Part 3: Measurement Errors &Uncertainity Analysis
Part 3: Measurement Errors & Uncertainity Analysis
Lecture 5Probability and Statistics
Lecture 5 Probability and Statistics
LearningObjectives()quantify the statistical characteristics of a data set as itrelates to the population of the measured variable.explain and use probability density functions to describethe behaviour of variables,create meaningful histograms of measured data,quantify a confidence interval about the measured meanvalue at a given probability
Learning Objectives (1) • quantify the statistical characteristics of a data set as it relates to the population of the measured variable, • explain and use probability density functions to describe the behaviour of variables, • create meaningful histograms of measured data, • quantify a confidence interval about the measured mean value at a given probability
LearningObjectives(2)perform regression analysis on a data set and quantifythe confidence intervals for theparameters of the resulting curve fit or response surface.identify outliers in a data set,specify the number of measurements required to achievea desired confidence interval, andexecute a Monte Carlo simulation that predicts thebehaviour expected in a result due tovariations in the variables involved in computing thatresult
Learning Objectives (2) • perform regression analysis on a data set and quantify the confidence intervals for the • parameters of the resulting curve fit or response surface, • identify outliers in a data set, • specify the number of measurements required to achieve a desired confidence interval, and • execute a Monte Carlo simulation that predicts the behaviour expected in a result due to • variations in the variables involved in computing that result
STATISTICALMEASUREMENTTHEORY Sampling refers to obtaining a set of data throughrepeated measurements of a variable under fixedoperating conditions. This variable is known asthe measured variable or, in statistical terms, themeasurand
STATISTICAL MEASUREMENT THEORY • Sampling refers to obtaining a set of data through repeated measurements of a variable under fixed operating conditions. This variable is known as the measured variable or, in statistical terms, the measurand
AssumptionsStudying the effects of random errors and how to quantifythem. Random errors are manifestedthrough data scatter and by the statistical limitations of a finite sampling topredict the behaviour of a population.We assume that any systematic errors in the measurementare negligible (average error in the data set is zero)
Assumptions Studying the effects of random errors and how to quantify them. Random errors are manifested • through data scatter and • by the statistical limitations of a finite sampling to predict the behaviour of a population. We assume that any systematic errors in the measurement are negligible (average error in the data set is zero)
SettingtheproblemEstimate the true value, xo, based on the informationderived from the repeated measurement of variable xIn the absence of systematic error, the true value of x is themean value of all possible values of x
Setting the problem Estimate the true value, x0, based on the information derived from the repeated measurement of variable x. In the absence of systematic error, the true value of x is the mean value of all possible values of x
Estimationoftruevalue From a statistical analysis of the data set and an analysisof sources of error that influence these data, we canestimate x'asx'=x+u(P%)x - most probable estimate of xUx - uncertainty intervalP% - probability level
Estimation of true value • From a statistical analysis of the data set and an analysis of sources of error that influence these data, we can estimate x’ as x x u (P%) x x - most probable estimate of x’ P% ux - uncertainty interval - probability level
ProbabilityDensityFunctions(1)During repeated measurements of a variable, each datapoint may tend to assume one preferred value or liewithin some interval about this value more often thannot, when all the data are compared.This tendency toward one central value about which allthe other values are scattered is known as a centraltendency of a random variable.The central value and those values scattered about it canbe determined from the probability density of themeasured variable
Probability Density Functions (1) • During repeated measurements of a variable, each data point may tend to assume one preferred value or lie within some interval about this value more often than not, when all the data are compared. • This tendency toward one central value about which all the other values are scattered is known as a central tendency of a random variable. • The central value and those values scattered about it can be determined from the probability density of the measured variable
ProbabilityDensityFunctions(2) The frequency with which the measured variableassumes a particular value or interval of values isdescribed by its probability density
Probability Density Functions (2) • The frequency with which the measured variable assumes a particular value or interval of values is described by its probability density