Error Analysis Repeating Measurements Calculation of Mean and Standard Deviation The Gaussian...

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 A 19%

Percentage with B 16%

Percentage with C 28%

Percentage with D 23%

Percentage with F 14%

Two Kinds of Statements1. Exact

2 + 3 = 5 (math) K = ½ m v2 (definition)

2. Approximate Fspring = –k x (any physical law) g = 9.80 m/s2 (all numerical measures

of the universe)

Today: approximate statements

Period of a Pendulum Procedure: Measure the time for 5

oscillations, t5. The period is calculated as T = t5 / 5. Did Harlow do anything wrong when

measuring t5?A. NoB. 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 t5 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 t5:7.53 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. 0.005 sB. 0.05 sC. 0.5 sD. 5 sE. Impossible to determine

Histogram: 4 Measurements

0

1

2

3

4

5

6

7

7.2 7.25 7.3 7.35 7.4 7.45 7.5 7.55 7.6 7.65 7.7

Measured Time (half second bins)

Nu

mb

er o

f M

easu

rem

ents

7.53 s

7.38 s

7.47 s

7.43 s

Histogram: 8 Measurements

0

1

2

3

4

5

6

7

7.2 7.25 7.3 7.35 7.4 7.45 7.5 7.55 7.6 7.65 7.7

Measured Time (half second bins)

Nu

mb

er o

f M

easu

rem

ents

7.53 s

7.38 s

7.47 s

7.43 s

7.44 s

7.56 s

7.48 s

7.40 s

Histogram: 12 Measurements

0

1

2

3

4

5

6

7

7.2 7.25 7.3 7.35 7.4 7.45 7.5 7.55 7.6 7.65 7.7

Measured Time (hafl second bins)

Nu

mb

er o

f M

easu

rem

ents

Histogram: 16 Measurements

0

1

2

3

4

5

6

7

7.2 7.25 7.3 7.35 7.4 7.45 7.5 7.55 7.6 7.65 7.7

Measured Time (half second bins)

Nu

mb

er o

f M

easu

rem

ents

Histogram: 16 Measurents

0

1

2

3

4

5

6

7

7.2 7.25 7.3 7.35 7.4 7.45 7.5 7.55 7.6 7.65 7.7

Measured Time (half second bins)

Nu

mb

er o

f M

easu

rem

ents

Gaussian Curve(best fit)

StopwatchMeasurements

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 t5 data

7.53 s7.38 s7.47 s7.43 s

+ 0.06 s+ 0.06 s+ 0.06 s+ 0.06 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 (7.056 +/- 0.705) m, it is better to report (7.1 +/- 0.7) m.