3. Measurement Error and Precision

Transcription

1 3.1 Measurement Error Definition 3. Measurement Error and Precision No physical measurement is completely exact or even completely precise. - Difference between a measured value and the true value - Absolute error(e) vs Percent(Relative) error(%e) e = true value measured value, e e true value - In most of real measurements the true value is unknown, thus the average of a sample replaces it Kinds of measurement error - Supposed there are no mistakes, * measured value(x) = true value(t) + measurement error(e) * measurement error(e) = random error(e r ) + systematic error(e s ) 1) Random Error(Accidental Error) - due to unpredictable variations in the experimental conditions or a deficiency in defining the variable. - does not have any consistent effects across the entire sample. - does not affect the average but only a distribution of measured values about the sample mean. scatter(precision), decrease the precision. - averaging of multiple measurements reduces random error. 2) Systematic Error - caused by systematic faults in the measuring instrument. - shifts the sample mean away from the true value by a fixed amount, that is, an offset (bias error) decrease the accuracy. - usually adjusting zero offset reduces systematic error. - can be estimated by comparison such as calibration, concomitant method. (A: Larger bias, smaller scatter, B: Smaller bias, larger scatter) PNU Prof. Ahn, Jung Hwan

2 Ex 3.1) 10 independent measurements with the true value known(fig 3.1) Fig independent measurements 3.2 Accuracy vs Precision Accuracy - Indicates how close to the truth the measured value is, in other words how small the bias error(=average value true value) is. - described as (% of reading) + offset. Ex 3.2) If the accuracy of a voltmeter is ±(0.3%+0.2), for a measured value of 73.6 volts the max. error is 0.42 V including 0.2 V of zero shift and for 100 V the max error is 0.5 V Precision - Indicates how closely a group of repeated measurements get together to the average the more they scatter, the worse the precision is. - Usually described as 2*S x (95%), standard deviation S x N i x i x - Is getting better as the resolution is smaller. cf. repeatability(or reproducibility) N Ex 3.3) Precision vs Accuracy (Fig 3.2) Fig 3.2 Comparison of accuracy, precision, bias errors; a:(low, high, large) b:(high, high, zero) c:(low, low, small) PNU Prof. Ahn, Jung Hwan

3 3.3 Precision vs the resolution of a measuring scale - How precisely can we measure? - a measured value is expressed by the significant (meaningful) digits, that is all the digits with certainty + one estimated(or uncertain) digit. 2.55/2.5, the last digit is therefore somewhat inaccurate(uncertain). - the precision of any measured value can be assumed to be 1 of the least significant digit. Measurement resolution is dependent on the graduation of a scale, which corresponds to the bit number in A/D converter. 3.4 Calibration Fig 3.3 Various graduations - Process of comparing an instrument with a more accurate known standard. - a calibration chart of relationship between input and output is drawn up. - Static calibration is usual; output variable(y) is measured each time when the input variable(x) is independently controlled to a certain value - A correlation equation, y=f(x), is determined through curve-fitting. dy - Static gain (static sensitivity): K dx - Output span (FSO: full-scale operating range) : r y m ax y m in - Input span : r i x m ax x m in cf. Dynamic calibration: When the variables of interest are time(or space) dependent, the dynamic behavior of the output variable is investigated against dynamic input signal such as a sinusoidal or a step signal. Fig 3.4 Representative static calibration curve PNU Prof. Ahn, Jung Hwan

4 3.5 Instrumental errors 1) Hysteresis error - differences between an upscale sequential test and a downscale one e h e h max y u pscale y down scale, Max hysteresis error : e h max r 2) Linearity error e L x yx yx (calibration: yx a a x yx measured) 3) sensitivity error( e K ): error in the slope of the calibration curve 4) zero shift error( e Z ): a drift in the zero intercept 5) instrument repeatability error( e R ) - The ability to indicate the same value upon repeated but independent application of the same input under the same equipment and environment S x - e R max r ( Sx : standard deviation) cf) Reproducibility: The results of multiple repeatability tests(replication) performed in different environment on a single unit. 6) Overall instrumental error: combining all elemental errors, RSS(root-sum squares). e c e e e M Fig 3.5 Instrumental errors PNU Prof. Ahn, Jung Hwan

5 3.6 Presentation of a measured value with significant digits Significant digits(significant figures) in measurement - Uncertainty for a group of data( a sample) = standard deviation - More significant digits means the value is more precise. - Scientific notation * = (two sig-fig) * = (six sig-fig) * 1300= (four sig-fig) or (two sig-fig) - How much precision is your instrument capable of? * 2.55(3 sig-fig) vs 2.5(2 sig-fig) * If your instrument are precise to 3 sig-fig, you can distinguish 2.55 and 2.56 cm long. cf. The units have nothing to do with the precision of a measurement. For example grams is not more precise than 12.1 kilograms. The first has 1 sig-fig and the second has 3 sig-figs Stating Results with Uncertainty - S x ± S x or x x with a percent uncertainty ( ) x e.g. 9.2 ± 0.3 g or 9.2 grams with an uncertainty of 3 % - The uncertainty in the final result should have, at most, 2 digits, and more commonly 1 digit. Ex. 9.5±0.3g, or 9.52 ±0.14g, but not 9.52±0.3g - for some reasons the data points varied so that the standard deviation is 2g. Then, the result should be reported as 9±2g. - If The uncertainty is much smaller than the least significant digit, then assume that is equal to "1" of the smallest digit. For example, 9.52±0.01g Comparing quantities with uncertainty To know whether two numbers obtained by two different methods but hypothetically referring to the same physical quantity agree. - State the quantities with their uncertainty and see if they overlap. If they do, they agree. If not, they don't. Ex 3.4) A theory predicts that the density of an object should be 10.0±0.1 g/cm 3. - If a measurement value is 9.8±0.3 g/cm 3, the two values agree within the experimental uncertainty. - If the measurement value is 9.81±0.02 g/cm 3, the two values did not agree PNU Prof. Ahn, Jung Hwan

6 3.7 Uncertainty in calculation (significance arithmetic) - Are the calculated results below "correct"? * 2.3 = , 3.7 / = Those are correct in a pure mathematical sense but in the real world, when we make measurements of anything, not correct because the value have uncertainty. - Rules for estimating the uncertainty in calculation using significant digits. 1) For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places. Ex 3.5) total weight of A,B,C: W= 1.23 N N N = N 9.5N Ex 3.6) ⁴ ³= = > = ) For multiplication and division, the result should have as many significant digits as the measured number with the smallest number of significant digits. Ex 3.7) Area of a rectangle: S= 1.25m 5.213m = m m 2 Ex 3.8) Volume of a sphere (D=20.05mm): D V = = = 2, ,374 cf. round-off errors: there is no reason to prematurely truncate a result, just because it is found to be uncertain. So keep your extra digits as you go (at most one or two extra, if calculating by hand), but make sure to adjust the final result when presenting your measurements for comparison PNU Prof. Ahn, Jung Hwan

7 3.8 Propagation of Uncertainty( Error propagation) - method of computing the uncertainty in a result which depends on several variables having its own uncertainty, respectively. - a relationship between a dependent variable y and a measured variable x: y f x x x n (3.1) From a number of measurements, the sample mean and the uncertainty interval for x are x, x x x n while for y they are - Expanding y as a Taylor series and taking a linear approximation, y, y f x. f f f f y x x x x xn xn xi xi : contribution to y by x i ) uncertainty of y: f y x f xn n - With no information on i, it is more practical to use the maximum error ; y f x x f x x f xn xn (3.2) - Measurement rules in an aspect of error propagation * Variables with a larger gain or power should be measured more accurately. * The variable with worst accuracy determines the whole accuracy. * It is effective to make all the variables even in accuracy. Ex 3.9) Find the maximum error in measuring the diameter and height of a cylinder so that the maximum error in calculating its volume falls within ±1%. V= d V h V d d h h * 1st term: no problem because can be approximated to any accuracy level. * 2nd term: affects two times as large as the third term does. Let the accuracy of each variable even, d d h h Substituting d= 2cm, l = 10cm, then d mm h mm the maximum measurement error: d ± mm h ± mm Function Formula Uncertainty formula Multiplication, Division f xy or f xy f x y Addition, Subtraction f x y or f x y f x y Product of Power fn. f x m y n f m x n y Constant multiplication f K x K constant f K x Logarithmic functions Exponential functions f log e x f log x f e x f x or f x f y x x y f log x x f x x log x x Table 3.1 Common formulas for propagating uncertainty( percent error PNU Prof. Ahn, Jung Hwan

8 Homework #3.1: In an experiment with an air track, an experimenter wishes to determine the average speed of an air track cart between two photo gates. The distance between the photo gates is given by ±, and the time of travel between these two points is ±. Calculate the average speed, the fractional uncertainty, and the absolute uncertainty given these data. Show your work, and state the results using correct significant digits, and following the format given in the section Stating Results with Uncertainty. Homework #3.2: To measure the density of a rectangular object, an experimenter measures the object's volume and mass. The volume is given by the formula, where is the length, is the width, and is the height. The density is given by, where is the object's mass. If the measurement of the mass is uncertain by 2%, and each of and is uncertain by 4%, what is the uncertainty, in percent, of the density? Show your work. Homework #3.3: The experimenter conducts the same density measurement with a second sample that is spherical in shape. The mass is again uncertain by 2%. The diameter of the sphere is measured to a precision of 4%. The volume of a sphere is given by the formula. What is the percent uncertainty of the density in this case? Show your work. Why is the uncertainty in this case different than in the case of a rectangular object? What is the underlying reason (not just how are the formulas different)? Homework #3.4: For a displacement transducer having a calibration curve, estimate the uncertainty in displacement for, if with ± and ± at 95% confidence PNU Prof. Ahn, Jung Hwan