Error Analysis How Do We Deal With Uncertainty In Science.

Transcription

1 How Do We Deal With Uncertainty In Science. 1 Error Analysis - the study and evaluation of uncertainty in measurement. 2 The word error does not mean mistake or blunder in science. 3 Experience shows no measurement, no matter how carefully made, is free of uncertainties.

2 How Do We Deal With Uncertainty In Science.

3 How Do We Deal With Uncertainty In Science.

4 Definitions 1 2 (measured value of x) = x best ± δx (1) x best = best estimate for x (2) 3 δx = (uncertainty in the measurement ) (3) 4 This statement expresses our confidence that the correct value of x probably lies in (or close to) the range from x best δx to x best + δx

5 Definitions 1 Discrepancy - difference between two measured values of the same quantity. 2 Fractional Uncertainty - if x is measured in the standard form x best ± δx, the fractional uncertainty in x is fractional uncertainty = δx x best (4) 3 Percent Uncertainty - percent uncertainty is just the fractional uncertainty expressed as a percentage (that is, multiplied by 100%).

6 Rules For Error Propagation - Rule I 1 Rule I (Provisional) - q = x z (u w) then (5) δq δx δz + δu δw (6) 2 Rule I (Random and Independent) - q = x z (u w) then (7) δq = δx δz 2 + δu δw 2 (8)

7 Rules For Error Propagation - Rule II 1 Rule II (Provisional) - q = x... z u... w δq q δx δz x z + δu δw +... u w 2 Rule II (Random and Independent) - δq q = (δx ) x q = x... z u... w ( ) δz 2 + z then (9) (10) then (11) ( ) δu u ( ) δw 2 (12) w

8 Rules For Error Propagation - Dealing With Powers 1 Raising To A Power - If n is an exact number q = x n (13) then 2 z.b. If then what is δq q = n δx x (14) q = h (15) δq q =? (16)

9 Rules For Error Propagation - General Rule General Formula for Error Propagation (Provisional): If q = q(x,..., z) is any function of x,..., z then δq q x δx q z δz (17)

10 Rules For Error Propagation - General Rule General Formula for Error Propagation: If q = q(x,..., z) is any function of x,..., z then ( q ) 2 ( ) q 2 δq = x δx z δz (18) provided all errors are independent and random.

11 Statistical Quantities - Mean Suppose that we make N measurements, x 1,..., x N of the same quantity x, all using the same method. Provided all uncertainties are random and small, the best estimate for x, based on these measurements, is their mean: x = 1 N N x i. (19) i=1

12 Rules For Error Propagation - Sample Standard Deviation The average uncertainty of the individual measurements x 1, x 2,..., x N is given by the sample standard deviation, or SD: 1 σ x = (xi x) N 1 2 (20) This definition of the SD is the most appropriate for our purposes. The population standard deviation is obtained by replacing the factor (N 1) in the denominator by N.

13 Rules For Error Propagation - Standard Deviation Significance Significance of the standard deviation σ x is that approximately 68% of the measurements of x (using the same method) should lie within a distance σ x of the true value. (This claim is justified in Section 5.4. of Taylor.) This result is what allows us to identify σ x as the uncertainty in any one measurement of x, δx = σ x, (21) and, with this choice, we can be 68% confident that any one measurement will fall within σ x of the correct answer.

14 Rules For Error Propagation - Standard Deviation of the Mean 1 As long as systematic uncertainties are negligible, the uncertainty in our best estimate for x (namely x) is the standard deviation of the mean, or SDM, σ x = σ x. (22) N 2 If there are appreciable systematic errors, then σ x gives the random component of the uncertainty in our best estimate for x: δx ran = σ x (23)

15 Rules For Error Propagation - Random & Systematic Rule If you have some way to estimate the systematic component δx sys a reasonable (but not rigorously justified) expression for the total uncertainty is the quadratic sum of δx ran and δx sys : δx tot = (δx ran ) 2 + (δx sys ) 2. (24)

16 Rules For Error Propagation - Gaussian Distribution 1 If f (x) is the limiting distribution for measurement of a continuous variable x, then f (x)dx = probability that any one measurement will give an answer between x and x + dx, 2 and b a f (x)dx = probability that any one measurement will give an answer between x = a and x = b. 3 The normalization condition is f (x)dx = 1. (25) 4 The mean value of x expected after many measurements is x = xf (x)dx. (26)

17 Rules For Error Propagation - Limiting Distributions 1 If the measurements of x are subject to many small random errors but negligible systematic error, their limiting distribution will be the Normal, or Gauss distribution: where 2 and G X,σ (x) = 1 σ )2/2σ2 e (x X, (27) 2π X = true value of x = center of distribution = mean value after many measurements, σ = width parameter of distribution = standard deviation after many measurements.

18 Rules For Error Propagation - Limiting Distributions The probability of a single measurement falling within t standard deviations of X is Prob(within tσ) = 1 2π t t e z2 /2 dz. This integral is often called the error function or the normal error integral. Its value as a function if t is tabulated in Appendix A of Taylor. In particular, Prob(within σ) = 68.27%.

19 Rules For Error Propagation - Estimating X and σ from N Measured Values 1 After N measurements of a normally distributed quantity x, x 1, x 2,..., x N, the best estimate for the true value X is the mean of our measurements, (best estimate for X ) = x = xi 2 and the best estimate for the width σ is the standard deviation of the measurements, (xi x) (best estimate for σ) = σ x = 2 (N 1) N (28) (29)

20 Rules For Error Propagation - Estimating X and σ from N Measured Values 1 The uncertainties in these estimates are as follows: The uncertainty in x as an estimate of X is (uncertainty in x) = SDOM = σ x N (30) 2 and the uncertainty in σ x as the estimate of the true width σ is given by 1 (fractional uncertainty in σ x ) = (31) 2(N 1)

21 Rules For Error Propagation - Weighted Average 1 If x 1, x 2,..., x N are measurements of a single quantity x, with know uncertainties σ 1, σ 2,..., σ N then the best estimate for the true value of x, is the weighted average x wav = wi x i wi, (32) where the sums are over all N measurements, i = 1,..., N, 2 and the weights w i are the reciprocal squares of the corresponding uncertainties, w i = 1/σ 2 i. (33)

22 Rules For Error Propagation - Weighted Average 1 The uncertainty in x wav is σ wav = 1/ wi, (34) where, again, the sum runs over all of the measurements i = 1,..., N.

23 Least Squares Fitting - A STRAIGHT LINE, y = A + Bx; EQUAL WEIGHTS 1 If y is expected to lie on a straight line y = A + Bx, and if the measurements of y all have the same uncertainties, then the best estimates for the constants A and B are: x 2 y x xy A = (35) 2 and B = N xy x y 3 where the denominator,, is (36) = N x 2 ( x ) 2. (37)

24 Least Squares Fitting - A STRAIGHT LINE, y = A + Bx; EQUAL WEIGHTS 1 Based on the observed points, the best estimate for the uncertainty in the measurements of y is σ y = 1 N (y i A Bx i ) 2. (38) N 2 i=1 2 The uncertainties in A and B are: 3 and x 2 σ A = σ y σ B = σ y N (39) (40)

25 Least Squares Fitting - Correlation Coefficient 1 Given N measurements (x 1, y 2 ),..., (x N, y N ) of two variables x and y, we define the correlation coefficient r as r = σ xy (xi x)(y i ȳ) = σ x σ y (xi x) 2 (y i ȳ). (41) 2 2 An equivalent form, which is sometimes more convenient, is r = xi y i N xȳ ( xi 2 N x 2 )(. (42) yi 2 Nȳ 2 )

26 Least Squares Fitting - Correlation Coefficient 1 Values of r near 1 or 1 indicate strong linear correlation; values near 0 indicate little or no correlation. 2 The probability Prob N ( r > r 0 ) that N measurements of two uncorrelated variables would give a value of r larger than any observed value r 0 is tabulated in Appendix C of Taylor. The smaller this probability, the better the evidence that the variables x and y really are correlated. 3 If the probability is less the 5%, we say the correlation is significant; if it is less than 1%, we say the correlation is highly significant.

27 Least Squares Fitting (Linear Regression) - A STRAIGHT LINE, y = A + Bx; EQUAL WEIGHTS 1 A good quality software package such as Minitab can do a LSQ for you. 2 EVEN Excel can do this but it is a struggle.