Assessing and Presenting Experimental Data

Transcription

1 Assessing and Presenting Eperimental Data Common Types of error Uncertainty and Precision Uncertainty Theory Based on the Population Theory Based on the Sample

2 Introduction How good are the data? Actual data Error The difference between the measured value and the true physical value

3 Error in Measuring Error : the difference between the measured value and the true value Error = ε m true bound (ε) ; uncertainty (u) ε u +u (n :1) m u + u true m (n :1)

4 Common Types of Error Bias errors : systematic errors The same way each time a measurement is made Eample: the scale on an instrument Precision errors : random errors Different for each successive measurement but have an average value of zero Eample: mechanical friction or vibration

5 Bias and Precision errors Bias errors > Precision errors Frequency of occurrence Bias error Total error Precision error true Measured value, m m

6 Bias and Precision errors Bias errors < Precision errors Frequency of occurrence Bias error Total error Precision error true m Measured value, m

7 Classification of Errors 1) Bias or systematic error a. Calibration errors b. Certain consistently recurring human errors c. Certain errors caused by defective equipment d. Loading errors e. Limitations of system resolution ) Precision or random error a. Certain human errors b. Errors caused by disturbances to th equipment c. Errors caused by fluctuating eperimental conditions d. Errors derived from insufficient measuring-system

8 Classification of Errors 3) Illegitimate error a. Blunders and mistakes during an eperiment b. Computational errors after an eperiment 4) Errors that are sometimes bias error and sometimes precision error a. From instrument backlash, friction, and hysteresis b. From calibration drift and variation in test or environmental conditions c. Resulting from variations procedure or definition among eperimental

9 Elements of instrument error Hysteresis error

10 Elements of instrument error Linearity error

11 Elements of instrument error Sensitivity error

12 Elements of instrument error Zero shift(null) error

13 Elements of instrument error Repeatability error

14 Calibration errors Ideal response : measured = true Actual response : measured =β true + offset Output, measured 1 Actual response β 1 1 Ideal response 1 offset Input, true

15 hysteresis error Backlash and mechanical friction Output, measured Ideal response Actual response Input, true

16 In Rating Instrument Performance Accuracy The difference between the measured and true values Maimum error as the accuracy The etent to which a reading might be wrong, and is often quoted as a percentage of the full-scale reading of an instrument For eample: ±1% of full-scale reading Accuracy: true value indicated value A = true value

17 Precision In Rating Instrument Performance The difference between the instrument s reported values during repeated measurements of the same quantity Determined by statistical analysis A term which describes an instrument s degree of freedom from random errors

18 Precision In Rating Instrument Performance

19 In Rating Instrument Performance Accuracy & Precision low accuracy ; low precision high accuracy ; low precision low accuracy ; high precision high accuracy ; high precision

20 Resolution In Rating Instrument Performance The smallest increment of change in the measured value that can be determined from the instrument s readout scale Same (or smaller) order as the precision Sometimes specified as an absolute value and sometimes as a percentage of full-scale deflection

21 Sensitivity In Rating Instrument Performance The change of an instrument or transducer s output per unit change in the measured quantity Higher sensitivity will also have finer resolution, better precision, and higher accuracy The sensitivity of measurement is therefore the slope of the straight line in measured quantity v.s. output reading characteristic chart

22 Sensitivity In Rating Instrument Performance Output reading Gradient= Sensitivity of Measurement Measured quantity

23 In Rating Instrument Performance Sensitivity to disturbance Scale reading Characteristic with zero drift Zero drift Nominal characteristic Measured quantity

24 In Rating Instrument Performance Sensitivity to disturbance Scale reading Characteristic with sensitivity drift Sensitivity drift Nominal characteristic Measured quantity

25 In Rating Instrument Performance Sensitivity to disturbance Scale reading Characteristic with zero and sensitivity drift Zero drift plus Sensitivity drift Nominal characteristic Measured quantity

26 Precision error and accuracy

27 Hysteresis In Rating Instrument Performance The non-coincidence between these loading and unloading curves

28 Dead space In Rating Instrument Performance As the rang of different input values over which there is no change in output value + Output reading - + Measured variable - Dead space

29 In Rating Instrument Range or span Performance An instrument defines the minimum and maimum values of a quantity that the instrument is designed to measure Input span: R input = ma - min Output span: R output =y ma -y min

30 Threshold In Rating Instrument Performance If the input to an instrument is gradually from zero, the input will have to reach a certain minimum level before the change in the instrument output reading is of a large enough magnitude to be detectable. This minimum level of input is known as threshold for instrument.

31 Uncertainty Bias uncertainty, B precision uncertainty, P Total uncertainty, U U = ( B + P )

32 Uncertainty eample A brass rod aial strain, yielding an average strain of ε=50 µ-strain(50 ppm). A precision uncertainty Pε=1 µ-strain with 95% confidence. The bias uncertainty is estimated to be Bε=9 µ-strain with odds of 19:1 (95% confidence). What is the total uncertainty of the strain? Solution. The total uncertainty for 95% coverage is Uε=(Bε +Pε ) 1/ =36 µ-strain (95%) In other word, with odds of 19:1, the true strain lies in the interval 50 ± 36 µ-strain: 484 µ-strain ε 556 µ-strain.

33 Sample versus Population Sample population

34 Sample versus Population 群體 (population) ( 所製造的所有品目 ) 1 n 樣本 (sample) 由母體取出之樣本

35 Probability Distributions Probability is an epression of the likelihood of a particular event taking place, measured eith reference to all possible events.

36 Probability Distributions The Gaussian, or normal, probability distribution Z-distribution Student s t-distribution Only a small sample of data is available The -distribution in predicting the width of a population s distribution, in comparing the uniformity of samples, and in checking the goodness of fit for assumed distributions

37 Theory Based on the population Normal distribution curve

38 Theory Based on the population probability density function, (PDF) Probabilit y ) = (1 f ( ) Gaussian probability density function f ( ) 1 ep π 1 ( µ ) σ = σ d = the magnitude of a particular measurement µ= the mean value of the entire population σ= the standard deviation of the entire population

39 Theory Based on the population the arithmetic average = n n = n i= 1 i n the deviation d = - µ µ : The most probable single value for the quantity the standard deviation σ d + d n d n

40 Standard normal distribution curve

41 Eample a.what is the area under the curve between z=-1.43 and z=1.43? b.what is the significance of this area? Solution. a. From Table 3., read This represent half the area sought. Therefore, the total area is 0.436= b. The significance is that for data following the normal distribution, 84.7% of the population lies within the range 1.43 < z < 1.43.

42 Eample What range will contain 90% of the data? Solution. We need to find z such that 90%/=45% of the data lie between zero and +z; the other 45% will lie between z and zero. Entering Table 3., we find z (by interpolation). Hence, since z=(-µ)/σ, 90% of the population should fall within the range (µ- z 0.45 ) < < (µ+ z 0.45 ) or (µ ) < < (µ+1.645)

43 Theorey Based on the Sample We deal with samples from a population and not the population itself to use average values from the sample to estimate the mean or standard deviation of the population the sample mean n = i= i = 1 n n n

44 Theorey Based on the Sample the sample standard deviation s = ( 1 ) + ( ) + + n 1 ( n ) = n ( i = 1 n i ) 1 n Difference between population and sample For population For sample Mean Standard Deviation µ or µ σ or σ S

45 An Eample of Sampling Results of a 1-hour pressure test Pressure p, in Mpa Number of results, m

46 Solution Histogram of the pressure data

47 Solution Sample mean and standard deviation Pressure p Number of results Deviation d d

48 Solution Sample mean and standard deviation p = d = n = m =100 5 p = /100 = Mpa S p = / 99 = Mpa

49 Goodness of Fit A given set of data may or may not abide by the assumed distribution and since, at best, the degree of adherence can be only approimate, some estimate of goodness of fit should be made before critical decisions are based on statistical error calculations.

50 Goodness of Fit Normal probability plot

51 Graphical effects Goodness of Fit

52 Propagation of Uncertainty What is that uncertainty? Finding the uncertainty in a result due to uncertainties in the independent variables is called finding the propagation of uncertainty. A linear function y of several independent variables i with standard deviations σi; The standard deviation of y is = n n y y y y σ σ σ σ

53 Propagation of Uncertainty We assume that each uncertainty is small enough that a first-order Taylor epansion of y( 1,,, 3 ) provides a reasonable approimation: n n n n n u y u y u y y u u u y ),...,, ( ),...,, (

54 Propagation of Uncertainty Under this approimation, y is linear function of the independent variable. The uncertainties are: = n n y u y u y u y u

55 Uncertainty eample Suppose that y has the form y=a1+b and that the uncertainties in 1 and are known with odds of n:1. What is the uncertainty in y? Solution. y 1 = Using A ; above Eq., y = B u y = ( Au ) + 1 ( Bu ) (n :1).

56 Uncertainty eample 例 : 兩個電阻值為 100Ω 之電阻, 每個電阻值之公差 ( 不確定度 ) 為 5% 試求將兩電阻串聯後之總電阻值及其電阻不確定度為何? Solution. 一般串聯總電阻為 R 每個電阻之不確定度為 = R 1 + R = 100 5% 00 Ω = 5 Ω 總共不確定度為 u = = 串聯總電阻為 00 Ω ± y R R 1 u ( 1 5) + ( 1 5) 7.07 Ω 1 R + R u = 7.07 Ω

57 The Uncertainty eample A cylindrical body of circular section has a normal length of 5000 ± 0.5mm, an outside diameter of 00 ± 0.05mm. Determine the uncertainty in calculated volume. Solution. π π V = d l ; v = u l 0.5 = = 0.01% ; l 5000 Using above Eq., or u v u v v = = uncertaint ( u d d ) y of the u l + ( l 5000 u d d ) = volume = = % is ( u 0.005% v = 8 mm %) 4 = mm (0.01 %) 3, or about = ± % %

58 Graphical Presentation of Data When used to present facts, interpretations of facts, or theoretical relationships, a graph usually serves to communicate knowledge from the author to his readers, and to help them visualize the features that he considers important. A graph should be used when it will convey information and portray significant features more efficiently than words or tabulations. According to the American Standards Association

59 Graphical Presentation of Data For eample: atmospheric pressure The data are tabled Time of Day Pressure (mbar) 10:00 A.M. 11:30 01:00 P.M. 0:15 03:40 04:40 05:

60 Graphical Presentation of Data For eample: atmospheric pressure The data are graphed

61 General Rules for Making graphs For eample: a temperature data

62 General Rules for Making graphs Minimum effort in understanding The aes should have clear labels Use scientific notation Use real logarithmic aes The aes should usually include zero The scales should be commensurate with the relative importance of the variations Use symbols for data points Other rules see tetbook pp.101~103

63 General Rules for Making A pool graph graphs

64 General Rules for Making graphs improved by graphing guidelines

65 Choosing Coordinates Linear coordinates

66 Choosing Coordinates Semi-logarithmic coordinates

67 Choosing Coordinates Semi-logarithmic coordinates

68 Choosing Coordinates Full logarithmic coordinates

69 Choosing Coordinates Full logarithmic coordinates

70 Choosing Coordinates Polar coordinates

71 Choosing Coordinates Polar coordinates

72 Choosing Coordinates Polar coordinates

73 Producing Straight Lines For eample: cooling data By linear coordinates

74 Producing Straight Lines For eample: cooling data By semi-logarithmic coordinates

75 Producing Straight Lines For eample: plot of y=1.0 + (.5/) As y versus

76 Producing Straight Lines For eample: plot of y=1.0 + (.5/) As y versus (1/)

77 Straight-line Transformations y=f() Y=A+BX F() Y y=a+b/ y y=1/(a+b) 1/y y=/(a+b) X/y y=ab log y y=ac b log y y=a b log y y=a+b n y X 1/ log n A a a a log a log a log a a B b b b log b b log c b b

78 Line Fitting The simplest approach is just to draw appears to be a good straight line through the data When this approach is used, the probable tendency is to draw a line that minimizes the total deviation of all points from the line

79 Bias and precision error in line fitting

80 Least Square for Line Fits y=a + b Correlation coefficient, r = ) ( i i i i i i i n y y a = ) ( i i i i i i n y y n b ( ) ( ) + = ) ( ) ( m i m i y y S y y r [ ] 1 ) ( the squared deviations = = n i i i y y S

81 Least Square for Line Fits Eample A cantilever beam deflects downword when a mass is attached to its free end. T deflection, δ(m), is a function of the beam stiffness, k(n/m), the applied mass, M(kg), and the gravitational body force, g=9.807m/s : k δ=mg To determine the stiffness of a small cantilevered steel beam, a student place various masses on the end of the beam and measures the corresponding deflections. The deflections are measured using a scale (a ruler) marked in 1mm increments. Each mass is measured in a balance. His results are as follow: Mass(g) Deflection(mm)

82 Least Square for Line Fits Solution : Setting y=δ and =M Eample n=9 ; Σ=1801g ; Σ = g ; Σy=33.50mm ; Σy =179.3mm ; Σy=9959g mm ; The least squares results are then y = a + b [or δ= a + (g/k)m ] ; a= mm ; b=g/k= mm/g ; r= ; The eperimental stiffness of the beam is k= g/b = 9.807/ = 516 N/m

83 Least Square for Line Fits Eample Beam deflection for various masses

84 Least Square for Line Fits Eample Solution : From the figure, these data do appear to fall on a straight line. The correlation coefficient, r, is nearly unity, but a better test is to consider (1-r ) 1/ = %. This value indicates that the vertical standard deviation of the data is only about 9% of the total vertical variation caused by the straight-line relationship between y and.

85 Solution : y= r =1.00 Least Square for Line Fits Eample For the following data, determine the equation for y=y() by graphical analysis y ) ( = = i i i i i i i n y y a ) ( = = i i i i i i n y y n b ( ) ( ) 1.00 ; ) ( ) ( = + = r y y S y y r m i m i

86 Least Square for Line Fits Solution : Eample 15 y