1. JRT-2 (v.04) 1 Data analysis An Introduction to Error Analysis The study of uncertainties in physical measurements

2. JRT-2 (v.04) 2 Data analysis CONTENTS Preliminary Description of Error Analysis How to Report and Use Uncertainties (Chap 2) Propagation of Uncertainties Statistical Analysis of Random Uncertainties The Normal Distribution

3. JRT-2 (v.04) 3 Data analysis 2. How to Report and Use Uncertainties 2.1 Best estimate ± Uncertainty Best estimate = average value = 2.4 probable range : 2.3 to 2.5 Measured value = 2.4  0.1 (measured value of x) = x best ±  x How to express ?

4. JRT-2 (v.04) 4 Data analysis 2.1 Best estimate ± Uncertainty (measured value of x) = x best ±  x x best x best -  xx best +  x  x : uncertainty, error  x>0 x best -  x : lowest probable value x best +  x : highest probable value Range of confidentiality

5. JRT-2 (v.04) 5 Data analysis 2. How to Report and Use Uncertainties 2.2 Significant Figures (measured g) = 9.82 ± 0.02385 m/s 2 4 figures Rule for stating Uncertainties Experimental uncertainties should almost always be rounded to one significant figure. (measured g) = 9.82 ± 0.02 m/s 2 Exception If the leading digit is 1 (one), two figures is better. Not  x=0.1 but  x=0.14

6. JRT-2 (v.04) 6 Data analysis 2.2 Significant Figures (measured speed) = 6051.78 ± 30 m/s >2 or <8 No meaning Rule for stating Answers The last significant figure in any stated answer should usually be of the same order of magnitude (in the same decimal position) as the uncertainty. (measured speed) = 6050 ± 30 m/s

7. JRT-2 (v.04) 7 Data analysis 2.2 Significant Figures Examples : 92.81 0.3 : 92.8  0.3 3 : 93  3 30 : 90  30 uncertainty

8. JRT-2 (v.04) 8 Data analysis 2.2 Significant Figures Notes : Any numbers to be used in subsequent calculations should normally retain at least one significant figure more than is finally justified. Unit should be placed after the uncertainty. In the scientific notation, the answer and the uncertainty should be in the same form. 6050 ± 30 m/s 6050 m/s ± 30 m/s (1.61 ± 0.05) x 10 -19 coulombs 1.61 x 10 -19 ± 5 x 10 -21 coulombs QC2-2

9. JRT-2 (v.04) 9 Data analysis 2. How to Report and Use Uncertainties 2.3 Discrepancy : difference between two measured values of the same quantity 0 10 20 30 discrepancy=10 significant 0 10 20 30 discrepancy=10 insignificant

10. JRT-2 (v.04) 10 Data analysis 2.3 Discrepancy Accepted value of gas constant accepted R = 8.31451  0.00007 J/molK Treatable as a true value Discrepancy from a true value : true error

11. JRT-2 (v.04) 11 Data analysis 2. How to Report and Use Uncertainties 2.4 Comparison of measured and accepted values 0 10 20 30 accepted value

12. JRT-2 (v.04) 12 Data analysis 2. How to Report and Use Uncertainties 2.5 Comparison of two measured values 0 10 20 30 0 10 20 30

13. JRT-2 (v.04) 13 Data analysis 2.5 Comparison of two measured values Repeating measurements ….. Momentum conservation 3.103.12 2.162.05

14. JRT-2 (v.04) 14 Data analysis 2.5 Comparison of two measured values Repeating measurements ….. Measured p = p best   p Measured q = q best   q Best estimate of p-q = p best - q best

15. JRT-2 (v.04) 15 Data analysis 2.5 Comparison of two measured values Repeating measurements ….. Measured p = p best   p Measured q = q best   q Best estimate of p-q = p best - q best Highest probable value = (p best – q best ) + (  p +  q) Lowest probable value = (p best – q best ) - (  p +  q) Uncertainty in difference = (  p +  q)

16. JRT-2 (v.04) 16 Data analysis 2.5 Comparison of two measured values Repeating measurements ….. Provisional rule : (2.18) If q=x-y, the uncertainty in q is the sum of the uncertainties in x and y;  q=  x+  y QC2-3

17. JRT-2 (v.04) 17 Data analysis 2. How to Report and Use Uncertainties 2.6 Checking relationships with a graph  Many physical laws are treated in linear theory. One quantity y is proportional to other quantity x. Hooke’s law  Straight line is easy to recognize. y = ax

18. JRT-2 (v.04) 18 Data analysis 2. How to Report and Use Uncertainties 2.6 Checking relationships with a graph 1.1  0.3

19. JRT-2 (v.04) 19 Data analysis 2. How to Report and Use Uncertainties 2.6 Checking relationships with a graph 1.1  0.3

20. JRT-2 (v.04) 20 Data analysis 2. How to Report and Use Uncertainties 2.6 Checking relationships with a graph

21. JRT-2 (v.04) 21 Data analysis 2. How to Report and Use Uncertainties 2.6 Checking relationships with a graph y=ae bx

22. JRT-2 (v.04) 22 Data analysis 2. How to Report and Use Uncertainties 2.6 Checking relationships with a graph y=ax 2

23. JRT-2 (v.04) 23 Data analysis 2. How to Report and Use Uncertainties 2.7 Fractional uncertainties (measured value of x) = x best ±  x  x : Reliability or precision of measurement QC2-4

24. JRT-2 (v.04) 24 Data analysis 2. How to Report and Use Uncertainties 2.8 Significant Figures and Fractional Uncertainties 21 20.5 ~ 21.5 111.6 ± 0.05 : [111.55, 111.65] 111.6 ± 0.1 : [111.5, 111.7] 111.6 +0/-0.1 : [111.5, 111.6]

25. JRT-2 (v.04) 25 Data analysis 2. How to Report and Use Uncertainties 2.8 Significant Figures and Fractional Uncertainties In general, a number with N significant figures has an uncertainty of 1 in the N th digit. 210.21 21±10.21±0.01 1/21=0.050.01/0.21=0.05 5% Example : SF = 2 Fractional uncertainty

26. JRT-2 (v.04) 26 Data analysis 2. How to Report and Use Uncertainties 2.9 Multiplying two measured numbers Momentum : p=mv m = m 0 ±  m v = v 0 ±  v p = p 0 ±  p ?

27. JRT-2 (v.04) 27 Data analysis 2.9 Multiplying two measured numbers

28. JRT-2 (v.04) 28 Data analysis 2.9 Multiplying two measured numbers q = q best (1 ±  q/|q best |)

29. JRT-2 (v.04) 29 Data analysis 2. How to Report and Use Uncertainties 2.9 Multiplying two measured numbers Uncertainty in a product If two quantities x and y have been measured with small fractional uncertainties  x/|x best | and  y/|y best |, and if the measured values x and y are used to calculate the product q=xy, then the fractional uncertainty in q is the sum of the fractional uncertainties in x and y, QC2-5

30. JRT-2 (v.04) 30 Data analysis Problem 2.27 Problem 2-27 A calculator gives the answer x= 6.1234, but it is known that x has a fractional uncertainty of 2%. State the answer in the standard form x best ±  x.