36 Chapter : Statistical Methods 1. Introduction. Characterizing statistical distribution 3. Interpretation of Results 4. Total Measurement System and Errors 5. Regression Analysis
37 1.Introduction The variability of experimental measurements of quantities (temperature, length, force, stress or strain) is fundamental to all measuring systems It is due to Two causes : 1.The quantity to be measured may exhibit a significant variation.the measuring system, which includes the tranducer, signal conditioning element, analog to digital converter, recording instrument, and the operator may introduce error into the measurement.
38 1.Introduction Array of results Distribution (Graphic Representation) Statistical Methods The statistical distribution is characterized with a measure of : 1. Its central value (the mean, the median or the mode). The spread or dispersion of the distribution (Variance or the standard deviation) X = X ± S x
39.Characterizing Statistical distributions Suppose that a quantity x has been measured n times. An experiment shows that : x 1 was recorded n 1 times x was recorded n times... and x n has been recorded n m times n = n 1 + n + +n m The coordinates of the relative frequency distribution: f n (x 1 ) = n 1 /n; f n (x )=n /n; ; f n (x m )=n m /n f n (x 1 )+f n (x )+ +f n (x m ) = 1
40.Characterizing Statistical distributions SAMPLE STRENGTH NUMBER STRENGTH KSI STRENGTH MPa SAMPLE NUMBER STRENGTH KSI STRENGTH MPa 1 65 448 11 79 545 68.3 478 1 79. 546 3 7. 498 13 79.9 551 4 73.5 507 14 80.3 554 5 74 510 15 81.1 559 6 75. 519 16 8.6 570 7 76.8 530 17 84 579 8 77.7 536 18 85.5 590 9 78.1 539 19 87.7 600 10 78.8 543 0 89.8 619
41.Characterizing Statistical distributions Group Intervals (Ksi) Group Intervals (Mpa) Observations in the group Relative Frequency Cumulative Frequency 65.0-69.9 448-48 0.1 0.1 70.0-74.9 483-516 3 0.15 0.5 75.0-79.9 517-551 8 0.4 0.65 80.0-84.9 55-585 4 0. 0.85 85.0-89.9 586-60 3 0.15 1.0 Total 0 1
.Characterizing Statistical distributions.1 Graphic Presentation of the distribution Histogram with a superimposed relative-frequency diagram 0,40 8 Relative Frequency 0,35 0,30 0,5 0,0 0,15 0,1 7 6 5 4 3 Number of observations 1 0,05 0 0 60 65 70 75 80 85 90 Results of Measurement 4
.Characterizing Statistical distributions Cumulative Frequency diagram 1. 1 0.8 0.6 0.4 0. Cumulative Frequency 43 0 60 65 70 75 80 85 90 Yield Length The cumulative frequency is a number of readings having a value less then a specified value of the quantity being measured divided by the total number of measurements: it is the running sum of the relative frequencies
44.Characterizing Statistical distributions. Measure of central tendency..1 The Mean x = n x i n i=1 n = i=1 x n i xi : the value of the quantity being measured n : the total number of measurements X = 78,4 Ksi (541 MPa).. The Median The Median is the central value in a group of ordered data Examples : In an ordered set of 1 readings: Median = X 11 In an ordered set of 0 readings: Median = (x 10 +x 11 )/ Median = (78.8+79)/=78.9 Ksi..3 The Mode The Mode is the most frequent value of data Mode = 77,5 Ksi It is a peak value on the relative frequency curve
45.Characterizing Statistical distributions If a typical set of data gives different values for the three measures of central tendency. There are two reasons: 1. The population from which the samples are drawn may be not Gaussian where the three measures are expected to coincide.. Even if the population is Gaussian, the number of measurements is usually small and deviations are to be expected.
46.Characterizing Statistical distributions.3 Measure of dispersion There are several measures of dispersion.3.1 The range R = X L - X S X L : the largest value in the dispersion X S : the smallest value in the dispersion.3. The mean deviation d 1 = n x n i= 1 x x i
47.Characterizing Statistical distributions.3 Measure of dispersion.3.3 The Variance S n 1 x = (x i n 1 - i=1 x ).3.4 The standard deviation S n 1 x = (x i - n 1 i=1 x ) For small populations, the standard deviation S x represents a good estimate of the true standard deviation s of the population.3.5 Coefficient of variation C v = x S x
.Characterizing Statistical distributions.4 Frequency distribution of measurement errors The spread of values about the mean is analyzed by first calculating the deviation of each value from the mean δ = x i - x Probability density function (F(E)) The error in each measurement is calculated as its deviation from the mean value As the number of measurements increases, smaller error bands can be defined for the error histogram 48 E 0 E p E 1 E Error n infini The histogram Magnitude E + becomes a smooth curve F(E).dE =1
.Characterizing Statistical distributions The probability that the error in any one particular measurement lies between two levels E 1 and E can be calculated by : P(E E 1 E E ) = E1 F(E).dE : The Error Function The cumulative distribution function (c.f.d) is defined as the probability of observing a value less or equal to E 0 P(E E ) = E0 0 F(E).dE Special types of frequency distributions: Gaussian, Binomial and Poisson Most data sets approach closely to one or other of them The distribution of relevance to data sets containing random 49 measurement errors is the Gaussian one.
50.Characterizing Statistical distributions.4.1 Gaussian Distribution 1 _ ( x _ µ F (x) = exp( σ π σ ) ) µ: the true value of the measurement set σ : the standard deviation of the measurement set The Gaussian curve is only applicable to data which has only random errors, where no systematic errors exist (E p =0) If the error is calculated as E = x µ F(E) is the Gaussian Curve known as the error frequency distribution curve F(x) = σ 1 π exp( _ E σ )
50.Characterizing Statistical distributions The probability that the error lies and band between levels E 1 and E can be expressed as: P(E E 1 E E ) = 1 _ E exp( ) σ π σ E1 de Substitution: z = E σ P(z z 1 _ z 1 z z ) = Exp ( ) π z1 dz The equation can not be evaluated by use of standard integrals Error function table give value of F(z) for various values of z
51.Characterizing Statistical distributions.5. Standard error of mean The foregoing analysis is only strictly true for measurement sets containing infinite populations It is of course not possible to obtain an infinite number of data values, and some error must therefore be expected in the calculated mean value of the prctical, finite data set available The error in the mean of a finite data set is usually expressed as the standard error of the mean, S x which is calculated as: S s x x = x n x = x ± S
5.Characterizing Statistical distributions.6. Application of intelligent instruments to reduce random errors If a measurement system is known to be subject to random errors, intelligent instruments can be programmed to take the same measurement a number of times within a short space of time and perform simple averaging or other statistical techniques on the readings before displaying an output measurement This is valid for all forms of random error, whether due to human observation deficiencies, electrical noise or any other fluctuation.
53 3. Total Measurement System Errors How the errors associated to each measurement system component combine together, so that a total error calculation can be made for the complete measurement system (addition, subtraction, multiplication and division) 3.1. Error in a Product P = y z lnp = lny +lnz dp P = dy y + dz z P y z = + P y z
54 3. Total Measurement System Errors 3.. Error in a Quotient Q = y z lnq = ln y lnz dq Q = dy y dz z Q y z = + Q y z 3.3. Error in a Sum S = x + y The most probable maximum error in S is e = x + y
55 3. Total Measurement System Errors 3.4. Error in a Difference D = x - y e = x + y 3.5. Total Errors when combining multiple measurements The final case to be covered is here the final measurement is calculated from several measurements which are combined together in a way which involves more then one type of arithmetic operations. The errors involved in each stage of arithmetic are cumulative
56 4. Regression analysis Many experiments involve the measurement of one dependent variable, Example y which may depend on ne or more independent variables x 1,x x k Regression analysis provide a statistical approach for condition in the data obtained from experiments in which two or more related quantities are measured
57 4. Regression analysis y 4.1 Linear regression analysis Y i = m x i + b x Y i is the predicted value of the dependant variable y i for a given value of the independent variable x i
58 4. Regression analysis The least squares method is a statistical procedure used to fit a straight line through scattered data points. With this method, the slope m and the intercept b are selected to minimize the total error. ( ) = y Y i i n ( i i ) i 1 n ( i i ) i 1 = y mx b = 0 b b = = y mx b = 0 m m = n ( i i ) ( ) i = 1 n ( i i ) ( i ) i = 1 y mx b 1 = 0 y mx b x = 0
60 4. Regression analysis m = x y n xy ( ) x n x b y m x x xy x y = = n x n x ( ) ρ = 1 n n 1 [ ] y mxy y y y = y [ xy ] = n xy x n y
61 4. Regression analysis 4. Validation of the regression analysis y y = m x+b The dispersion of the distribution of y is a measure of the correlation Small dispersion Good correlation ρ ~ 1 The regression analysis is effective Large dispersion Poor correlation ρ ~ 0 The regression analysis is not adequate