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.