Class Objectives MISTAKES AND ERRORS MISTAKES AND ERRORS CE 211 SURVEYING ENGINEERING FALL 2011 CLASS 03: THEORY OF ERROS 8/26/2011

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1 8/6/011 Class Objectives CE 11 SURVEYING ENGINEERING FALL 011 CLASS 03: THEORY OF ERROS Ahmed Abdel-Rahim, Ph.D, P.E. Associate Professor, Civil Engineering Define mistakes and errors in measurements and relate them to different sources Compute the most probable value, standard deviation, and 95 % error Understand how errors propagate through multiple measurements Define accuracy and precision MISTAKES AND ERRORS No measurement can be perfect or exact because of the physical limitations of the measuring instruments as well as limits in human perception. The difference between a measured distance or angle and its true value may be due to mistakes and/or errors. These are two distinct terms. MISTAKES AND ERRORS Blunders A blunder is a significant mistake caused by human error. It may also be called a gross error. Systematic and Accidental Errors Systematic Errors Accidental (Random) Errors Analytical Procedures Most Probable Value The 95 Percent Error How Accidental Errors Add Up 1

2 8/6/011 Types of Errors Errors in Observations Systematic Errors Repeated if identified and modeled, can be easily corrected (Natural errors and Instrumental errors) Random Errors Occur randomly remain in the measurements after mistakes and systematic errors are corrected. Magnitude and direction of the error are subject to chance Mistakes versus Observation errors Mistakes can be easily identified and isolated (examples of mistakes???) Our study and analysis will focus of observation errors Sources of Errors Natural errors [Systematic] Instrumental errors [Systematic] Personal errors [Random] Again, Which source of error is harder correct? Errors in Observations Theory of Probability X = Observed Value X = True Value E = Error Facts: No observation is exact Every observation contains an error True value (and thus the error) is never known E X X Most probable value represent the best estimate of the true value as the true value of any observation is never known M = individual measurements n = total number of observations Average (mean) value M M n

3 Frequancy Frequancy 8/6/011 Theory of Probability Residuals or errors in individual measurements : the difference between any observation M and the its most probable value. For a group of observations, residuals can be assumed to be normally distributed with an average value of zero M M Residual Distribution (Length) Residuals Residual Distribution (Width) Residuals More about residuals Characteristics of Normal Distribution It can be assumed that residuals for a group of measurements are normally distributed with the following characteristics: Small residuals (errors) are more probable, they occur more often that large ones Large residuals may be mistakes rather than random errors (again, what s the difference between random errors and mistakes?) Positive and negative errors of the same size happen with equal frequency 3

4 8/6/011 Normal Distribution Normal Distribution Standard deviation Defines the abscissa width for the distribution curve. SD is a measure of precision n 1 Percent Errors Defines the probability of an error of any percentage Engineers typically look at 95% percent Ep C P E E ( ) n(n 1) Error Propagation Error Propagation Case I: Sum of Errors Z a b c... n E z E E E E a b c... n Case II: Error of a serious (group of the same observations E z Z na E E E... E E z n E E 1 n 4

5 8/6/011 Error Propagation Case III: Error in a product Z AB E z A E b B E a ACCURACY AND PRECISION Precision Degree of perfection used in the survey Accuracy Degree of perfection obtained in the results FIGURE -4. It is important to understand the difference between accuracy and precision in surveying measurements. Concepts and Definitions Precision: An indication of the uniformity or reproducibility of a result. Accuracy: the degree of conformity with a standard (the "truth"). Concepts and Definitions The accuracy of an analytical measurement is how close a result comes to the true value. Determining the accuracy of a measurement usually requires calibration of the analytical method with a known standard. Precision is the reproducibility of multiple measurements and is usually described by the standard deviation, standard error, or confidence interval. 5

6 8/6/011 Concepts and Definitions Any measurement has three numbers associated with it: Estimate of the quantity Range of error or uncertainty Level of certainty (confidence level) ACCURACY AND PRECISION Error of Closure and Relative Accuracy Relative Accuracy Standards of Accuracy Choice of Survey Procedure Relative Accuracy Relative accuracy = 1 : D/C D = Distance measures C = Error of closure ACCURACY AND PRECISION Table -1. Selected Federal Standards for Traverse Surveys 6

7 8/6/011 HW Assignment #1 Chapter, problems: 7, 8, 9, 1, 13, 19, 0, 1,, 5, 7, 9, 31, 35 Due at the beginning of class on Monday 08/9/011 7