high school mathematics at the research frontier

High School Mathematics at the

Research Frontier

Don Lincoln

Fermilab

http://www-d0.fnal.gov/~lucifer/PowerPoint/HSMath.ppt

What is Particle Physics?

High Energy Particle Physics is a study of the smallest pieces of matter.

It investigates (among other things) the nature of the universe immediately after the Big Bang.

It also explores physics at temperatures not common for the past 15 billion years (or so).

It’s a lot of fun.

Now (15 billion years)

Stars form(1 billion years)

Atoms form (300,000 years)

Nuclei form (180 seconds)

??? (Before that)

4x10-12 seconds

Nucleons form (10-10 seconds)

DØ Detector: Run II

30’

30’

50’

• Weighs 5000 tons• Can inspect

3,000,000 collisions/second

• Will record 50 collisions/second

• Records approximately 10,000,000 bytes/second

• Will record 1015 (1,000,000,000,000,000) bytes in the next run (1 PetaByte).

Remarkable Photos

This collision is the most violentever recorded. It required thatparticles hit within 10-19 m or 1/10,000 the size of a proton

In this collision, a top and anti-top quark were created,helping establish their existence

How Do You Measure Energy?• Go to Walmart and buy an energy detector?

• Ask the guy sitting the next seat over and hope the teacher doesn’t notice?

• Ignore the problem and spend the day on the beach?

• Design and build your equipment and calibrate it yourself.

Build an Electronic Scale

150 lbs

?? Volts

Volts are a unit of electricity

Car battery = 12 VoltsWalkman battery = 1.5 Volts

Calibrating the Scale

120 lb girl = 9 V (120, 9)

180 lb guy = 12 V (150, 12)

Scale Calibration

0

5

10

15

0 50 100 150 200

Weight (lb)

Voltag

e (V

))

Make a line, solve slope and intercept

y = m x + b

Voltage = (0.05) weight + 3

Implies

Weight = 20 (Voltage – 3)

This implies that you can know the voltage for any weight.

For instance, a weight of 60 lbswill give a voltage of 6 V.

Now you have a calibrated scale.

(Or do you?)

Issues with calibrating.Scale Calibration

0

10

20

0 50 100 150 200

Weight (lb)

Volta

ge (V

))

FitValue at

60 lb

Purple 6

Blue 10

Red -70

Green 11.5

All four of these functions go through the two calibration points. Yet all give very different predictions for a weight of 60 lbs.

What can we do to resolve this?

Approach: Take More DataScale Calibration

0

10

20

0 50 100 150 200

Weight (lb)Vo

ltage

(V))

Easy

Scale Calibration

0

10

20

0 50 100 150 200

Weight (lb)

Volta

ge (V

))Hard

Solution: Pick Two Points

Scale Calibration

0

10

20

0 20 40 60 80 100 120 140 160 180 200

Weight (lb)

Voltag

e (V))

Dreadful representation of data

Solution: Pick Two Points

Scale Calibration

0

10

20

0 50 100 150 200

Weight (lb)

Volta

ge (V

))

Better, but still poor, representation of data

Why don’t all the data lie on a line?

• Error associated with each calibration point.

• Must account for that in data analysis.

• How do we determine errors?

• What if some points have larger errors than others? How do we deal with this?

First Retake Calibration Data• Remeasure the 120 lb point

• Note that the data doesn’t always repeat.

• You get voltages near the 9 Volt ideal, but with substantial variation.

• From this, estimate the error.

Attempt Voltage

1 9.26

2 9.35

3 9.08

4 8.72

5 8.58

6 9.02

7 9.25

8 8.86

9 8.94

10 9.12

11 8.72

12 9.33

DataCalibration Data for 120 lb

01

2

79

1618

1416

9

53

00

5

10

15

20

5 7 9 11 13Voltages

#

Bin Frequency6 06.5 17 27.5 78 98.5 169 189.5 1410 1610.5 911 511.5 312 0

More 0

While the data clusters around 9 volts, it has a range. How we estimate the error is somewhat technical, but we can say

9 1 Volts

Redo for All Calibration Points

Weight Voltage

60 4.2 0.5

120 9.4 1.0

150 10 0.7

180 13.2 1.2

300 13.2 8.4

Redo for All Calibration Points

Weight Voltage

60 4.2 0.5

120 9.4 1.0

150 10 0.7

180 13.2 1.2

300 13.2 8.4

Scale Calibration Data

0510152025

0 100 200 300 400

Weight

Vol

tage

0

10

20

30

0 100 200 300 400

Weight

Volta

geScale Calibration Data

0

10

20

30

0 50 100 150 200 250 300 350

Volta

ge


0510152025

0 100 200 300 400

Weight

Volta

ge

Both lines go through the data.

How to pick the best one?

State the Problem• How to use mathematical techniques to

determine which line is best?

• How to estimate the amount of variability allowed in the found slope and intercept that will also allow for a reasonable fit?

• Answer will be m m and b b

The Problem

• Given a set of five data points, denoted (xi,yi,i) [i.e. weight, voltage, uncertainty in voltage]

• Also given a fit function f(xi) = m xi + b

• Define

5

12

2

25

255

21

2112

)]([

)]([)]([

i i

ii xfy

xfyxfy

Looks Intimidating!

Forget the math, what does it mean?


051015202530

0 100 200 300 400

Weight

Volta

ge


10

12

14

16

18

170 175 180 185 190

Weight

Volta

gef(xi)

xi

yi

yi - f(xi) i

2

2)]([

i

ii xfy

Each term in the sum is simply the separation between the data and fit in units of error bars. In this case, the separation is about 3.

More TranslationSo

Means

2

2

2

2

2

2

bars)error in 5 separation(





Since f(xi) = m xi + b, find m and b that minimizes the 2.

25

255

24

244

23

233

22

222

21

2112

)]([

)]([

)]([

)]([

)]([

bmxy

bmxy

bmxy

bmxy

bmxy

5

12

22 )]([

i i

ii xfy

Approach

5

12

22 ][

i i

ii bmxy

Find m and b that minimizes 2

Calculus

5

12

2

5

12

2

)1]([20

)]([20

i i

ii

i i

iii

bmxy

b

xbmxy

m

Back to algebra

Note the common term (-2). Factor it out.

Approach #2

5

12

5

12

][0

)]([0

i i

ii

i i

iii

bmxy

xbmxy

Now distribute the terms

5

12

5

12

2

][0

][0

i i

ii

i i

iiii

bmxy

bxmxxy

Rewrite as separate sums

5

12

5

12

5

12

5

12

5

12

25

12

0

0

i ii i

i

i i

i

i i

i

i i

i

i i

ii

bmxy

bxmxxy

Move terms to LHS

5

12

5

12

5

12

5

12

5

12

25

12

i ii i

i

i i

i

i i

i

i i

i

i i

ii

bmxy

bxmxxy

Factor out m and b terms

5

12

5

12

5

12

5

12

5

12

25

12

1

i ii i

i

i i

i

i i

i

i i

i

i i

ii

bx

my

xb

xm

xy

5

12

5

12

5

12

5

12

5

12

25

12

1

i ii i

i

i i

i

i i

i

i i

i

i i

ii

bx

my

xb

xm

xy

Approach #3

Notice that this is simply two equations with two unknowns. Very similar to

yExDF

yBxAC

You know how to solve this

BDAE

CDFAy

BDAE

BFCEx

,

]][[]1

][[

]][[]][[

]][[]1

][[

]][[]1

][[

5

12

5

12

5

12

5

12

2

5

12

5

12

5

12

25

12

5

12

5

12

5

12

5

12

2

5

12

5

12

5

12

5

12

i i

i

i i

i

i ii i

i

i i

i

i i

ii

i i

i

i i

i

i i

i

i i

i

i ii i

i

i i

i

i i

i

i ii i

ii

xxx

xxyxy

b

xxx

yxxy

m

Substitution

Note the common term in the denominator

5

12

5

12

5

12

5

12

5

12

25

12

1

i ii i

i

i i

i

i i

i

i i

i

i i

ii

bx

my

xb

xm

xy

ohmigod….

yougottabekiddingme

i x i y i

i x i y i /( i )2

1 60 4.2 0.5 1008.02 120 9.4 1 1128.03 150 10 0.7 3061.24 180 13.2 1.2 1650.05 300 19.8 8.4 84.2

6931.4

4.6931

2.8416502.306111281008

25

5524

4423

3322

2221

115

12

xyxyxyxyxyxy

i i

ii

So each numberisn’t bad

Approach #4

25

12

5

12

5

12

2

5

12

5

12

5

12

25

12

25

12

5

12

5

12

2

5

12

5

12

5

12

5

12

][]1

][[

]][[]][[

][]1

][[

]][[]1

][[

i i

i

i ii i

i

i i

i

i i

ii

i i

i

i i

i

i i

i

i ii i

i

i i

i

i i

i

i ii i

ii

xx

xxyxy

b

xx

yxxy

m

Inserting and evaluating, we get

m = 0.068781, b = 0.161967

What about significant figures?

2

2

2

2

2

2






0.8178

0.300145

0.468521

0.996884

031568.0

2

2nd and 5th terms give biggest contribution to 2 = 2.587

Best FitScale Calibration Data

0

10

20

30

0 100 200 300 400

Weight

Volta

ge

Best vs. Good

Best

Doesn’t always

mean good

Goodness of Fit


0

10

20

30

0 100 200 300 400

Weight

Volta

ge

Our old buddy, in which the data and the fit seem to agree


012345

0 100 200 300 400Weight

Vo

ltag

e

A new hypothetical set of data with the best line (as determined by the same 2 method) overlaid

New Important Concept

• If you have 2 data points and a polynomial of order 1 (line, parameters m & b), then your line will exactly go through your data

• If you have 3 data points and a polynomial of order 2 (parabola, parameters A, B & C), then your curve will exactly go through your data

• To actually test your fit, you need more data than the curve can naturally accommodate.

• This is the so-called degrees of freedom.

Degrees of Freedom (dof )

• The dof of any problem is defined to be the number of data points minus the number of parameters.

• In our case,

• dof = 5 – 2 = 3

• Need to define the 2/dof

Goodness of Fit


0

10

20

30

0 100 200 300 400

Weight

Volta

geScale Calibration Data

012345

0 100 200 300 400Weight

Vo

ltag

e

2/dof = 2.587/(5-2) = 0.862

2/dof = 22.52/(5-2) = 7.51

2/dof near 1 means the fit is good.

Too high bad fitToo small errors were over estimated

Can calculate probability that data is represented by the given fit. In this case:

Top: < 0.1%

Bottom: 68%

In the interests of time, we will skip how to do this.

2 Distribution for b

2

2.5

3

3.5

4

4.5

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

b

2

The error in b is indicated by the spot at which the 2 is changed by 1.

Uncertainty in m and b #1Recall that we found

m = 0.068781, b = 0.161967

What about uncertainty and significant figures?

If we take the derived value for one variable (say m), we can derive the 2 function for the other variable (b).

CBxAx

mxymxybb

bmxy

i i

ii

i i

ii

i i

i i

ii

2

5

12

25

12

5

12

2

5

12

22

~

)()()2(

1

])[(

So 0.35

2 Distribution for m

2

2.5

3

3.5

4

0.064 0.066 0.068 0.07 0.072 0.074m

2

Uncertainty in m and b #2Recall that we found

m = 0.068781, b = 0.161967

What about uncertainty and significant figures?

If we take the derived value for one variable (say b), we can derive the 2 function for the other variable (m).

CBxAx

bybyxm

xm

mxby

i i

i

i i

ii

i i

i

i i

ii

2

5

12

25

12

5

12

22

5

12

22

~

)()()2(

])[(

The error in m is indicated by the spot at which the 2 is changed by 1.

So 0.003

Uncertainty in m and b #3So now we know a lot of the story

m = 0.068781 0.003b = 0.161967 0.35

So we see that significant figures are an issue.

Finally we can see

Voltage = (0.069 0.003) × Weight + (0.16 0.35)


0

10

20

30

40

0 100 200 300 400

Weight

Volta

ge

Final complication: When we evaluated the error for m and b, we treated the other variable as constant. As we know, this wasn’t correct.

Error Ellipse

xyFyExDCyBxA

mbFbEmDCbBmA

xmb

yb

yxmb

xm

y

bmxy

i i i i i i

i

i

i

i i

ii

ii

i

i

i

i i

ii

22

22

5

1

5

1

5

1

5

1

5

122

5

122

22

22

2

2

5

12

22

2221

][

m

bBest b & m

2min + 1

2min + 2

2min + 3

More complicated, but shows that uncertainty in one variable also affects the uncertainty seen in another variable.


0510152025

0 100 200 300 400

Weight

Volta

ge

Increase intercept, keep slo

pe the same

To remain ‘good’, if you increase the intercept, you must decrease the slope

Increase intercept, keep slo

pe the same


0510152025

0 100 200 300 400

Weight

Volta

ge

Decrease slope, keep intercept the same

Similarly, if you decrease the slope, you must increase the intercept

Error Ellipse

m

b

Best b & m

mbest

bbest

new m

within errors

new b

within errors

When one has an m below mbest, the range of preferred b’s tends to be above bbest.

From both physical principles and strict mathematics, you can see that if you make a mistake estimating one parameter, the other must move to compensate. In this case, they are anti-correlated (i.e. if b, then m and if b, then m.)

Back to Physics

Data and error analysis is crucial, whether you work in a high school lab…

Or the Frontier!!!!

http://hubblesite.org/newscenter/newsdesk/archive/releases/1996/01/image/a

http://hubblesite.org/newscenter/newsdesk/archive/releases/1996/01/image/a

References• P. Bevington and D. Robinson, Data Reduction

and Error Analysis for the Physical Sciences, 2nd Edition, McGraw-Hill, Inc. New York, 1992.

• J. Taylor, An Introduction to Error Analysis, Oxford University Press, 1982.

• Rotated ellipses – http://www.mecca.org/~halfacre/MATH/rotation.htm

http://www-d0.fnal.gov/~lucifer/PowerPoint/HSMath.ppt

http://worldscientific.com/books/physics/5430.html

high school mathematics at the research frontier

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