Introduction
Example
When does a line cross an error bar?
Large and Small Scatter
Wilfrid Laurier University
Terry Sturtevant
Wilfrid Laurier University
April 30, 2015
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Introduction
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Introduction
When you have points on a linear graph, sometimes they will
line up pretty well.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Introduction
When you have points on a linear graph, sometimes they will
line up pretty well.
In this case, the uncertainties in the slope and y -intercept will
be determined by the error bars on the graph.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Least squares fit clearly crosses all error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Least squares fit clearly crosses all error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Least squares fit clearly crosses all error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Least squares fit maybe crosses all error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Least squares fit doesn’t cross all error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
Least squares fit definitely doesn’t cross all error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
When does a line cross an error bar?
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
y = mx + b
(xi , yi )
The line clearly crosses the vertical error bar.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
y = mx + b
(xi , yi )
Actually, the error bars form a box.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
∆yi
∆xi
This is the box close up.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
y = mx + b
(xi , yi )
The line goes through the box, even though it doesn’t cross either
bar.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
∆yi
∆xi
Here’s the box again.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
∆yi
∆xi
We can define it by diagonal lines.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
∆yi
slope is
∆yi
∆xi
∆xi
Here’s one.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
∆yi
When this doesn’t work
Another case where this doesn’t work
K
K
K
K
K
K
K
i
slope is − ∆y
∆xi
K
K
K
∆xi
K
K
Here’s the other.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
y = mx + b
(xi , yi )
A line only needs to cross either diagonal.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
The slope of the diagonal with the positive slope is
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
The slope of the diagonal with the positive slope is
m+i =
Terry Sturtevant
∆yi
∆xi
(1)
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
The slope of the diagonal with the positive slope is
m+i =
∆yi
∆xi
(1)
The y -intercept of the diagonal with the positive slope is
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
The slope of the diagonal with the positive slope is
m+i =
∆yi
∆xi
(1)
The y -intercept of the diagonal with the positive slope is
b+i = yi − m+i xi
Terry Sturtevant
(2)
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
The slope of the diagonal with the positive slope is
m+i =
∆yi
∆xi
(1)
The y -intercept of the diagonal with the positive slope is
b+i = yi − m+i xi
(2)
since, by definition, (xi , yi ) must be on the line.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Similarly, the slope of the diagonal with the negative slope is
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Similarly, the slope of the diagonal with the negative slope is
m−i = −
Terry Sturtevant
∆yi
∆xi
(3)
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Similarly, the slope of the diagonal with the negative slope is
m−i = −
∆yi
∆xi
(3)
The y -intercept of the diagonal with the negative slope is
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Similarly, the slope of the diagonal with the negative slope is
m−i = −
∆yi
∆xi
(3)
The y -intercept of the diagonal with the negative slope is
b−i = yi − m−i xi
Terry Sturtevant
(4)
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Similarly, the slope of the diagonal with the negative slope is
m−i = −
∆yi
∆xi
(3)
The y -intercept of the diagonal with the negative slope is
b−i = yi − m−i xi
(4)
since, by definition, (xi , yi ) must be on the line.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
For two lines y = m1 x + b1 and y = m2 x + b2 , they cross when
there is a single point (xc , yc ) that satisfies both equations.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Thus
yc = m1 xc + b1
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Thus
yc = m1 xc + b1
and
yc = m2 xc + b2
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Thus
yc = m1 xc + b1
and
yc = m2 xc + b2
Combining this gives
m1 xc + b1 = m2 xc + b2
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Thus
yc = m1 xc + b1
and
yc = m2 xc + b2
Combining this gives
m1 xc + b1 = m2 xc + b2
and so
xc =
Terry Sturtevant
b2 − b1
m1 − m2
(5)
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
To see where the best fit line y = mx + b crosses one of the
diagonal lines, we just use Equation 5 and so
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
To see where the best fit line y = mx + b crosses one of the
diagonal lines, we just use Equation 5 and so
xci+ =
Terry Sturtevant
b − b+i
m+i − m
(6)
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
To see where the best fit line y = mx + b crosses one of the
diagonal lines, we just use Equation 5 and so
xci+ =
b − b+i
m+i − m
(6)
xci− =
b − b−i
m−i − m
(7)
and
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
If either of xci+ or xci− is between xi + ∆xi and xi − ∆xi
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
If either of xci+ or xci− is between xi + ∆xi and xi − ∆xi then the
best fit line crosses the error bars for point i.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
If either of xci+ or xci− is between xi + ∆xi and xi − ∆xi then the
best fit line crosses the error bars for point i. If this is true for all
of the data points, then this is a case of small scatter.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
When this doesn’t work
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
When this doesn’t work
This tests whether a given line crosses all of the error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
When this doesn’t work
This tests whether a given line crosses all of the error bars. So, if
the least squares fit line crosses all of the error bars, then this test
is easy to use.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
When this doesn’t work
This tests whether a given line crosses all of the error bars. So, if
the least squares fit line crosses all of the error bars, then this test
is easy to use. It’s possible, though, that the least squares fit line
doesn’t cross all of the error bars, but another line can be drawn
which does.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
When this doesn’t work
This tests whether a given line crosses all of the error bars. So, if
the least squares fit line crosses all of the error bars, then this test
is easy to use. It’s possible, though, that the least squares fit line
doesn’t cross all of the error bars, but another line can be drawn
which does. Unless we can determine the equation of that line, we
can’t apply this test.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Another case where this doesn’t work
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Another case where this doesn’t work
If the x co-ordinate has no uncertainty, (i.e. there are only error
bars in the y direction), then this won’t work either.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
Another case where this doesn’t work
If the x co-ordinate has no uncertainty, (i.e. there are only error
bars in the y direction), then this won’t work either. However, in
that case, it easy to apply a similar test to see if the line crosses all
of the (vertical) error bars.
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
In this case, for each xi , it must be that
yi − ∆yi ≤ mxi + b ≤ yi + ∆yi
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University
Introduction
Example
When does a line cross an error bar?
When this doesn’t work
Another case where this doesn’t work
In this case, for each xi , it must be that
yi − ∆yi ≤ mxi + b ≤ yi + ∆yi
or
|yi − (mxi + b)| ≤ ∆yi
Terry Sturtevant
Large and Small Scatter Wilfrid Laurier University