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

� Example : e/m measurement

Assignments

• E-Class survey ends today at 11pm

• Read Taylor Ch 6

• This week: first lab starts.

• Radiation safety CAPA must be done before

your first lab this week! You’ll need to bring

print-out of completed CAPA to your first lab.

• First lab write-up due on Friday 9/16, 4pm.

Two types of uncertainty:

Statistical Systematic

sometimes high, sometimes low

centered on the true value

� ‘random’ distribution

always high, or always low

“systematically” off-center

We try to find “true value” X

This Lecture: How to deal with random

uncertainties

Defined as: Measure an infinite number of

times and take the average.

(can’t do in practice, but a very large number of

measurements get’s us pretty close to the same value)

1. Repeat the measurement many times

2. Calculate the average:

3. Estimate uncertainty (how far is this

average value off from the real value)

In practice we can and do:

Warning: Systematic uncertainty is NOT addressed

by statistical methods! Dealing with it requires

knowledge of the experimental setup and its

limitations.

∆xN

µ

Calculating uncertainty using statistical

analysis

Step 1: Make a histogram.

a) Split up the x-axis into n intervals. (n<<N)

b) Record the number of times a value falls

into a given range when you repeat the

measurement over and over.

c) Normalize: Divide the number in each bin

by the total number of measurements

Example: Exam score of a large class

160

120

80

40

0

Num

ber

of

Stu

de

tns

10080604020

Exam Score

76 students got a

score between 44

and 53 points

The class average is

µ = 68.7

(not normalized)

N = 608

What is the sum of the histogram

entries (ni) after Nmeasurements?

As recorded (i.e. before normalization):

After normalization:

bin i in timesofnumber : ; th

i

bins

i nNn =∑

1/ N ni

bins

∑ =1

160

120

80

40

0

Num

ber

of S

tude

tns

10080604020

Exam Score

ni

ni+

1

N=608 students

Example: Measure the circumference of 150 cows

3 4 5 6Bovine Circumference (m)

# o

f co

ws/

0.2

m

σ

✷The values appear to be

centered on the average value µ(‘mean’) .

✷The values appear to occur

more frequently near the average

and less frequently further away.

✷The mean (µ) and width

(“standard deviation” σ) describe the

relevant features of the distribution.

µ

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.

x

Probability density function:

F(x) =1

2πexp −

x 2

2

F(x)dx =1−∞

+∞

∫

x

• 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)

– You cannot change its integral, or else it

is not a probability distribution anymore.

As with any probability density

function, this still integrates to 1.

F(x) =1

σ 2πexp −

(x − µ)2

2σ 2

Gaussian Distribution

Gaussian probability distribution:

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 6Bovine Circumference (m)

Co

ws/

0.2

m

Read up in Taylor, Chapter 5, on definitions and uses of variance,

standard deviation and standard deviation on mean!

μ=4.39 m,

σ=0.70 m

� = �̅ = 1� � ��

�= 4.39 m

�� = ∑ (�� − �)������ − 1

= 0.7 m

Circumference of Cow data

3 4 5 6Bovine Circumference (m)

Co

ws/

0.2

m

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

So

• 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 normalized

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:

�� = ��� = 0.06 m

µ = (4.39 ± 0.06) m

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 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, 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 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 and sigma directly as discussed.

• Uncertainty on the mean is now

• Note: More measurements → smaller error on

the mean!

μ

σ

σ µ =σN

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. (Taylor Appx. A&B)

• 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.

� ��,! � "�#

$

So far we discuss “statistical uncertainty”.

Now what about uncertainty involved with

measurement itself ?

(Systematic Uncertainty)

Analyzing the error on a quantity

• You measure the voltage of a Josephson voltage standard with a $4

volt meter off e-bay

• You measure it 2983 times.

• The histogram of your results is at right.

• You calculate the mean to be 0.89 V, and the standard deviation to be

0.21 V.

• The 1σ uncertainty on the mean is

0.0038 V.

• If we assume the underlying

distribution is Gaussian and

centered on the true value, we can

turn this into a confidence level:

The true voltage of the Josephson

device is in the rage of 0.886 V to

0.894 V with a 68% confidence. 0 0.3 0.6 0.9 1.2 [V]

Analyzing the error on a quantity

0 0.3 0.6 0.9 1.2 [V]

• You measure the same Josephson voltage standard with a NIST

calibrated voltmeter.

• You find the true value is 1.200 V

• We are off by 75 sigma, which is extremely unlikely to be

correct!

• No matter how many times you measure, the systematic error

will not go away!

Statistical (random) vs. Systematic Uncertainties

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

g = [965 ± 3(stat) ± 2(syst)] cm/s2

• Add in quadrature (note that this assumes Gaussian distribution) to compare

with known values: g = [965 ± 4(total)] cm/s2

STATISTICAL SYSTEMATIC

NO PREFERRED DIRECTIONBIAS 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 directly measuring 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 physical quantity?

• 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

From PHYS 1140: 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.. variables with random

uncertainties δx, δy, δz…

Multiplication: if q=xyz,

Thus, fractional error is

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

Examples:

Example: e/m Experiment

• Electrons accelerated with 40, 60, or 80 V

• Acted on by magnetic field perpendicular to velocity

• r F = e

r v ×

r 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 1

2mv 2 = eV

• Magnetic field from Helmholtz coils is B =8µ0NI

a 125

• After putting together, e

m= 3.906

Va2

µ0

2N 2I 2r2

e/m Experiment e

m= 3.906

Va2

µ0

2N 2I2r2

Variable Definition 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

e

m= 3.906

Va2

µ0

2N 2I2r2e/m Compute Statistical Uncertainty

• Mean: x =xi

i=1

N

∑N

=1.946 × 1011C/kg

• Standard deviation: σ x =(xi

i=1

N

∑ − x )2

N −1= 0.072 ×1011C/kg

• Uncertainty on mean: σ x =σ x

N=

0.072

14= 0.019 ×1011C/kg

Result with statistical uncertainty only:

e

m= (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

I: net current

IT: total current

I0: cancellation current

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???

e/m measurement: Summary of Results

✸What is the probability of value being further than 2.8σ

from the mean for a gaussian distribution?

x

Probability of being inside of 2.8σ is

99.49% (look up the table p.287 Taylor)

Thus , probability of being outside of

2.8σ is 0.51%!!!

➙ Very Unlikely that a reasonable

measurement be that far from the

mean!

➙ Either mistakes or underestimates

systematic uncertainties.

✹If this happen to you, you will need to discuss more about

what sources of error might have caused this.

Assignments

• E-Class survey ends today at 11pm

• Read Taylor Ch 6

• This week: first lab starts.

• Radiation safety CAPA must be done before

your first lab this week! You’ll need to bring

print-out of completed CAPA to your first lab.

• First lab write-up due on Friday 9/16, 4pm.