ; ; ; ; ; EEEEEEEE FFFFFFFF BBBBBBBBBBBBBBBBCCCCCCCC FFFFFFFFAAAAAAAA7 BBBBBBBBB B BBBBBBB7 CCCCCCCC7 F FFFFFFF  ,  c D-DDDDDDDDDDDDDDDDDDDDDDD" D.DDDDDDDDDDDDDDD" D/DDDDDDDDDDDDDDD" D0DDDDDDDDDDDDDDD"""""""" 1 2BX(((((((((0000(( ;! ;" w# w$ ;% & w' ;( ;) ;* ;+ ;, ;- ;. ;/ ;0 ;1 ;2 ;3 w4 w5 w6 7 8 w9 : w; < w= w> ? w 3! " # $4 %5 &6 '7 ( )* +8, -9. /: 0 1;1 2<2 334 4=5 5>6 6?7 7@8 8A9 9B :C: ;D; <E< =F= >G> ?H?Dl    @ A B C D E wF G H I wJ K L M N ;O ;P Q ;R ;S T ;U ;V 'W vX ;Y ;Z ;[ ;\ ;] ;^ ;_ ; @ I AJ B K CL D M EN F O GP H Q I JR K LS M!TN OU PVPQ RWR SXST UU VV~ W? W W W W~ X@ X X X X~ Y@ Y Y Y Y~ Z@ Z Z Z Z~ [@ [ [ [ [\ \ \ \ \]^_Dl    FFFFFB` ;a ;b ;c ;d ;e ;f wg h i j k l ;m n o ;p ;q ;r ;t u ;v ;w ;x ;y ;z ;{ ;| ;} ;~ ; ` a b c dY d ef ffg g gh hhi i ij j jk k kl llm m m n n o o pZp q[q r\r t^ t] u_u v`v waw x yby z { | } ~@8D"&"&&&"& ;4P (  v2  < @{=h]`@~v2  < @=]`~v2  < @=]`~v2  < @=]`~v2  < @{43h4]`$0~v2  < @{53h5]`t0~v2  < @{6/h6]`d1~v2  < @{7/h7]`1~v2   < @{83h8] `2~v2   < @{9/h9] `2>@r   7 ^-:t7< ;! ;" w# w$ ;% & w' ;( ;) ;* ;+ ;, ;- ;. ;/ ;0 ;1 ;2 ;3 w4 w5 w6 7 8 w9 : w; < w= w> ? wC 4Mean Value, Standard Deviation, and Standard Error 2!)"A measurement is repeated N times, resulting in a set (or list) of values Ai (where the index  i identifies each specific measurement -  i takes values between 1 and N). If the variations of the values in this set are entirely due to random errors, we KLY ce ~ #The Mean Value  is the average of the measured values. The mean is calculated from the sum all Ai (from i = 1 to i = N) and then division of the sum by N. Our above example of five mass measurements has a mean value of 9.102 g. This is our best estimabc$The Sample Standard Deviation is obtained by taking the square root of the mean of the squared deviations (the rms-deviation). This is calculated in the following steps:? @K LW X%First, for each measured value a squared deviation is calculated using the expression (Ai  )2. The set of these for our example is {0.002304, 0.000784, 0.020164, 0.006084, and 0.000144 grams2}. XY^_&Second, a sum of the squared deviations is calculated and divided by N-1. In our example the sum is 0.02948 g2 and this is divided by 5-1 = 4, to obtain an average-squared-deviation of 0.00737 g2.no'Third, the sample standard deviation SD is found from the square root of the previous result. For our example, the sample standard deviation is equal to 0.08585 grams.(The Standard Error represents an estimate of our uncertainty for the measured mean value (as determined by the number of measurements and the variations in our set of values). The standard error S is calculated by dividing the sample standard deviation  )The Result of a Set of Measurements: The best estimate of the measured value (and its estimated uncertainty) is given by  S. For our example, the result of our measurements is 9.10 0.04 grams (note that mean value is rounded off to the same digit as$*+Propagation of Error,-Values calculated from several measured quantities will inherit their uncertainties. Final results should reflect the appropriate precision. Intermediate calculations should be done with extra digits (to avoid the accumulation of round-off errors)..;/2Obtaining a Linear Fit to Data (Linear Regression) 01Often measurements are taken by changing one variable X, measuring its new value, and then subsequently measuring a second variable Y to determine its dependence on the first variable. In many cases, Y will depend on X with a linear relationship given by12 Y = MX + B,23where M and B are constants. The constant M is the slope of the line and B is the Y-intercept (the value of Y when X = 0). To verify this relationship, a set of N measurements of the variables will produce a set of X and Y pairs which we may denote as (34s4.the sum of all Xi values, denoted below as Xi-5s5.the sum of all Yi values, denoted below as Yi-662the sum of all (Xi)2 values, denoted below as Xi201772the sum of all (Yi)2 values, denoted below as Yi201885the sum of all (XiYi) values, denoted below as XiYi234993the quantity [N(Xi2 )-(Xi)2], denoted below as D.:zthe quantity [N(XiYi)-(Xi)(Yi)], denoted below as E. The ratio E/D is the value of the optimally fitting line slope M. :C;the quantity [(Yi )(Xi2 )-(Xi)( XiYi)], denoted below as F. The ratio F/D is the value of the optimally fitting line intercept B.  %&'(;<The uncertainties for the slope and intercept can be obtained by evaluating the propagation of errors from the measured values (Xi and Yi ) to the calculated slope and intercept (see the section on error propagation). Using this approach:<=mthe square of the uncertainty for Xi is obtained from the sum of the squared uncertainties for each Xi value$%fg=O>(Xi)2 = (Xi)2, >?hthe square of the uncertainty for Yi is equal to the sum of the squared uncertainties for each Yi value$%ab?DlG - !  ?"(-Q ]@ A B C D E wF G H I wJ K L M N ;O ;P Q ;R ;S T ;U ;V 'W vX ;Y ;Z ;[ ;\ ;] ;^ ;_ ;O@ (Yi)2 = (Yi)2, ;Athe square of the uncertainty for Xi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv_B (Xi2)2 = 2(XiXi)2, ;Cthe square of the uncertainty for Yi2 is equal to the sum of the squared products of each Xi with its uncertainty Xi, multiplied by two,$%&\]tv_D (Yi2)2 = 2(YiYi)2,  Ethe square of the uncertainty for the quantity XiYi is equal to the sum of the squared products of each Yi with the corresponding uncertainty Xi plus the sum of the squared products of each Xi with the corresponding uncertainty Yi 1234jkF "(XiYi)2 = (YiXi)2 + (XiYi)2,  !Gthe square of the uncertainty for the quantity D (defined as [N(Xi2 )-(Xi)2]) is equal to N multiplied by the uncertainty for Xi2 (equal to 2N(XiXi)2) added to the twice the squared product of Xi with its uncertainty Xi BCDJKLMH $D2 = 2N(XiXi)2 + 2(Xi2)(Xi)2, !"#UIthe square of the uncertainty for the quantity E, with E = [N(XiYi)-(Xi)( Yi)], is equal to N multiplied by the squared uncertainty for XiYi added to the squared product of Xi with the uncertainty Yi plus the squared product of Yi with the uncert@ABCHINOJAE2 = N[(YiXi)2 + (XiYi)2] + (Xi2)(Yi)2 + (Yi2)(Xi)2  %&'+-./567;=>?Kthe square of the uncertainty for value F, with F = (Yi)(Xi2 )-(Xi)( XiYi), is equal to the sum of the squared product of Xi2 with the uncertainty for Yi added to the squared product of Yi with the uncertainty of Xi2 plus the sum of the squared p!78<=>DEJKLM  '()*MNOPkL+F2 = (Xi2)2(Yi)2 + (Yi)2[2(XiXi)2] +     "$%&'(M!1(Xi)2[(YiXi)2 + (XiYi)2] + (XiYi)2[(Xi)2]   #$%&'(,./0NOConsequently, we can determine the uncertainties for the slope M. The square of the uncertainty for the slope is related to the above uncertainties for quantities E and D. Since M = E/D we havePM2 = E2/D2 + D2(E2/D4).    PQRLikewise, we can obtain the uncertainty for the Y-intercept B from the uncertainties for quantities F and D. Since B = F/D we haveRSB2 = F2/D2 + D2(F2/D4).    STUHere is a specific example: Suppose a car is moving with constant velocity down a road and its position is accurately measured at five different times. A graph of the measured position vs. measured time explicitly shows the motion of the car. In such aUV MeasurementV~ W?W Time (Xi)WXiW Position (Yi)WYi~ X@X1.5 secX0.1 secX12.0 cmX0.1 cm~ Y@Y3.0 secY0.1 secY14.4 cmY0.1 cm~ Z@Z4.5 secZ0.1 secZ16.7 cmZ0.1 cm~ [@[6.0 sec[0.1 sec[19.2 cm[0.1 cm\\7.5 sec\0.1 sec\21.5 cm\0.1 cm]^_DlS?c?c #Y *p]]]]Y` ;a ;b ;c ;d ;e ;f wg h i j k l ;m n o ;p ;q ;r ;t u ;v ;w ;x ;y ;z ;{ ;| ;} ;~ ; ` a b c'dUsing these measured values and the formulae given above we can obtain values (shown with extra digits  to avoid subsequent round-off errors): d ef1fXi = 22.5 sec fg-g Yi = 83.8 cmqg!and (Xi)2 = (Xi)2 = 0.05 sec2     hEhXi2 = 123.75 sec2 hi?iYi2 = 1461.14 cm2i%and (Xi2)2 = 2(XiXi)2 = 2.48 sec4    $jCjXiYi = 412.8 cmsecj$and (Yi2)2 = 2(YiYi)2 = 29.2 cm4    #kkD = 112.5 sec2 _kand (XiYi)2 = 15.9 cm2sec2   llE = 178.5 cmseclm"mF = 1082.25 cmsec2Mmand E2 = 159 cm2sec2 nQnand F2 = 34689 cm2sec4oFrom these values we may determine the best estimates for the car s average velocity and its original position. The optimally fitted slope and Y-intercept (shown with five digits  the uncertainties will determine the actual number of significant figures o#pM = E/D = 1.5867 cm/sec. pqB = F/D = 9.6200 cmqQrHAlso we will obtain the squared uncertainties (shown with three digits):rt2B2 = F2/D2 + D2(F2/D4) = 2.92 cm2 = (1.71 cm)2   $%1t>M2 = E2/D2 + D2(E2/D4) = 0.0175 cm2/sec2 = (0.132 cm/sec)2   &'+,=uConsequently, by taking the square roots of these values we can express our final best fitting results with uncertainties (rounding-off appropriately):u]v*The car s velocity was 1.59 0.13 cm/sec.vcw-The car s original position was 9.6 1.7 cm.w x y.y z { | } ~@D9I]1c1*_kq ;40 (  v2  < @{=h]` ~v2  < @=]`h ~v2  < @=]`X~v2  < @=]`~v2  < @{43h4]`~v2  < @{53h5]`~v2  < @{6/h6]`~v2  < @{7/h7]`(~v2   < @{83h8] `~v2   < @{9/h9] `h>@r   7 ՜.+,0@ PXp x Fort Worth, TX Linear RegressionSheet1Sheet2Example ChartCar Motion I (2) Car Motion I  WorksheetsChartsOh+'0   8x>=hL!WX {1^ӛ?]mE8<Xgb^$˷enVdW'PwP?tw=3|fǷ8!*4tYFep6om _iڿ(H: .\>ygB\f讁{CZLLg/jK6B%"s{}\K>|bۜi#l\At1ccT~1v|S.1y|S{E[L9i9ܷ1ǹ:?'kWcq7 u| nǺ5'hOgLuH=G __ fk:wԢ~oߔsI5'g}StOfȾZ5a5RTlԟs}y^tφw7kjj[϶~!MkdD%w鿣DTN"*'3Dk;JwqVg("{;,Nś. gzV_ޜhkhgc*|Qo9՛-\#j$$If!vh55#v#v:V lJp6,554j$$If!vh55#v#v:V lJp6,554j$$If!vh55#v#v:V lJp6,554j$$If!vh55#v#v:V lJp6,554j$$If!vh55#v#v:V lJp6,554j$$If!vh55#v#v:V lJp6,554J$$If!vh5#v:V l654$$If!vh555c55G#v#v#vc#v#vG:V l 6! 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