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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x

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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

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sio

nle

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)

Jeff Hosea : University of Surrey, Physics Dept, Level 1 Labs, Oct 20072

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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

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x (dimensionless)

y (

dim

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sio

nle

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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

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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 ……

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y (

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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

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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

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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

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• 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

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• 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

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• 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

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• 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)

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• 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

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