Chapter 6Statistical Analysis of Experimental DataMeasures of Central TendencyStandard Deviation of Population“Normal” or Gaussian Distribution“Normal” Distribution Information:Standard Deviation of SampleExercise #1aExercise #1bExercise #1cRejecting Questionable Data – p.142Table 6.8 Values of Thompson’s tExercise #2aExercise #2bExercise #2cExercise #2bCorrelation of Experimental DataUse Excel to plot X-Y dataR2 – “Goodness of Fit”Excel - TrendlineWarning!Plot X-Y data with uncertaintiesAdd Error Bars for UncertaintyPlot with Error BarsWhat do you need to know how to do in Excel?Chapter 6Statistical Analysis of Experimental DataMeasures of Central TendencyGiven a set of data: { x1, x2, x3, … xN}►Mean (average):►Median: Put data in ascending order, pick the ______ value. (If even-numbered, pick the _______ of the two middle values.)=µStandard Deviation of Population►Deviation: How far a point is from __►Standard deviation of a population, σ, given by Eq. 6.5:►Population: _______________=σ=id“Normal” or Gaussian Distribution()=xf“Normal” Distribution Information:►____ of population values are within ±__of mean value, µ►____ of population values are within _____ of mean value, µ►_____ of population values are within ____ of mean value, µNote – the “normal” distribution is NOT the only oneused in engineering (binomial, Poisson, Weibull, etc.)Standard Deviation of Sample►Standard deviation of n samples, S, given by Eq. 6.6►sample is a ______ of all possible values=id=SSampleSamplemeanmeanExercise #1aWhat is the expected valueand the 95% confidence uncertainty interval for this data?xTensileSampleStrength, MN/m212.4822.7632.9642.7252.6262.65Exercise #1bTensileSampleStrength, MN/m^2 di (di)^212.4822.7632.9642.7252.6262.65Avg: Sum:Exercise #1c=XS∈x(95% confidence interval)∈x≤≤xRejecting Questionable Data – p.142Modified Thomson τ technique1. Determine mean and std. dev. SX2. Find ______ deviation from mean,3. If then _____ data point4. If data point ________________ mean and standard deviation5. ______ process with new mean and standard deviationx()xxii−=δixrejectrejectTable 6.8 Values of Thompson’s τProblem #20.102 sec0.106 sec0.100 sec0.101 sec0.105 sec0.119 sec0.103 sec0.104 secExercise #2aFind the expected value and the standard deviation for these experimental time constant data=x=XSShould any of these experimental data values be eliminated?Exercise #2bLargest deviation is=iδFrom Table 6.8 we find for n = __, τ = _____=⋅XSτProblem #20.102 sec0.106 sec0.100 sec0.101 sec0.105 sec0.103 sec0.104 secExercise #2cFind the expected value and the standard deviation for these experimental time constant data=xShould any of these experimental data values be eliminated?=XSExercise #2bLargest deviation is=iδFrom Table 6.8 we find for n = __, τ = _____=⋅XSτCorrelation of Experimental Data►Least-squares fit to a straight line is the most common Fits to other curves are often more appropriate, i.e., exponential►Some data must be transformed before fitting to straight line (section 6.6.3)►Excel’s Trendlinefunction can be usedUse Excel to plot X-Y dataTime Voltage1351.5 232173124957668410 3R2– “Goodness of Fit”()()()()∑∑∑===−−−−=n1i2in1i2in1iiixyyy*xxyyxxr1r1xy≤≤−Excel - Trendline►in Excel - select the data series click Chart\Add Trendline►The numerical values for the fitted line can be displayed by selecting Display Equation on Chart under Chart\Add Trendline\Options►The R2(“goodness of fit) value can also be displayedWarning!►Note that Trendline WILL fit a line of the desired type to the data - even if it is not appropriate!►The data we plotted are NOT well represented by a straight lineRR22should be close to 1!should be close to 1!Plot X-Y data with uncertaintiesTime Voltage Uncert.1351.51.5 23 1.022170.783120.584 9 0.465 7 0.386 6 0.348 4 0.2610 3 0.22Add Error Bars for UncertaintySelect CustomPlot with Error Bars051015202530354001234567891011Time, secVoltageWhat do you need to know how to do in Excel?► Remove the default gray background from Excel graphs► Add error bars to plots of experimental data► Change axis label locations (should almost always be along the bottom of the graph)► Re-label “x” and “y” values after a trendline curve fit► Reduce number of significant digits displayed► Remove labels from legend box (without removing the data from the graph!)► Rescale axis to fit data (no long gaps where no data occurs)► Create logarithmic axes with gridlines► Re-size the fonts in a graph► Cut and paste (as a picture) from Excel to