Introduction to Instrumentation Engineering Chapter 1: Measurement Error Analysis By Sintayehu Challa

Goals of this Chapter Differentiate the types of error 1/16/2019 Goals of this Chapter Differentiate the types of error Every measurement involves an error Give an overview of data analysis techniques in instrumentation systems Understand basic mathematical tools required for this purpose Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Overview Measurement error analysis Types of errors and uncertainty Statistical analysis Gaussian and Binomial distributions Method of Least Squares Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Measurement Error Types of errors: Systematic and random errors 1/16/2019 Measurement Error Types of errors: Systematic and random errors Systematic error Cause repeated readings to be in error by the same amount Consistent, or fixed error component May arise due to instrument short coming & environmental effects Related to calibration errors and can be eliminated by correct calibration Or human error such as consistent misreading and arithmetic error such as incorrect rounding off Or by using an inadequate measurement methods Example unjustified extrapolation of experimental data Accuracy is related to such type of errors Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

Measurement Error …. Random errors 1/16/2019 Measurement Error …. Random errors Due to unknown cause and occurs when all systematic errors have been accounted for Caused by random electronic fluctuations in instruments, unpredictable behavior of the instrument, influences of friction, etc… Random fluctuations usually follow certain statistical distribution Treated by statistical methods Characterized by positive and negative errors Such errors are related to precision Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Measurement Error …. Systematic errors analysis can be divided into Worst-cases analysis and RMS error analysis Worst-case analysis: Let Qm be the measured quantity and Qt be true quantity Error: Relative error: E.g., if the measured value is 10.1 when the true value is 10.0, the error is -0.1. If the measured value is 9.9 when the true value is 10.0, the error is +0.1 Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Error & Uncertainty Uncertainty 1/16/2019 Error & Uncertainty Uncertainty Since the true value cannot be known, the error of a measurement is also unknown Thus, the closeness of the value obtained through a measurement to the true value is unknown We are uncertain how well our measured value represents the true value Uncertainty characterizes the dispersion of values ±Ea is the assigned uncertainty of Ea Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Error and Uncertainty … 1/16/2019 Error and Uncertainty … Differentiate between error and uncertainty Error indicate knowledge of the correct value May be either positive or negative! Uncertainty indicate lack of knowledge of the correct value or may be either positive or negative! Is always a positive quantity, like standard deviation Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

Combined Uncertainty – Commonsense Basis 1/16/2019 Combined Uncertainty – Commonsense Basis Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Combined Uncertainty … 1/16/2019 Combined Uncertainty … Given a function The RMS error is given as And Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Uncertainty of Measurements 1/16/2019 Uncertainty of Measurements Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Overview Measurement error analysis Types of errors and uncertainty Statistical analysis Binomial and Gaussian distributions Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Statistical Analysis Allows analytical determination of uncertainty of test result Arithmetic mean (or most probable value) of n readings x1 to xn is given by With a large sample, frequency distribution of the individual xi’s can be used to save time If a particular value of xi occurs in the sample fj times, the mean value can be determined as The sample frequency fj/n is an estimate of the probability Pj that x has the value of xj in the population sample Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Statistical Analysis … Deviation xi-xm is the difference of all readings or observations from the mean reading Is a good indicator of the uncertainty of the instrument Average deviation: Sum of the absolute value of all deviations, i.e., Tends to zero and gives an indication of the precision of the instrument (low value shows that the instrument is highly precise) Standard deviation: deviation from the mean & is given as σ is called the population or biased standard deviation Measure the extent of expected error in any observation Variance: σ2 Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

Statistical Analysis … Using the probability distribution Pj and noting that Pj=fj/n For most distributions (both real and theoretical) met in statistical work, more than 94% of all observations in the population are within the interval xm  2 Generally, it is desirable to have about 20 observations in order to obtain reliable estimate of  For smaller set of data, the expression for  modifies to Called unbiased or sample standard deviation Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Cumulative Frequency Distribution Sometimes the investigator is interested in estimating the proportion of the data whose values exceed some stated level or fall short of the level E.g., for the random number, if the cumulative frequency distribution for drawing digits less than 3 is 0.75 and the cumulative frequency distribution for drawing digits >= 3 is 0.25 See the cumulative frequency distribution shown in the figure Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Overview Measurement error analysis Types of errors and uncertainty Statistical analysis Gaussian and Binomial distributions Method of Least Squares Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Gaussian Distribution Measurements will always have random errors For a large number of data, these errors will have a normal distribution which follows P(x) is the probability density function It gives the probability that the data x will lie between x and x+dx Is called the Gaussian or Normal error distribution Xm is the mean and  is the standard deviation Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Gaussian Distribution … Gaussian error distribution for σ=0.5 and 1 and xm=3 Probability density function has the property Xm is the most probable reading The value of the maximum probability density function is Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Gaussian Distribution … The standard deviation is a measure of the width of the distribution curve about the mean Smaller σ produces larger value of the maximum probability For a measurement, this tends to go to more precision The probability that a measurement will fall within a certain range x1 of the mean reading is given by Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Binomial Distribution In many statistical analysis, the samples may consist of only two kinds of elements E.g., odd or even, pass or fail, male or female, infested or free, dead or alive, etc. We may be interested in the proportion, percentage, or number of data in one of the two classes The head and tail case of a coin throw is very typical in this respect The chances of having a head or a tail in one throw is 50% if the coin is not weighted In other words the frequency of occurrence is the same for both heads and tails Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

Binomial Distribution … Supposing we toss the same coin twice (or toss two coins once), the outcomes could be any of the following: H H H T T H T T ½ x ½ ½ x ½ ½ x ½ ½ x ½ ¼ ¼ ¼ ¼ The chance of getting one H and one T is (¼ + ¼)= ½ Sum of probabilities is ¼+ ½+¼=1 For three throws, there will be 8 outcomes HHH HHT HTT HTH TTH THT THH TTT Each occurrence has a chance of (½ x ½ x ½ =⅛) Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

Binomial Distribution … 3H- only one → gives the chance of ⅛ 2H and 1T-3 of them → gives ⅜ 2T and 1H-3 of them → gives ⅜ 3T- only one → gives ⅛ The sum of the probabilities is ⅛+ ⅜ + ⅜ +⅛ = 1 To generalize, if the outcomes are successes (S) and failures (F) with probability of success being (p) and that of failure (q) where (p+q=1), for a sample size of N 2 Each of the success (events) can be determined from where n is the number of success Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data

Binomial Distribution … Requires the number of mutually exclusive ways in which the n successes and the (N-n) failures can be arranged Statistically, this term is called the number of combinations of n letters out of N letters and is given by The probability that n events will result in success stories is The right one is a binomial expansion expression, hence the distribution called the binomial distribution Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Binomial Distribution … E.g., for events of 0 and 1 designated by xj, mean of the binomial distribution is Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Overview Measurement error analysis Types of errors and uncertainty Statistical analysis Gaussian and Binomial distributions Method of Least Squares Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Method of Least Squares In the operation of an instrument, input parameter is varied over some range Could be in increments or decrements Happens during calibration or measurement Least square can be applied to determine an equation for a measured data Used to fit the data into a line (cure) to give a working relation between input and output This relation will help to determine the characteristics of the instrument Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Method of Least Squares … Example: Linear Least Square Analysis (LLSA) Suppose xi and yi be the input and measured values, respectively such that the data points (x1 , y1), (x2 , y2), ….. (xn , yn) are obtained If the expected straight line is of the form y = mx + b where m is the slope and b is the intercept The error, which is the difference between the actual and measured data, summed for all points is given as Minimizing S using Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Method of Least Squares … Will give And Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Method of Least Squares … 1/16/2019 Method of Least Squares … Example: Assume that the input and output are related by a second order equation y = b2x2 + b1x +b0 The error will take the form Minimizing the error with respect to b0, b1, and b2 yields n = total number of data points Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Method of Least Squares … 1/16/2019 Method of Least Squares … Assignment: The iron losses (L) in a ferromagnetic material, which is used to construct a transformer, vary with frequency (f) of the supply driving the transformer For a particular transformer, these losses were determined at various frequencies with a constant flux density in the ferromagnetic material Assume that the iron losses have a general form Using LLSA, determine the constants A and B. Frequency (Hz) 1100 1400 1700 2000 Iron losses (mW) 46 62 94 122 Introduction to Instrumentation Engineering - Ch. 2 Analysis of Experimental Data

Method of Least Squares … 1/16/2019 Method of Least Squares … Example: Consider data of high school versus college GPA, given as x and y, respectively. Compute the equation of linear least square regression line. Student x y x2 y2 xy 1 2.0 1.6 4.00 2.56 3.20 2 2.2 4.84 4.40 3 2.6 1.8 6.76 3.24 4.68 4 2.7 2.8 7.29 7.84 7.56 5 2.1 4.41 5.88 6 3.1 9.61 6.20 7 2.9 8.41 7.54 8 3.2 10.24 7.04 9 3.3 10.89 8.58 10 3.6 3.0 12.96 9.00 10.80 Totals 28.4 22.7 82.84 53.41 65.88 Introduction to Instrumentation Engineering- Ch. 2 Analysis of Experimental Data