PHYSICS 2150 �Experimental Modern Physics–Fall2011

Lecturer : Minhyea Lee Lab Instructors:

Minhyea Lee (tue 10am,thu1pm) Youngwoo Yi (tue 1pm, thu 10am)

Lab Coordinators: Jerry Leigh, Scott Pinegar

Lecture 2 �August 30, 2011 �

** Need to complete the Radiation Certification by this Friday 4pm (see me if having problems)

-last slide from Lec1. UNCERTAINTY •  As used by physicists, “error” is a synonym of “uncertainty.” It is

distinct from “discrepancy” or “mistake.” •  A result is meaningless without an uncertainty. ALL results should

be quoted with an error! •  The uncertainty can result from inaccurate equipment, limited

statistics, or other factors beyond your control •  Uncertainties should have 1-2 significant digits (generally 1 if the

first digit is 4 or more). The measurement result (“central value”) should have the same final digit as the uncertainty:

GOOD � BAD�1.41 ± 0.07� 1.408 ± 0.07�

6.7 ± 1.3� 6.7 ± 1�

0.1006 ± 0.0022� 0.1006 ± 0.00225�

PHYS2150 Lecture 2

•  The Gaussian distribution – Chapter 5 in Taylor –  What it looks like –  Where and why it shows up –  Mean, sigma, and all that

•  Statistical and systematic error and error propagation •  Examples : e/m measurement

Uncertainty in Scientific Measurements:

Let’s Measure the height of a doorway

-- Estimate : 210 cm -- Tape measure : 211.3cm -- Laser measure : 211.3065 cm

What is the exact height? What does that eve mean? No physical quantity can be measure with absolute certainty!!

Uncertainty in Scientific Measurements:

Is King’s crown made of gold (ρg = 15.5g/cm3) or alloy(ρg = 13.8g/cm3) ?

-- George’s measurement : “ρ = 15.0 and probably btw 13.5 and 16.5 g/cm3

-- Anne’s measurement : “ρ = 13.9 and probably btw 13.7 and 14.1 g/cm3

ρ(g/cm3) 13 14 15 16 17 Alloy Gold

George Anne

  Both may be reasonable measurements.   George is less precise, can’t conclude. If no uncertainty, it could be gold!   Anne must justify her range for us to believe her conclusion of alloy!

Uncertainty in Scientific Measurements:

3 4 5 6 Bovine Circumference (m)

Cow

s/0.2

m

µ

σ

✷Measure circumference of 150 cows.

✷The values appears to be centered on the average value (mean, µ) .

✷The values appears to occur more frequently near the average and less frequently further away (symmetry distribution) .

✷Mean (µ) and width (standard deviation, σ) describes distribution!

GAUSSIAN DISTRIBUTION

•  Shows up just about everywhere •  Synonyms: Normal Distribution, Bell

Curve •  Most basic form is “unit Gaussian”:

centered at zero, unit integral, unit σ:

•  This is the probability density for a continuous, normally distributed random variable with mean zero and standard deviation of 1.

F(x)

x

With any probability density function,

!

!

F(x) =12"exp # x

2

2$

% &

'

( )

!

F(x)dx =1"#

+#

$

F(x)

x

GAUSSIAN DISTRIBUTION

•  What can you do to the unit gaussian, and still keep it a gaussian?

–  Change its mean from zero to µ –  Change its width from 1 to σ (while

increasing height by 1/σ)

–  Change its integral — but then it’s not a normalized probability distribution anymore. So this isn’t allowed here.

As with any probability density function, this still integrates to 1.

!

F(x) =1

" 2#exp $ (x $µ)2

2" 2

%

& '

(

) *

MEAN, SIGMA, AND ALL THAT

•  Let’s go back to our Cow histogram : Circumference of 150 cows

•  Calculate the mean circumference

•  Standard deviation is calculated by:

3 4 5 6 Bovine Circumference (m)

Cow

s/0.2

m

• Read up in Taylor on definitions and uses of variance, standard deviation and standard deviation on mean.

µ=4.39 m, σ=0.70 m

Circumference of Cow data

•  Mean: 4.39 m, STD: 0.70 m

•  68% of cows have circumference between (4.39−0.70) and (4.39+0.70) m. This is because the integral of the (unit) gaussian from µ−σ to µ +σ is 0.68. (68% of confidence level)

•  How well do we know the mean circumference? Need std. dev. on the mean:

3 4 5 6 Bovine Circumference (m)

Cow

s/0.2

m

Note that we know the mean to much better than σ of individual measurements.

So

WHERE IT SHOWS UP

•  If you don’t have a clue what the probability distribution of a random quantity is (say, the circumferences of cows at the Hbar Ranch), it’s highly likely to be approximately gaussian!

•  Central Limit Theorem: a sum of a large enough number of random numbers has a gaussian distribution, no matter what the initial distribution shapes might have been.

•  Aside: Distribution of counts of a process with a uniform rate in a finite amount of time is Poisson-distributed (see a later lecture) but is approximately gaussian in the high-number limit. This is another example of the Central Limit Theorem.

•  If there is no systematic bias (more on that later), then the mean of measurements of a quantity is the best estimate of its true value.

USING THE GAUSSIAN

•  Can use the same mathematics to describe the results of repeated measurements of the same quantity, where there is random error/resolution in the instrument.

•  The distribution will be centered on a mean (assume for now that this is the correct value)

•  The distribution will have a standard deviation

•  Can fit this to a gaussian (Lecture 4,5) or just calculate mean, sigma directly

•  Uncertainty on the mean is now

•  Note: More measurements → smaller error on the mean! Also means better determination of error.

μ

σ

!

" µ ="

N

WHAT DOES SIGMA MEAN?

•  First if the measurements are from a gaussian Fµ,σ(x), then the probability of measuring a value in the range (a,b) is

•  For a normalized gaussian,

•  The general integral can’t be expressed analytically. Use error function (erf(x)) tables for values other than 1σ.

•  So, saying “J=5.4±0.9” means one can say the true value of J is between 4.5 and 6.3 with 68% confidence level.

Looking up Probabilities - Page 286/287 in Taylor’s Book

Loo

king

up

Prob

abili

ties

-

Page

286

/287

in T

aylo

r’s

Book

So far we discuss “statistical uncertainty”. Now what about uncertainty involved with

measurement itself ? (Systematic Uncertainty)

1σ error on mean

ANALYZING ERROR ON A QUANTITY •  You are in a car on a bumpy road on a rainy day, and are trying to measure the

length of the moving windshield wiper with a shaky ruler.

•  You measure it 37 times.

•  The histogram of your results is at right. It doesn’t look very gaussian. But, with only 37 measurements plotted in lots of bins, distributions often look ratty.

•  You calculate the mean to be 56.2 cm, and the standard deviation to be 6.8 cm.

• The 1σ uncertainty on the mean is 1.1 cm.

• If we assume the underlying distribution is nevertheless gaussian and centered on the true value, we can turn this into a confidence level: the wiper length is between 55.1 and 57.3 cm with 68% confidence.

1σ error on mean

ANALYZING ERROR ON A QUANTITY

•  Stop the car, go outside and measure the wiper properly: it is 61.0 cm long!

•  We said the wiper length was between 55.1 and 57.3 cm with 68% confidence.

•  We are off by over 4 sigma. This is shockingly unlikely!

•  Clearly there is a systematic shift. The distribution does not center on the true value.

•  More data points won’t get us any closer to the true value. We need to make better measurements, or find the source of the error and apply a correction to the data.

true

STATISTICAL (RANDOM) vs. SYSTEMATIC UNCERTAINTIES

•  Keep statistical, systematic errors separate. Report results as something like:

g = [965 ± 30(stat) ± 12(syst)] cm/s2 •  Add in quadrature (note that this assumes gaussian distribution) to compare

with known values: g = [965 ± 32(total)] cm/s2

STATISTICAL SYSTEMATIC

NO PREFERRED DIRECTION BIAS ON THE MEASUREMENT: ONLY ONE

DIRECTION (THOUGH OFTEN DON’T KNOW WHICH)

CHANGES WITH EACH DATA POINT: TAKNG MORE DATA REDUCES ERROR ON THE MEAN

STAYS THE SAME FOR EACH MEASUREMENT: MORE DATA WON’T HELP YOU!

GAUSSIAN MODEL IS USUALLY GOOD (EXCEPT COUNTING EXPERIMENTS WITH FEW EVENTS)

GAUSSIAN MODEL IS USUALLY TERRIBLE. BUT WE USE IT ANYWAY IF DON’T HAVE A BETTER

MODEL.

PROPAGATION OF ERRORS

•  Often, we aren’t measuring directly the quantity our experiment is after: we measure some lab quantities and our final “physics result” is a function of them. –  Kaon experiment: we measure curvature of tracks, and from

them calculate the momentum of the pions, and then calculate the mass of the kaon from that.

•  We know the errors on the “lab quantities.” How do we find the error on the final “physics result?”

•  This is a specific case of the general problem of finding the error on a quantity that is a function of random (uncertain) variables.

PROPAGATION OF ERRORS The general formula for errors on a function f(x,y,z),

Addition : if (q=x+y) or (q=x-y), The systematic error in q is

x,y,z.. “random variables”

Multiplication: if q=xyz,

Thus, fractional error is

Power:if q=x2y/z4, fractional error is

COMPARING WITH ‘KNOWN’ VALUE:

•  Measure: g = [965 ± 32] cm/s2 = x±δx

•  Known value: 980.665 cm/s2 = x0±δx0

•  Discrepancy is (x-x0)±δ(x-x0), where

–  [δ(x-x0)]2 = (δx)2 + (δx0)2 (add in quadrature)

–  Discrepancy in units of sigma (often called significance of discrepancy) is

•  Discrepancy here is (16 ± 32) cm/s2, or 0.5σ.

•  Use erf table to determine agreement confidence level: 62% agreement: good!

usually negligible

Example: e/m Experiment

!

• Electrons accelerated with 40, 60, or 80 V• Acted on by magnetic field perpendicular to velocity

• ! F = e! v "

! B forces electrons into circular paths

• Measuring radius of circle give centripetal force by F =mv 2

r

• Energy of electrons given by accelerating potential 12

mv 2 = eV

• Magnetic field from Helmholtz coils is B =8µ0NIa 125

• After putting together, em

= 3.906 Va2

µ02N 2I2r2

e/m Experiment

!

em

= 3.906 Va2

µ02N 2I2r2

Variable Defini+on How  Determined V Accelerating Potential measured

a Helmholtz Coil Radius measured

μ0 Permeability of Free Space constant

N Number of turns in each coil given

r Electron beam radius given

I=IT-­‐I0 Net Current calculated

IT Total Current measured

I0 Cancellation Current measured

✹Want an answer with systematic and statistical uncertainties !!

Data for e/m example …..

Entry   I(A)   r(m)   Voltage  (V)   e/m  (C/kg  x  1011)  

1   1.75   0.0572   40.0   2.08  

2   1.99   0.0509   40.0   2.08  

3   2.28   0.0447   40.0   2.00  

4   2.67   0.0384   40.0   1.98  

5   3.20   0.0321   40.0   1.97  

6   2.20   0.0572   60.0   1.97  

7   2.49   0.0509   60.0   1.94  

8   2.85   0.0447   60.0   1.92  

9   3.34   0.0384   60.0   1.90  

10   4.03   0.0321   60.0   1.86  

11   2.58   0.0572   80.0   1.91  

12   2.90   0.0509   80.0   1.91  

13   3.30   0.0447   80.0   1.88  

14   3.39   0.0384   80.0   1.85  

Measure I0 three times and find I0 = 0.17, 0.20, and 0.23; giving a mean of 0.20 and an uncertainty of 0.03, i.e. I0 = 0.20±0.03 A

Compute e/m for 14 different voltages and pin radii !

em

= 3.906 Va2

µ02N 2I2r2

e/m Compute Statistical Uncertainty

• Mean: x =xii=1

N!N

=1.946 " 1011C/kg

• Standard deviation: ! x =(xii=1

N! # x )2

N #1= 0.072"1011C/kg

• Uncertainty on mean: ! x =! x

N=

0.07214

= 0.019"1011C/kg

Result with statistical uncertainty only:

em= (1.946± 0.019)!1011C/kg

e/m : Compute Systematic Uncertainty

from statistical uncertainty: I0 = 0.17, 0.20, and 0.23; I0 = 0.20±0.03 A

from meter (0.003x0.20+0.01) = 0.011A

from meter (0.003x2.85+0.01) = 0.019A

e/m Systematic Uncertainty

e/m measurement: Summary of Results

✸ Compare Measured value [1.946 ± 0.019(stat.) ±0.064(sys.)] x 1011 C/kg to accepted value 1.75882 x 1011 C/kg

✸ Discrepancy from the accepted value [1.946-1.75882] x 1011 C/kg = 0.187 x 1011 C/kg

✸ Significance of Discrepancy : Ratio of discrepancy and σtotal

often stated as “off” by 2.8σ

Is this a good or bad measurement???