Lecture 4: Fitting Straight Lines by the
Least-Squares Method
M.Shattuck,
CUNY,
Ideal Particle
Image
Physics 3719
Spring Semester 2011
Problem of Straight-Line Fitting
●
We have discussed and seen examples of
linearization of data, in which we recast
mathematical relations between variables
in straight-line form
–
Easy to recognize by eye
–
Easy to fit
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Transform to Linear: Basic Idea
x = independent variable
y = dependent variable
slope
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intercept
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Transform to Linear: Example
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Problem of Straight-Line Fitting
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Remaining problem: I have a set of points
(xi,yi ± i)
which I expect to follow a linear
relationship
y = A + Bx
What is my best estimate for the
coefficients A, B?
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st
1 : Mechanical Analog
●
Suppose we have a set of
points (xi,yi).
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st
1 : Mechanical Analog
●
●
Suppose we have a set of
points (xi,yi).
To each point, connect a
spring. (Imagine spring
can only move in y
direction.)
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st
1 : Mechanical Analog
●
●
●
Suppose we have a set of
points (xi,yi).
To each point, connect a
spring. (Imagine spring
can only move in y
direction.)
To the other end of the
springs attach a rigid
rod.
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st
1 : Mechanical Analog
●
●
●
●
Suppose we have a set of
points (xi,yi).
To each point, connect a
spring. (Imagine spring
can only move in y
direction.)
To the other end of the
springs attach a rigid
rod.
What will determine
where rod comes to rest?
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Equilibrium position of rod will
minimize potential energy.
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define
where
In terms of stretch of string
then
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Discussion
●
●
To what does the spring constant k
correspond in our linear fitting problem?
Must the ki all be the same?
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replace
thus replace
With the “chi-squared”
function:
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Minimization of 
2
Solve two equations in two unknowns
See handouts for results.
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Application left
for homework
exercise!
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Accuracy and Precision/
Statistical and Systematic
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Uncertainties in Experimental Physics
●
●
Freshman physics:
–
90% of time is spent measuring
central value
–
Compare to “accepted value”
–
Tack on experimental uncertainty as an
afterthought, if at all.
Undergrad Lab (and the rest of your careers):
–
Spend most of your time understanding errors
–
Before you even touch equipment!
–
Is experiment worth doing? Will someone fund it?
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Terminology...
●
●
Accuracy: How
close to True
value?
Precision: What is
the spread you'd
see in repeated
measurements?
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More
terminology...
●
●
●
Random or Statistical
Error: Can be reduced
by taking more
measurements
Systematic Error:
Cannot be reduced by
taking more
measurements
Q: When is an
experiment
“finished”?
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