Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The.
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Level 1 Laboratories
Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007
Estimating Uncertaintiesin Simple Straight-Line Graphs
The Parallelogram & Related Methods
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0
5
10
15
20
0 5 10 15
x
y
Example of Parallelogram Method to obtain errors in gradient & intercept
Figure 1 : Give the graph a title to which you can refer ! & Add a descriptive caption. Blah, blah ……
(dimensionless)
(dim
en
sio
nle
ss
)
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20072
0
5
10
15
20
0 5 10 15
x (dimensionless)
y (
dim
en
sio
nle
ss
)
Estimate”best fit”
line
gradient m
[ e.g. herem = 0.59 ]
intercept c
[ e.g. herec = 6.5 ]
Example of Parallelogram Method to obtain errors in gradient & intercept
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20073
0
5
10
15
20
0 5 10 15
x (dimensionless)
y (
dim
en
sio
nle
ss
)
Draw 2 linesparallel to
best line soas to encloseroughly 2/3
of data points
Example of Parallelogram Method to obtain errors in gradient & intercept
Why 2/3? - It’s because we assume the data obey a Normal Distributionin which there is a 68.3% ( 66.7% = 2/3) confidence that the
“true” value lies within of the measured value
2 68.3% ofarea under
Normal curve
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20074
0
5
10
15
20
0 5 10 15
x (dimensionless)
Draw “extreme lines”
betweenopposite cornersof parallelogram
Max gradient mH
Min intercept
CL
Min gradient mL
Max intercept
CH
Example of Parallelogram Method to obtain errors in gradient & intercept
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20075
0
5
10
15
20
0 5 10 15
x (dimensionless)
Draw “extreme lines”
betweenopposite cornersof parallelogram
Max gradient mH
Min intercept
CL
Min gradient mL
Max intercept
CH
Final Results including errors
n
mmm LH
mmasgradientQuote :
n
ccc LH ccasinterceptQuote :
Example of Parallelogram Method to obtain errors in gradient & intercept
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20076
0
5
10
15
20
0 5 10 15
x (dimensionless)
y (
dim
en
sio
nle
ss
)
Add expt.“error bars”
Example of Parallelogram Method to obtain errors in gradient & intercept
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007
NB : these error bars are estimated from the scatter in the data.
Here, they play no part in getting the errors in the gradient and intercept.7
Recommended final appearance of graph for Diary or Reports, if using Parallelogram Method
Figure 1 : Descriptive caption. Blah, blah ……
0
5
10
15
20
0 5 10 15
x (dimensionless)
y (
dim
en
sio
nle
ss
)
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20078
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
x
y
9
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
• Draw the best fit line and determine equationyfit = m1x + c1
x
y
10
• Draw the best fit line and determine equationyfit = m1x + c1
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
x
y
0
• The small difference (ydata – yfit ) will be dominated by the random scatter
11
x
(yda
ta –
yfi
t )
0
• Replot (ydata – yfit ) on an expanded scale.
• Draw the best fit line and determine equationyfit = m1x + c1
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
0
• The small difference (ydata – yfit ) will be dominated by the random scatterx
y
12
• Draw the best fit line and determine equationyfit = m1x + c1
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
x
y
0
• The small difference (ydata – yfit ) will be dominated by the random scatter
x
(yda
ta –
yfi
t )
0
• Replot (ydata – yfit ) on an expanded scale.
• Fit the best line (ydata – yfit ) = m2x + c2
13
• Draw the best fit line and determine equationyfit = m1x + c1
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
x
y
0
• The small difference (ydata – yfit ) will be dominated by the random scatter
x
(yda
ta –
yfi
t )
0
• Replot (ydata – yfit ) on an expanded scale.
• Fit the best line (ydata – yfit ) = m2x + c2
• Form the parallelogram enclosing 2/3 of points
14
• Draw the best fit line and determine equationyfit = m1x + c1
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
x
y
0
• The small difference (ydata – yfit ) will be dominated by the random scatter
x
(yda
ta –
yfi
t )
0
• Replot (ydata – yfit ) on an expanded scale.
• Fit the best line (ydata – yfit ) = m2x + c2
• Use the extreme lines to find the max and min values of m2 and c2
• Form the parallelogram enclosing 2/3 of points
15
• Replot (ydata – yfit ) on an expanded scale.
• Fit the best line (ydata – yfit ) = m2x + c2
• Draw the best fit line and determine equationyfit = m1x + c1
What if the scatter is so small that it is difficult to draw the max. and min. lines by eye?
Method 2 : Modified Parallelogram Method
• Plot the experimental points (x, ydata)
x
y
0
• The small difference (ydata – yfit ) will be dominated by the random scatter
x
(yda
ta –
yfi
t )
0
• Use the extreme lines to find the max and min values of m2 and c2
• Form the parallelogram enclosing 2/3 of points
Final Results including errors
nmm
m LH 222
221: mmmasgradientQuote
ncc
c LH 222
221: cccasinterceptQuote
16
There is another method commonly used when the scatter is small
Method 3 : Using Predetermined Error Bars(e.g. by multiple measurements at a single value of x)
x
y
0
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 200717
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007
• Add predetermined error bars to the plotted points
x
y
0
There is another method commonly used when the scatter is small
Method 3 : Using Predetermined Error Bars(e.g. by multiple measurements at a single value of x)
18
• Add predetermined error bars to the plotted points
x
y
• Draw best line through points(if predetermined error is consistent with the scatter in the points, the line should go through ~2/3 of error bars & miss remaining ~1/3 )
0
There is another method commonly used when the scatter is small
Method 3 : Using Predetermined Error Bars(e.g. by multiple measurements at a single value of x)
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 200719
• Add predetermined error bars to the plotted points
x
y
• Put in extreme lines so as to still pass through ~2/3 of error bars
• Draw best line through points(if predetermined error is consistent with the scatter in the points, the line should go through ~2/3 of error bars & miss remaining ~1/3 )
0
There is another method commonly used when the scatter is small
Method 3 : Using Predetermined Error Bars(e.g. by multiple measurements at a single value of x)
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 200720
• Add predetermined error bars to the plotted points
• Put in extreme lines so as to still pass through ~2/3 of error bars
• Draw best line through points(if predetermined error is consistent with the scatter in the points, the line should go through ~2/3 of error bars & miss remaining ~1/3 )
There is another method commonly used when the scatter is small
Method 3 : Using Predetermined Error Bars(e.g. by multiple measurements at a single value of x)
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007
x
y
0
Final Results including errors
n
mmm LH
mmasgradientQuote :
n
ccc LH ccasinterceptQuote :
21
1. If the “scatter” in plotted data looks different to the size of error bars (much smaller or larger), something has gone wrong!
2. Example shows all y-axis error bars of same length. This might not be true in any given case, so do not assume this unless you have confirmed it!
3. Example also shows no error bars on the horizontal x-axis : there might be errors in this direction too!
Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 2007
Caution with Method 3
x
y
0
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