level 1 laboratories jeff hosea, university of surrey, physics dept, level 1 labs, oct 2007...
TRANSCRIPT
![Page 1: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/1.jpg)
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
1
![Page 2: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/2.jpg)
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
![Page 3: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/3.jpg)
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
![Page 4: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/4.jpg)
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
![Page 5: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/5.jpg)
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
![Page 6: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/6.jpg)
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
![Page 7: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/7.jpg)
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
![Page 8: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/8.jpg)
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
![Page 9: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/9.jpg)
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
![Page 10: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/10.jpg)
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
![Page 11: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/11.jpg)
• 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
![Page 12: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/12.jpg)
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
![Page 13: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/13.jpg)
• 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
![Page 14: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/14.jpg)
• 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
![Page 15: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/15.jpg)
• 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
![Page 16: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/16.jpg)
• 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
![Page 17: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/17.jpg)
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
![Page 18: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/18.jpg)
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
![Page 19: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/19.jpg)
• 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
![Page 20: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/20.jpg)
• 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
![Page 21: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/21.jpg)
• 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
![Page 22: Level 1 Laboratories Jeff Hosea, University of Surrey, Physics Dept, Level 1 Labs, Oct 2007 Estimating Uncertainties in Simple Straight-Line Graphs The](https://reader036.vdocuments.us/reader036/viewer/2022062301/5697bf721a28abf838c7eafd/html5/thumbnails/22.jpg)
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
22