The Nature of Scientific Progress,
More Error Analysis,
Exp #1
Session 3
Physics 2BL
Summer 2009
Outline
• How scientific knowledge progresses:
– Replacing models
– Restricting models
• What you should know about error analysis
(so far) and more
• Limiting Gaussian distribution
• More on Exp. 1
• Reminder
Models
• Invented
• Properties correspond closely to real world
• Must be testable
How Models Fit Into Process of
Doing Science
• Science is a process that studies the world by:
–
–
–
–
–
–
Limiting the focus to a specific topic (making a choice)
Observing (making a measurement)
Refining Intuitions (making sense) Creating
Extending (seeking implications) Predicting
Demanding consistency (making it fit) Refining or Replacing
Community evaluation and critique
• Start with simple model
How Models Change
• If models disagree with observation,
we change the model
– Refine - add to existing structure
– Restrict - limit scope of utility
– Replace - start over
Refining
• Original model consistent with
observations, but not complete
• Extend model to account for new
observations
• May include new concepts
e.g. Model of interaction between charged
objects; to include interactions between
charged & uncharged add concept of
induced charge
Restriction
• New model correct in situations where
old isn’t
• New model agrees w/ old over some
range
Ö Old still useful in limited range
e.g. General relativity vs. classical
gravitational theory
Replacement
• Old model can’t be extended
consistently
• Replace entire model
Ö Earlier observations provide limits for
new model
e.g. Geocentric vs heliocentric models
for solar system
Random and independent?
•
•
•
o
o
Yes
Estimating between
marks on ruler or meter
Releasing object from
‘rest’
Mechanical vibration
Judgment
Problems of definition
•
•
•
o
o
No
End of ruler screwy
Reading meter from
the side (speedometer
effect)
Scale not zeroed
Reaction time delay
Calibration
Zero
Random & independent errors:
q = Bx
q = x+ y−z
δq = (δx) + (δy ) + (δz )
2
2
δq = B δx
2
δq
q
q = x× y ÷ z
δx
x
q = q ( x, y , z )
2
δq
⎛ δx ⎞ ⎛ δy ⎞ ⎛ δz ⎞
= ⎜ ⎟ + ⎜⎜ ⎟⎟ + ⎜ ⎟
q
⎝ x⎠ ⎝ y⎠ ⎝ z⎠
2
=
2
2
⎛ ∂q ⎞ ⎛ ∂q ⎞ ⎛ ∂q ⎞
δq = ⎜ δx ⎟ + ⎜⎜ δy ⎟⎟ + ⎜ δz ⎟
⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎝ ∂z ⎠
2
2
Propagation in formulas
Independent
Propagate error in steps
For example:
x
q=
y−z
• Then
• First find
p= y−z
δp = (δy ) 2 + (δz ) 2
x
q=
p
δq
⎛ δx ⎞ ⎛ δp ⎞
= ⎜ ⎟ + ⎜⎜ ⎟⎟
q
⎝ x⎠ ⎝ p⎠
2
2
An Important Simplifying Point
1 2
h = gt
2
g = 2h t 2 , δh h = 5%, δt / t = 0.1% Requires random & ind. errors!
δg
⎛ δh ⎞ ⎛ δt ⎞
= ⎜ ⎟ + ⎜2 ⎟
g
⎝ h⎠ ⎝ t ⎠
2
2
δg g = 5% 2 + (2 × 0.1% )2
δg g = 0.050039984 = 5%
ÖSimplifies calc.
ÖSuggests improvements
in experiment
• Often the error is
dominated by error in
least accurate
measurement
More Complicated Example
What is error in
q = Bx – yn
Start with
δq2 = (δ(Βx))2 + (δ(yn))2
δ(Bx) = B δ(x)
Example continued
δ(yn) = ε(yn) * yn
= n ε(y) yn = n yn-1 δy
Then,
δq2 = (B δx)2 – (n yn-1 δy)2
From Yagil
From Yagil
From Yagil
Analyzing Multiple Measurements
• Repeat measurement of x many times
• Best estimate of x is average (mean)
x1 , x2 ,⋅ ⋅ ⋅, x N
xbest
x1 + x2 + ⋅ ⋅ ⋅ + x N
=x=
N
∑ xi
x=
N
Repeated Measurements
Number measurements
16
14
12
10
8
6
4
2
0
69.4
69.6
69.8
70.0
70.2
height (inches)
70.4
• If errors are random
and independent:
– Expect most values
near true value
– Expect few values far
from true value
ÖAssume values are
distributed normally
Number measurements
How are Measured Values
Distributed?
69.88
4
3
70.04
69.72
2
1
0
69.4
69.6
69.8
70.0
70.2
height (inches)
1
G X ,σ ( x) =
exp(− ( x − x) 2 2σ 2 )
σ 2π
70.4
Normal Distribution
Number measurements
x
69.88
4
x−σ
x+σ
3
70.04
69.72
2
1
0
69.4
69.6
69.8
70.0
70.2
70.4
height (inches)
1
2
2
G X ,σ ( x) =
exp(− ( x − x) 2σ )
σ 2π
Error of an Individual Measurement
• How precise are
measurements of x?
• Start with each value’s
deviations from mean
• Deviations average to
zero, so square, then
average, then take
square root
• ~68% of time, xi will
be w/in σx of true value
d i ≡ xi − x
d =0
σx ≡
=
(d i )2
1 N
2
( xi − x )
∑
N − 1 i =1
Take σx as error in
individual measurement called standard deviation
Number measurements
Standard Deviation
69.88
4
3
70.04
69.72
2
1
0
69.4
69.6
69.8
70.0
70.2
70.4
height (inches)
Measurement
16
12
8
4
0
69.4
69.6
69.8
70.0
height (inches)
70.2
70.4
Error of the Mean
• Expect error of mean
to be lower than error
of the measurements
it’s calculated from
• Divide SD by square
root of number of
measurements
• Decreases slowly with
more measurements
Standard Deviation of
the Mean (SDOM)
or
Standard Error
or
Standard Error of the
Mean
σx = σx
N
Summary
∑ xi
x=
N
• Average
• Standard deviation
σ =
x
1 N
2
(
x
−
x
)
∑
i
N − 1 i =1
• Standard deviation of the mean
σ x= σ x / N
Lab Objectives, This Week
• Finish with analyses
– Estimate radius of earth using ∆t between sunsets at sea
level and on cliff
– Determine gravitational constant g.
– Use established models for earth and gravitational
attraction
• Physics
– Pendulum
– Measurement
• More complicated analysis
Eratosthenes
From Yagil
From Yagil
From Yagil
From Yagil
From Yagil
From Yagil
Remember
•
•
•
•
Experiment writeups due this time
Prelab questions due next time
Read Taylor through Chapter 5
Homework for lab 2 (Problems 4.6, 4.14,
5.2, 5.36)