Error Analysis Repeating Measurements Calculation of Mean and Standard Deviation The Gaussian distribution Propagation of Errors Significant Figures.

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Presentation transcript:

Error Analysis Repeating Measurements Calculation of Mean and Standard Deviation The Gaussian distribution Propagation of Errors Significant Figures

Kirk is sitting in the right-hand passenger seat of a car. The car makes a right-hand turn at constant speed. If Kirk stays in his seat as the car turns, there is A.no force on Kirk. B.a horizontal force directed forward on Kirk. C.a horizontal force directed to the left on Kirk. D.a horizontal force directed to the right on Kirk. E.a horizontal force in a direction between forward and left on Kirk. Review quiz

About the Mechanics Test Test average was 64%. You will receive your marked test near the END of tutorial this week. If you find a mistake in the marking you must notify Dr. Savaria in MP129 before next Friday, November 16 by 5:00PM. This guy is responsible for calculating your mark!

Test 1 Histogram Percentage with A19% Percentage with B16% Percentage with C28% Percentage with D23% Percentage with F14%

Two Kinds of Statements 1.Exact  = 5 (math)  K = ½ m v 2 (definition) 2.Approximate  F spring = –k x (any physical law)  g = 9.80 m/s 2 (all numerical measures of the universe) Today: approximate statements

Period of a Pendulum Procedure: Measure the time for 5 oscillations, t 5. The period is calculated as T = t 5 / 5. Did Harlow do anything wrong when measuring t 5 ? A.No B.Yes, he should have counted “Zero” when he started the stopwatch. C.Yes, he should have started the stopwatch when it was at the bottom of its swing, not at the top.

The t 5 data 7.53 s 7.38 s 7.47 s 7.43 s

Repeated Measurements of Period Consider a single measurement, in a group of measurements that follow a normal distribution. What is the probability that this measurement lies within + or – one standard deviation σ of the mean? A.0% B.50% C.68% D.95% E.100%

Here were Harlow’s measurements of t 5 : 7.53 s 7.38 s 7.47 s 7.43 s Which of the following might be a good estimate for the error in Harlow’s first measurement of 7.53 seconds? A s B.0.05 s C.0.5 s D.5 s E.Impossible to determine

7.53 s 7.38 s 7.47 s 7.43 s

7.53 s 7.38 s 7.47 s 7.43 s 7.44 s 7.56 s 7.48 s 7.40 s

The Gaussian 68% of data between the dotted lines on the graph.

Heights of some People (London, 1886) inches

Random Walk Where does an object end up, if it takes N steps randomly left or right? The final distribution is described by a Gaussian function!

The t 5 data 7.53 s 7.38 s 7.47 s 7.43 s s s Numerically:

Propagation of Errors z = A x Δz = A Δx

Repeated Measurements Repeated n times Each individual measurement has an error of precision  x

Significant Figures Discussed in Section 1.9 of Knight Ch.1 Rules for significant figures follow from error propagation  Assume error in a quoted value is half the value of the last digit.  Errors should be quoted to 1 or 2 significant figures  Error should be in final displayed digit in number. Example: If a calculated result is ( / ) m, it is better to report (7.1 +/- 0.7) m.