judah levine jila s-460 [email protected] 303 …judah levine, 2150l9 2 announcements this...

Judah Levine, 2150L9 1

physicscourses.colorado.edu/phys2150

Judah Levine JILA S-460

[email protected] 303 492-7785 (2-7785)

Monday, 2-4 pm Also by appointment

Usually ok without an appointment

mailto:[email protected]


Announcements This is the last lecture of the class Laboratory periods and office hours

will continue Please check clicker grades on Canvas Please respond to FCQ request later in

the term Meeting of the American Physical

Society, 12-16 April, – Sheraton downtown, 16th & Court St.


Grading system - 1 6 experiments, max 20 points each

– 120 points maximum – Check with your TA if you think there is a

mistake Clickers: count as one experiment

– Maximum is 20 points Total possible points= 140 No pre-defined link to letter grades No quotas


Grading system - 2

Clicker calculation:

𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑔𝑔𝑐𝑐𝑔𝑔𝑔𝑔𝑐𝑐 =𝑠𝑠𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑔𝑔𝑠𝑠𝑠𝑠 𝑝𝑝𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝𝑠𝑠

𝑝𝑝𝑜𝑜𝑝𝑝𝑔𝑔𝑐𝑐 𝑝𝑝𝑜𝑜𝑠𝑠𝑠𝑠𝑐𝑐𝑝𝑝𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑔𝑔𝑠𝑠𝑠𝑠 𝑝𝑝𝑜𝑜𝑐𝑐𝑝𝑝𝑝𝑝𝑠𝑠× 20

I will not include the points of your first absence in the denominator

Send me an e-mail if you think your score is wrong. Include your clicker number


Hypothesis testing Are observations consistent with the

predictions of a model? Observations and model are functions

of some independent variable(s) Observations are limited only by

statistical uncertainty


General use of χ2

χ2 = �𝑜𝑜𝑝𝑝𝑠𝑠𝑐𝑐𝑐𝑐𝑜𝑜𝑐𝑐𝑔𝑔 − 𝑝𝑝𝑐𝑐𝑐𝑐𝑔𝑔𝑐𝑐𝑐𝑐𝑝𝑝𝑐𝑐𝑔𝑔

𝑠𝑠𝑝𝑝𝑔𝑔.𝑔𝑔𝑐𝑐𝑜𝑜

2

predicted can be any function of independent variable

Common problem: what is standard deviation?

Sum is over all data points (possibly grouped)


χ2 for continuous variable

χ2 = �(𝑂𝑂𝑗𝑗 − 𝐸𝐸𝑗𝑗)2

𝜎𝜎𝑗𝑗2𝑗𝑗

If the width of the blocks is too small then many will have too few members If the width of blocks is too large then the test is very weak

𝐸𝐸𝑗𝑗 𝑥𝑥𝑗𝑗 ,∆ = 𝑁𝑁 � 𝑝𝑝 𝑥𝑥 𝑔𝑔𝑥𝑥

𝑥𝑥𝑗𝑗+∆𝑗𝑗/2

𝑥𝑥𝑗𝑗−∆𝑗𝑗/2

Oj =number of data values in bin xj ± ∆j/2


What is value for σ? Repeat experiment multiple times,

estimate σ from data at each point Assume expected number in each bin

satisfies Poisson statistics – σ2= predicted number

Estimate σ from specifications of apparatus

Systematic errors do not contribute Calibration uncertainty is ambiguous


χ2 for discrete variable Observations are counting experiments,

measured exactly – Fluctuations are purely statistical

Probability from model: p(xj) – Mean for N data: Ej= Np(xj)

Variance is equal to the mean – Poisson statistics assumed.

χ2 = �

(𝑂𝑂𝑗𝑗 − 𝐸𝐸𝑗𝑗)2

𝐸𝐸𝑗𝑗2𝑗𝑗


χ2 for Least Squares - 1

Observe variation in some output, o, as some parameter, x, is changed

Model: relationship between x and output is of the form 𝑦𝑦𝑖𝑖 = 𝑔𝑔1𝑜𝑜1 𝑥𝑥𝑖𝑖 + 𝑔𝑔2𝑜𝑜2 𝑥𝑥𝑖𝑖 + … + 𝑔𝑔𝑘𝑘𝑜𝑜𝑘𝑘(𝑥𝑥𝑖𝑖)

a1, a2, …, ak are determined by minimizing:

𝑅𝑅2 = �𝑐𝑐𝑖𝑖2 = � 𝑜𝑜 𝑥𝑥𝑖𝑖 − 𝑦𝑦(𝑥𝑥𝑖𝑖) 2

𝑖𝑖𝑖𝑖


χ2 for Least Squares - 2

χ2 = �𝑂𝑂 𝑥𝑥𝑖𝑖 − 𝑦𝑦(𝑥𝑥𝑖𝑖) 2

𝜎𝜎𝑖𝑖2= �

𝑐𝑐𝑖𝑖2

𝜎𝜎𝑖𝑖2

Usual problem: What is σ ? Typical solutions: repeat experiment group residuals into bins, assume σ estimated from residuals in bin


χ2

If the model fits the data, we expect that the contributions to the sum will be of order 1 or less

The sum will be of order the number of terms in the sum= number of bins = N

Reduced χ2=χ2/N will be of order 1

χ2 = �𝑜𝑜𝑝𝑝𝑠𝑠𝑐𝑐𝑐𝑐𝑜𝑜𝑐𝑐𝑔𝑔 − 𝑝𝑝𝑐𝑐𝑐𝑐𝑔𝑔𝑐𝑐𝑐𝑐𝑝𝑝𝑐𝑐𝑔𝑔

𝑠𝑠𝑝𝑝𝑔𝑔.𝑔𝑔𝑐𝑐𝑜𝑜

2


Clicker question UMC sales:

– Coke= 935, Pepsi= 930, Dr. Pepper= 860 – Sprite=850 coffee: 925

If differences are purely statistical, compute χ2

A: 3.17 B: 7.62 C: 10.23


Clicker question discussion UMC sales:

– Coke= 935, Pepsi= 930, Dr. Pepper= 860 – Sprite=850 coffee: 925

Assume all equal, Poisson statistics – (935+930+860+850+925)/5=4500/5= 900

χ2 =935 − 900 2

900+

930 − 900 2

900+ …

χ2 = 1.36 + 1 + 1.78 + 2.78 + 0.7 = 7.62

Reduced χ2 = χ2/N, but what is N?


Statistics of χ2 N= Number of degrees of freedom

– Number of data points – number of parameters computed from the data

Expected mean value usually computed from data – Degrees of freedom= N-1

If std. dev. also computed from data – Degrees of freedom= N-2

Least-squares parameters also reduce degrees of freedom


Clicker question, discussion χ2 = 7.62

5 values in sum, but only 4 are independent 4 degrees of freedom Reduced χ2=7.62/4= 1.9 Expected value of order 1

Text, Appendix D, p. 293: Reduced χ2=1.9, 4 degrees of freedom Probability about 9%

If observations matched model, we would expect a smaller value 91% of the time


Problem 12.15

Roll 2 dice 360 times, observe sum each time

6×6 = 36 possibilities, all equally likely – Compute p(2), p(3), … p(12) – Expected number= N(j)=360×p(j)

χ2 = �𝑁𝑁 𝑗𝑗 − 𝑜𝑜(𝑗𝑗) 2

𝑁𝑁(𝑗𝑗)= 19.79

12

𝑗𝑗=2

See class notes


Last Words A statistic is a tool It is only as good as the assumptions

and model that are used There is no substitute for keeping your

brains turned on You are unlikely to get out more than

you put in

judah levine jila s-460 [email protected] 303 …judah levine, 2150l9 2 announcements this...

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