InfiniteStatistics.docx 9/24/2008 1:34 PM Page 1 Probability and Statistics Measurement Theory Probability theory provides a rational basis for assessing the precision of a measurement. Using measurement theory, we can make statements such as: "The probability is less than 5% that X lies outside of the interval .45 < X < .55." If we measure any quantity repeatedly, we will never get exactly the same value twice. Why not? Resolution Repeatability Noise Interference Spatial variation Temporal variation We would like to quantify average of the data set variation of the data set interval about the average in which the “true value” is expected to beInfiniteStatistics.docx 9/24/2008 1:34 PM Page 2 Notation True mean valuex Data set mean valuex From a statistical analysis of the data set and an analysis of sources of error that influence these data, we can estimate the true mean value as ( %)xx x u P where ux is the confidence interval and P% is the probability level. A convenient way of visualizing repeated measurements is by means of a histogram.InfiniteStatistics.docx 9/24/2008 1:34 PM Page 3InfiniteStatistics.docx 9/24/2008 1:34 PM Page 4 Measurement Theory Probability Density Function Here is an example of repeated measurements of the same quantity: TABLE 4.1 Sample of Variable x i xi i xi 1 0.98 11 1.02 2 1.07 12 1.26 3 0.86 13 1.08 4 1.16 14 1.02 5 0.96 15 0.94 6 0.68 16 1.11 7 1.34 17 0.99 8 1.04 18 0.78 9 1.21 19 1.06 10 0.86 20 0.96InfiniteStatistics.docx 9/24/2008 1:34 PM Page 5 From Lab 2 – Static Calibration Thermocouple Data Point Dial Thermometer Temp C DMM microVolts DMM Volts 1 0 -859.5 -0.00086 2 28 289.5 0.00029 3 18 -118 -0.00012 4 37 630 0.00063 5 7 -598 -0.0006 6 18 -149 -0.00015 7 0 -830 -0.00083 8 26 223.5 0.000224 9 8 -556 -0.00056 10 32 511 0.000511 11 18 -109 -0.00011 12 25 200 0.0002 13 0 -785 -0.00079 14 32 448 0.000448 15 9 -525 -0.00053 00.511.522.533.50369121518212427303336394245Number of Data PointsTemperature CStatic Calibration Data DistributionInfiniteStatistics.docx 9/24/2008 1:34 PM Page 6 As the number of measurements increases, the width of the bars on the histogram becomes narrower, and the histogram approaches a smooth curve: In the absence of systematic, or bias, error, repeated measurements of the same quantity can be regarded as randomly distributed.InfiniteStatistics.docx 9/24/2008 1:34 PM Page 7 The number of intervals for data set with more that 40 points can estimated with the formula: 0.401.87( 1) 1KN where N is the number of data points Probability Density Function limorNote errors in both editions of textjjnP(x) = (4.3)N , x 0N xn = P(x) xN The total area is normalized to 1.InfiniteStatistics.docx 9/24/2008 1:34 PM Page 8 The shaded area under the curve between 0 and z1 is the probability that a single measurement will lie between 0 and z1 Random Variables Mean and Standard Deviation Regardless of the shape of P(x), we can define a mean value: 0limTT1x = x(t)dt (4.4a)T and a variance:InfiniteStatistics.docx 9/24/2008 1:34 PM Page 9 2201(4.6 )limTT= x(t)- x dt aT If we are dealing with discrete data the corresponding definitions are limNiN i=11x = (4.5)xN limN22iN i=11 = (4.7)( - x )xN The standard deviation is the square root of the variance. Typical Distributions Random variables are generally described by some simple probability density functionsInfiniteStatistics.docx 9/24/2008 1:34 PM Page 10InfiniteStatistics.docx 9/24/2008 1:34 PM Page 11InfiniteStatistics.docx 9/24/2008 1:34 PM Page 12 The Normal Distribution or Normal Error Function For a normal distribution, if we know the mean and standard deviation, we can estimate the probability that a single measurement will lie within a band around the mean:InfiniteStatistics.docx 9/24/2008 1:34 PM Page
View Full Document
Unlocking...