1. Introduction to Error Analysis and Propagation Dr Umi Fazara Md Ali umifazara@unimap.edu.my 7 January 2008

2. Error Analysis and Propagation Uncertainties • Measurement • Systematic, Statistical and Accidental • Error propagation • Presentation of results • Graphics 8 February 2007

3. Uncertainty of a single measurement: mm ruler Generally half of the smallest division of scale analog digital Pressure gauges Measurement All measurements, no matter how simple, have an associated uncertainty (or error). All measurements need units (unless dimensionless) Ruler: 0.5 mm Analog: 50 psi Digital: 0.5 torr (need indicate criterion)

4. Single Measurement All measurements, no matter how simple, have an associated uncertainty (or error). All measurements need units (unless dimensionless) mm L=7.0±0.5 mm

5. Single Measurement All measurements, no matter how simple, have an associated uncertainty (or error). All measurements need units (unless dimensionless) P=1700±50 psi P=(170±5) 101 psi analog

6. Single Measurement All measurements, no matter how simple, have an associated uncertainty (or error). All measurements need units (unless dimensionless) P= 760.0±0.5 torr (need indicate criterion) digital

7. Systematic Statistical Accidental Bathroom scale parallax Uncertainties • Can be known exactly (and therefore eliminated) • Result from a systematic bias: miscalibration, instrument malfunction, method, or operator bias

8. Systematic Statistical Accidental Uncertainties • Can be known exactly (and therefore eliminated) • Result from a systematic bias: miscalibration, instrument malfunction, method, or operator bias • (or Random) result from independent, uncorrelated events. Can be reduced but NOT eliminated. • Result from fluctuations, equipment limitations and reading uncertainties.

9. Systematic Statistical Accidental L=100 mm Uncertainties • Can be known exactly (and therefore eliminated) • Result from a systematic bias: miscalibration, instrument malfunction, method, or operator bias • (or Random) result from independent, uncorrelated events. Can be reduced but NOT eliminated. • Result from fluctuations, equipment limitations and reading uncertainties. • Outlier point, possible to detect

10. Systematic Statistical Accidental What is uncertainty? • Measure of the degree of confidence in result [integral part of the measurement & cannot be eliminated; essential for comparing results of validating theories] • Estimate improves with number of measurements (ie reduce error) Uncertainties • Can be known exactly (and therefore eliminated) • Result from a systematic bias: miscalibration, instrument malfunction, method, or operator bias • (or Random) result from independent, uncorrelated events. Can be reduced but NOT eliminated. • Result from fluctuations, equipment limitations and reading uncertainties. • Outlier point, possible to detect

11. “(…), a value of 12.5 bn years is obtained for the age of the Cosmos, with an uncertainty of about 3 bn years.“ A1= 12.5 ± 3.0 bn years A2= 13.7 ± 0.2 bn years “13.7 bn years, with an uncertainty of about 0.2 bn years.“ Age of Universe?

12. assume line is ‘exact’ high precision low accuracy low precision high accuracy Estimate improves with number of measurements (ie reduce error) Series of Measurements Nomenclature: Accuracy & Precision Accuracy: free from mistake Precision: small error (s)

13. Mean (or arithmetic average) (n: number of measurements) Max error (few measurements, n) Error estimates: Statistical error (large n ≥ 10) Series of Measurements

14. Example: r = 0.91; 0.92; 0.94; 0.98 g/cm3 <r> = 0.9375 g/cm3 <r> = (0.94 ± 0.04) g/cm3 d<r> = 0.0425 g/cm3 <0.5 → 0 (unchanged) • Rounding up: >0.5 → +1 =0.5 → closest even number (eg. 17.250 ± 0.322 → 17.2 ± 0.3) Max error (few n) Mean (or arithmetic average) Error: • Error should have one non-zero digit • Measured quantity should have same decimals as error

15. Standard deviation: (of course, ) (describes the scatter around the average) 68% 1 s: 68% 2 s: 95% 3 s: 99.7 % of the results fall within these intervals (Usually. Always indicate criterion for error) Statistical error (n≥10) Gaussian Distribution: function describing the sum large number of uncorrelated, independent measurements with finite probability (central limit theorem.

16. Example 1: L= L1+L2 Example 2: A= L1x L2 dA? L1 dL1 + L2dL1 dA= L1dL2 L1 L2 dL2 Error propagation Calculate a quantity derived from several measured quantities dL=dL1+dL2 must have units of area L2

17. General error propagation formula Relative error Error propagation (2) Calculate a quantity derived from several measured quantities Example 3: Velocity v = L / t (m s-1) Dv ? Example 1: L= L1+L2 dL=dL1+dL2 dA= L1dL2 + L2dL1 Example 2: A= L1x L2 dv= dL/t + L/t2dt Example 3: v = L/t

18. Example 1: f= x1+x2 df=dx1+dx2 Example 3: f = x1/x2 Error propagation (3) General error propagation formula General error propagation formula (max) (stat) Systematic errors (N≤10) Statistical errors (N≥10)

19. Example: gravitational potential energy m Ep = mgh m: mass h g: gravitation constant (9.81 m s-2) h: height Error propagation (4) • Make reasonable assumptions in propagation: • Constants ‘do not have’ error and • Some variables may have negligible errors • Explain your assumptions. dEp = mg dh + gh dm

20. Fit is obtaining by minimising: Quality of fit: R2 →1 Graphics Data points must have error bars, dy (sometimes dx) Slopes also have error (can be determined graphically) dm = |mmax-m| http://office.microsoft.com/en-us/excel-help/add-change-or-remove-error-bars-in-a-chart-HP010007462.aspx#BMadderrorbars

21. Error Analysis and Propagation Uncertainties • Measurement • Systematic, Statistical and Accidental • Error propagation • Presentation of results • Graphics 8 February 2007