Coming to the End

Final Report

Summary

Correlation/Covariance

c2 Testing

Lecture # 7

Physics 2CL

Summer 2011

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Formal Report• 5-6 pages

• Title page

• Introduction

• Theory

• Experiment

• Results

• Discussion

Title page

– Title page

• Experiment title

• Name, institution

• Date

• Abstract

– 150 word limit

– Succinct summary of experimental goals,

results and significance of results

Introduction

• Overall description

• Background

• Objectives

Theory

• Review theory – give references

• Predictions

• Diagrams

Experiment

• Apparatus - diagrams

• Procedure

• Make clear all tools

• Easily understood

Results

• Raw data

– Tables

– Graphs

– organization

• Analysis of data

– Sample and clear calculations

– Uncertainty analysis

– Tables, graphs

– units

Discussion

• Summarize key data

• Quantitative comparison with theory

• Discuss in terms of d, s, and t analyses

Summary – best value and error

• Average

• Standard deviation

• Standard deviation of the mean

• Weighted averages

N

xx i

N

i

i xxN 1

2)(1

1

xs

xs Nx /s

i

ii

w

xwx

Summary of propagation formulae

222 )()()( zyxq

zyxq

dddd

222 )()()( zyxq

zyxq

eeee

xq

xBq

Bxq

ee

dd

summation

multiplication

constants

From Yagil

Distributions - Normal

69.4 69.6 69.8 70.0 70.2 70.40

1

2

3

4

Num

ber

mea

sure

men

ts

height (inches)

± 1 s

± 2 s

68 %

95 %

)2)(exp(2

1)( 22

, ss

s xxxGX

Probability within 1.47 s – App. A

Rejection of data

Area under part of curve -

integration

69.4 69.6 69.8 70.0 70.2 70.40

1

2

3

4

Num

ber

mea

sure

men

ts

height (inches)

)2)(exp(2

1)( 22

, ss

s xxxGX

a

a

b

b

Area under part of curve – two

steps

69.4 69.6 69.8 70.0 70.2 70.40

1

2

3

4

Num

ber

mea

sure

men

ts

height (inches)

)2)(exp(2

1( 22 s

sxxx

x-b

a

b

-a

-b

))2)(exp(2

1 22 ss

xx

x-a

x+ax+b

12

Clicker Question # 10You measure two lens in series, each with a power of 10 ± 2 diopters (i.e. P1=P2 = 10 ± 2 diopters). What is the t-value is your measurement of the combination is 22 ± 2 diopters and the expected value is given by PTOTAL = P1

+P2?

(A) 0.5 s

(B) 1.0 s

(C) 1.5 s

(D) 2.0 s

OVERVIEW Latter Chapters of Taylor

Chapter 9 – Covariance and Correlation

Are two measurements related?

How significant is the relation?

Chapter 10 – Binomial Distributions

Probabilities of two outcomes (flip dice)

Can be approximated by Gaussian

distribution when N is large

Chapter 11 – Poisson Distribution

Events are random but occur at an

average rate (radioactivity)

Can also be approximated by Gaussian

distribution when N is large

Chapter 12 – c2 Testing

Test as to how well a distribution fits the

observed measurements

Correlation and Covariance(Taylor Chapter 9)

c2 Testing(Taylor Chapter 12)

Degrees of Freedom

d

Fitting Summary

Example - Dice

Application of c2 – Use of Table D

~

c2 Test for a Fit

If Prob(c2 > c2) is less than 5% - disagreement “significant”~

~

If Prob(c2 > c2) is less than 1% - disagreement “highly significant”~

~

Reminder

• Formal Report due next week

• No more Homework

• No more lectures