L 4 — Uncertainty and Error Analysis Outline

L 4 - Uncertainty and Error Analysis Outline 1. Errors / Uncertainties in Measurements 2. Accuracy and precision 3. Random and Systematic Errors 4. Estimating Random Errors 5. Absolute and Fractional Uncertainties 6. Combining Experimental Errors in Formula 7. Significant Figures in Labs 1

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1. Error / Uncertainty in Measurements In Experimental Physics, EVERY measurement must be stated with an estimate of its error (or uncertainty) An error is not a mistake but a measure of how good your measurement is L=(56. 4 ± 0. 2) cm means 56. 2 cm ≤ L ≤ 56. 6 cm A measurement and its error must have the same number of decimal places Ø CORRECT: L = (56. 41 ± 0. 20) cm, L = (56. 400 ± 0. 200) cm 3 Ø INCORRECT: L = (56. 41 ± 0. 2) cm, L = (56. 40 ± 0. 200) cm

Why are errors important? Consider two measurements of body temperature before and after a drug is administered to a patient Tbefore = 38. 2 °C Tafter = 38. 6 °C Question: Is the temperature rise significant? Answer: It depends on the measurement error. Tbefore = (38. 2 ± 0. 1) °C and Tafter = (38. 6 ± 0. 1) °C => Significant rise Tbefore = (38. 2 ± 0. 5) °C and Tafter = (38. 6 ± 0. 5) °C => Not significant rise 4

2. Accuracy and precision Accuracy: The degree to which the result of a measurement, calculation conforms to the correct value or a standard => accuracy is the measure of exactness Precision: Refinement in a measurement, calculation, as represented by the number of digits given => precision is the measure of reproducibility or consistency 5

3. Random and Systematic Errors Random Error ü Varies between successive measurements of the same quantity ü Is equally likely to be positive or negative ü Can be reduced by measuring the same quantity several times and taking the average (or the mean) Systematic Error Ø Ø Affects each reading in the same way Can result from incorrectly calibrated equipment Cannot be reduced by repeating the same measurement Difficult to identify → you can suggest a possible source 7

Random and systematic errors - example True value Random errors only Random + systematic § A result is said to be accurate if it is relatively free from systematic errors § A result is said to be precise if the random error is small 8

4. Estimating Random Errors (Single Reading) § Digital meter: reading error is usually taken as the smallest division that can be accurately read. V = ( 3. 36 ± 0. 01) V § This is correct if the reading is steady and easily read. It will be less accurate, with larger error, if the last digit is changing. 9

Estimating random errors (single reading) § Linear Scale: reading error is usually taken as one half the smallest scale division that can be accurately read. 16 17 L = (16. 75 ± 0. 05) cm 10

Estimating random errors (multiple readings) § When you have several measurements of the same quantity, the best estimate is the average (arithmetic mean) § The random error can be estimated from the minimum and maximum value 11

Estimating random errors (multiple readings) T (s) 1. 853 1. 861 1. 831 1. 842 1. 854 1. 847 1. 859 1. 850 1. 852 1. 833 Best estimate is the average of the 10 readings of T which gives T = 1. 848 s The uncertainty, ΔT, can be found from Hence, T = Tave ± ∆T = (1. 848 ± 0. 015) s Note: Tave is also written as and T. 12

Estimating random errors from a graph § When two quantities are proportional, you can estimate (visually or using Excel) the random error of the gradient of the best-fit line as follows: § 1) Draw a best-fit line and calculate the gradient G § 2) Draw two lines of minimum (Gmin) and maximum (Gmax) gradients § 3) Estimate the uncertainty in the gradient ΔG as: § Note that gradients usually have a dimension (hence a unit) 13

Estimating random errors from a graph Simple pendulum. Estimation of uncertainty in gradient (g/4π) 35 Maximum Gradient 30 Length L (m) 25 20 Best fit line 15 Minimum Gradient 10 5 0 0 20 40 60 Period T 2 (s 2) 80 100 120

5. Absolute and Fractional Uncertainties An uncertainty can be expressed in two ways: 1) As an absolute error (± Δx). Δx has the same dimension as the quantity x. 2) As a fractional error or a percentage error Fractional and percentage errors are dimensionless 15

6. Formula for Error Propagation § When variables are added or subtracted, absolute uncertainties are added - the absolute uncertainty in A 16

Combining Experimental Errors § When variables are multiplied or divided, the fractional uncertainties are added in quadrature: - the fractional uncertainty in A 17

Example 1 (Simple Pendulum) § Suppose we want to estimate the acceleration due to gravity, g, using a simple pendulum and we estimated: - the period T = (1. 848 ± 0. 015) s from multiple readings - the length L = (0. 950 ± 0. 005) m § Calculate the value of g and its estimated absolute random uncertainty 18

Example 1, cont. Knowing the true value of g at the Earth’s surface, what can you conclude about random and systematic errors in this estimation? 19

Example 1, cont. § You then realized that you did not use the ruler appropriately and that the length should be L = (0. 850 ± 0. 005) m § Recalculate g and conclude 20

7. Significant Figures in Labs § When recording data, you must record the entire number that the device gives you, along with the correct unit. § Intermediate answers can have as many significant figures or decimal places as common sense allows, since they are not results. § When combining numbers, such as finding voltage (V) = current (I) × resistance (R), if I and R are measured, the voltage can be only as accurate as the least accurate of the 21 inputs.

Significant Figures in Labs § Keep in mind that when measuring various values of a quantity, the number of significant figures may vary. § Use the same number of decimal places for all values of that quantity. # 1 2 3 I (m. A) 1. 47 0. 50 0. 31 4 5 6 7 8 9 10 0. 22 0. 17 0. 13 0. 11 0. 10 0. 09 0. 07 ln (I/I 0) -11. 1 -12. 2 -12. 7 -13. 0 -13. 3 -13. 6 -13. 7 -13. 8 -13. 9 -14. 2 I is measured quantity and I 0= 1. 00 x 105 A 22

CHECK LIST © q Further READING 1. Ch. 2. 6 and 2. 7 in Adams & Allday’s Advanced Physics; 2. Ch 1. 4 in Serway & Vuille’s Essentials of College Physics; Other (not required, but) useful books: 1. J. R. Taylor (1997) Introduction to error analysis: the study of uncertainties in physical measurements - 2 nd Edition; 2. L. Lyons (1991) A practical guide to data analysis for physical science students. q SUMMARY § Understand the concept of errors (uncertainties) in measurements § Know the definitions of accuracy and precision § Know the definitions of random and systematic errors § Be able to estimate random experimental errors § Know what are absolute and fractional uncertainties § Be able to perform error analysis having set of experimental data § Know how to draw a conclusion from an error analysis § Be able to estimate uncertainties by error propagation § Know the rules for determining significant figures

Numerical Answers to Example • Ex 1 ) a) g = (11. 0 ± 0. 2) ms-2 b) Systematic error c) (9. 83 ± 0. 17) ms-2 Answers to Example if length is measured with classical ruler, tape measure or meter stick: q g = (10. 98 ± 0. 18) m/s 2 q There is a systematic error q g = (9. 826 ± 0. 160) m/s 2 Note: the ruler uncertainty is ± 0. 5 mm or ± 0. 0005 m (four decimal places)