Error Analysis

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Error Analysis

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

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Review quiz

• 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
• no force on Kirk.
• a horizontal force directed forward on Kirk.
• a horizontal force directed to the left on Kirk.
• a horizontal force directed to the right on Kirk.
• a horizontal force in a direction between forward
  and left on Kirk.

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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 500PM.

This guy is responsible for calculating your mark!
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Test 1 Histogram
Percentage with A 19
Percentage with B 16
Percentage with C 28
Percentage with D 23
Percentage with F 14
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Two Kinds of Statements

• Exact
• 2 3 5 (math)
• K ½ m v2 (definition)
• Approximate
• Fspring k x (any physical law)
• g 9.80 m/s2 (all numerical measures of the
  universe)
• Today approximate statements

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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?
• No
• Yes, he should have counted Zero when he
  started the stopwatch.
• Yes, he should have started the stopwatch when it
  was at the bottom of its swing, not at the top.

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The t5 data

• 7.53 s
• 7.38 s
• 7.47 s
• 7.43 s

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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 s of
  the mean?
• 0
• 50
• 68
• 95
• 100

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Here were Harlows 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 Harlows first measurement of
  7.53 seconds?
• 0.005 s
• 0.05 s
• 0.5 s
• 5 s
• Impossible to determine

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7.53 s
7.38 s
7.47 s
7.43 s
11
7.53 s
7.38 s
7.47 s
7.43 s
7.44 s
7.56 s
7.48 s
7.40 s
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The Gaussian
68 of data between the dotted lines on the graph.
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Heights of some People(London, 1886)
inches
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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!
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The t5 data
0.06 s 0.06 s 0.06 s 0.06 s

• 7.53 s7.38 s7.47 s7.43 s

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Propagation of Errors
z A x
?z A ?x
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Repeated Measurements

• Repeated n times
• Each individual measurement has an error of
  precision Dx

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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.