




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.



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!

Percentage with A 19%
Percentage with B 16%
Percentage with C 28%
Percentage with D 23%
Percentage with F 14%

1. Exact


2 + 3 = 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

 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.

5
7.53 s
7.38 s
7.47 s
7.43 s

 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. 0.005 s
B. 0.05 s
C. 0.5 s
D. 5 s
E. Impossible to determine

Histogram: 4 Measurements
7
6
5
7.53 s
7.38 s
7.47 s
7.43 s
2
1
4
3
0
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)

Histogram: 8 Measurements
7
6
5
4
3
2
1
7.53 s
7.38 s
7.47 s
7.43 s
7.44 s
7.56 s
7.48 s
7.40 s
0
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)

Histogram: 12 Measurements
7
6
5
4
3
2
1
0
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)

Histogram: 16 Measurements
5
4
3
2
7
6
1
0
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)

Histogram: 16 Measurents
4
3
7
6
5
Gaussian Curve
(best fit)
Stopwatch
Measurements
2
1
0
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)

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

inches

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

5
7.53 s
7.38 s
7.47 s
7.43 s
+ 0.06 s
+ 0.06 s
+ 0.06 s
+ 0.06 s
Numerically:

z = A x Δz = A Δx

 Repeated n times
 Each individual measurement has an error of precision
D x

 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.