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1. Data Handling/Statistics
There is no substitute for books—
— you need professional help!
My personal favorites, from which this lecture is drawn:
•The Cartoon Guide to Statistics, L. Gonick & W. Smith
•Data Reduction in the Physical Sciences, P. R. Bevington
•Workshop Statistics, A. J. Rossman & B. L. Chance
•Numerical Recipes, W.H. Press, B.P. Flannery, S.A. Teukolsky
and W.T.Vetterling
•Origin 6.1 Users Manual, MicroCal Corporation

2. Outline
•Our motto
•What those books look like
•Stuff you need to be able to look up
•Samples & Populations
•Mean, Standard Deviation, Standard Error
•Probability
•Random Variables
•Propagation of Errors
•Stuff you must be able to do on a daily basis
•Plot
•Fit
•Interpret

3. Our Motto
That which can be taught can be learned.
The “progress” of civilization relies being able to
do more and more things while thinking less and
less about them.
An opposing, non-CMC IGERT
viewpoint

4. What those books look like
The Cartoon
Guide to
Statistics

5. The Cartoon
Guide to
Statistics
In this example, the author
provides step-by-step analysis
of the statistics of a poll.
Similar logic and style tell
you how to tell two populations
apart, whether your measley
five replicate runs truly
represent the situation, etc.
The Cartoon Guide gives an
enjoyable account of statistics in
scientific and everyday life.

6. Bevington
Bevington is really good at introducing
basic concepts, along with simple code
that really, really works. Our lab uses a
lot of Bevington code, often translated from
Fortran to Visual Basic.

7. “Workshop Statistics”
This book has a website full of data that it
tells you how to analyze. The test cases are
often pretty interesting, too.
Many little shadow boxes provide info.

8. “Numerical Recipes”
A more modern and thicker version of Bevington.
Code comes in Fortran, C, Basic (others?). Includes
advanced topics like digital filtering, but harder to read
on the simpler things. With this plus Bevington and a
lot of time, you can fit, smooth, filter practically
anything.

9. Stuff you need to be able to look up
Samples vs. Populations
The world as we
understand it, based
on science.
The world as God
understands it, based
on omniscience.
Statistics is not art but artifice–a bridge to help us
understand phenomena, based on limited observations.

10. Our problem
Sitting behind the target, can
we say with some specific level of
confidence whether a circle
drawn around this single arrow
(a measurement) hits the
bullseye (the population mean)?
Measuring a molecular weight by
one Zimm plot, can we say with
any certainty that we have obtained
the same answer God would have
gotten?

11. Sample
View
Population
View
Average
Variance
Standard
deviation
Standard
error of
mean






n
i
i
n
x
n
n
x
x
x
x
x
1
3
2
1 1
...





n
i
i x
x
n
s
1
2
2
)
(
1
1
2
s
s 
)
(x
E


n
x)
(

 
2

n
s
SEM 

12. Sample View: direct, experimental, tangible
The single most important thing about this is the reduction
In standard deviation or standard error of the mean according
To inverse root n.
)
large
(for
1
~
2
n
n
s
s 

Three times better takes 9 times longer (or costs
9 times more, or takes 9 times more disk space).
If you remembered nothing else from this lecture, it
would be a success!

13. Population View: conceptual,
layered with arcana!
The purple equation in the table is an expression of the central
limit theorem. If we measure many averages, we do not
always get the same average:
).
"
Cartoon...
"
(from
"
deviation
standard
and
mean
with
on
distributi
normal
a
approaches
itself
)
large
(for
then
,
deviation
standard
and
mean
with
population
a
from
size
of
samples
random
takes
one
if
“
variable!
random
a
itself
is
n
x
n
n
x






14. Huh? It means…if you want to estimate , which only
God really knows, you should measure many averages, each
involving n data points, figure their standard deviation,
and multiply by n1/2. This is hard work!
A lot of times,  is approximated by s.
If you wanted to estimate the population average ,
the best you can do is to measure many averages and
averaging those.
A lot of times  is approximated by x.
IT’S HARD TO KNOW WHAT GOD DOES.
I think the  in the purple equation should be an s, but the equation only works in the limit
of large n anyhow, so there is no difference.

15. You got to compromise, fool!
The t-distribution was invented by
a statistician named Gosset, who was forced
by his employer (the Guinness brewery!)
to publish under a pseudonym.
He chose “Student” and his t-distribution is
known as student’s t.
The student’s t distribution helps us assign confidence in
our imperfect experiments on small samples.
Input: desired confidence level, estimate of population
mean (or estimated probability),
estimated error of the mean (or probability).
Output: ± something

16. Probability
…is another arcane concept in the “population” category: something
we would like to know but cannot. As a concept, it’s
wonderful. The true mean of a distribution of mass is given as the
probability of that mass times the mass. The standard deviation
follows a similarly simple rule. In what follows, F means a
normalized frequency (think mole fraction!) and P is a probability
density. P(x)dx represents the number of things (think molecules)
with property x (think mass) between x+dx/2 and x-dx/2.





x
all
x
all
x
F
x
x
xF
2
2
)
(
)
(
)
(



Discrete system











dx
x
P
x
dx
x
xP
)
(
)
(
)
(
2
2



Continuous system

17. Here’s a normal probability density distribution from
“Workshop…” where you use actual data to discover.
   68% of results
  2 95% of results

18. What it means
Although you don’t usually know the distribution,
(either  or ) about 68% of your measurements will
fall within  1 of ….if the distribution is a “normal”,
bell-shaped curve. t-tests allow you to kinda play this
backwards: given a finite sample size, with some
average, x, and standard deviation, s—inferior to
 and , respectively—how far away do we think the true
 is?

19. Details
No way I could do it better than “Cartoon…”
or “Workshop…”
Remember…this is the part of the lecture
entitled “things you must be able to look up.”

20. Propagation of errors
Suppose you give 30 people a ruler and ask them to measure
the length and width of a room. Owing to general
incompetence, otherwise known as human nature,
you will get not one answer but many. Your averages
will be L and W, and standard deviations sW and sL.
Now, you want to buy carpet, so need area A = L·W.
What is the uncertainty in A due to the measurement errors
in L and W?
Answer! There is no telling….but you have several options
to estimate it.

21. A = L·W example
Here are your measured data:
ft
W
ft
L
2
19
1
30




2
2
2
average
2
2
min
2
2
max
65)
(560
:
area
reported
65
2
490
-
620
:
y
uncertaint
estimated
557
2
490
620
490
17
29
620
20
31
ft
ft
ft
A
ft
ft
W
L
A
ft
ft
W
L
A















You can consider “most” and “least” cases:

22. Another way
We can use a formula for how  propagates.
Suppose some function y (think area) depends on
two measured quantities t and s (think length &
width). Then the variance in y follows this rule:
2
2
2
2
2
s
t
y
s
y
t
y


 

















Aren’t you glad you took partial differential equations?
What??!! You didn’t? Well, sign up. PDE is the bare
minimum math for scientists.

23. Translation in our case, where A = L·W:
2
2
2
2
2
2
2
2
2
W
L
W
L
A
L
W
W
A
L
A

























Problem: we don’t know W, L, L or W! These are
population numbers we could only get if we had the
entire planet measure this particular room. We therefore
assume that our measurement set is large enough (n=30)
That we can use our measured averages for W and L and
our standard deviations for L and W.

24. 2
2
2
2
2
4
2
2
2
2
2
)
65
560
(
:
n
calculatio
most/least
empirical,
our
to
this
Compare
)
63
570
(
or....
63
30)
(19
:
is
report
to
value
the
So
63
3961
)
2
(
)
30
(
)
1
(
)
19
(
ft
A
ft
A
ft
ft
ft
ft
ft
ft
ft
ft
A
A













25. Error propagation caveats
The equation, 2
2
2
2
2
s
t
y
s
y
t
y


 
















 , assumes
normal behavior. Large systematic errors—for example,
3 euroguys who report their values in metric units—are not
taken into consideration properly. In many cases, there
will be good knowledge a priori about the uncertainty in
one or more parameters: in photon counting, if N is
the number of photons detected, then N = (N)1/2 . Systematic
error that is not included in this estimate, so photon folk are
well advised to just repeat experiments to determine
real standard deviations that do take systematic errors into
account.

26. Stuff you must know how
to do on daily basis
0 2 4 6 8 10
0
5000
10000
15000
20000
25000
Larger Particle
30.9 g/ml
Parameter Value Error
------------------------------------------------------------
A -0.00267 44.94619
B 2.25237E-7 8.46749E-10
------------------------------------------------------------
R SD N P
------------------------------------------------------------
0.99987 118.8859 21 <0.0001
------------------------------------------------------------

/Hz
q2/1010cm-2
Plot!!!
r=0.99987
r2=0.9997
99.97% of the trend can be explained
by the fitted relation.
Intercept = 0.003 ± 45
(i.e., zero!)

27. The same data
0 2 4 6 8 10 12
0.0
0.5
1.0
1.5
2.0
2.5
3.0 Larger Particle
30.9 g/ml
twilight users rcueto e739
Parameter Value Error
------------------------------------------------------------
A 2.2725E-7 7.62107E-10
B -3.09723E-20 1.43575E-20
------------------------------------------------------------
R SD N P
------------------------------------------------------------
-0.44355 2.01583E-9 21 0.044
------------------------------------------------------------
D
app
/
cm
2
s
-1
q2/1010cm-2
How to find
this file!
r=0.444
r2=0.20
Only 20% of the data can be
explained by the line! While
 depended on q2, Dapp does not!

28. 0 10 20 30
0
5
10
15
20
25
[6/7/01 13:44 "/Rhapp" (2452067)]
Linear Regression for BigSilk_Ravgnm:
Y = A + B * X
Parameter Value Error
------------------------------------------------------------
A 20.88925 0.19213
B 0.01762 0.01105
------------------------------------------------------------
R SD N P
------------------------------------------------------------
0.62332 0.28434 6 0.18611
------------------------------------------------------------
Range of Rg
values obsreved in MALLS
(3/5)
1/2
Rh
R
h
/nm
c/g-ml
-1
Plot showing 95% confidence limits.
Excel doesn’t excel at this!

29. Interpreting data: Life on the bleeding edge of
cutting technology. Or is that bleating edge?
1E7
10
2E7
3E6
n = 0.324 +/- 0.04
df
= 3.12 +/- 0.44
R
g
/nm
M
The noise level in individual runs is much less than
The run-to-run variation. That’s why many runs are
a good idea. More would be good here, but we are
still overcoming the shock that we can do this at all!

30. Correlation Caveat!
Correlation  Cause. No, Correlation=Association.
Chart Title y = 35.441x + 57.996
R2
= 0.5782
0
10
20
30
40
50
60
70
80
90
0.0000 0.2000 0.4000 0.6000 0.8000 1.0000
TV's per person
Life
Expectancy
Country Life Expectancy People per TV TV's per person
Angola 44 200 0.0050
Australia 76.5 2 0.5000
Cambodia 49.5 177 0.0056
Canada 76.5 1.7 0.5882
China 70 8 0.1250
Egypt 60.5 15 0.0667
France 78 2.6 0.3846
Haiti 53.5 234 0.0043
Iraq 67 18 0.0556
Japan 79 1.8 0.5556
Madagascar 52.5 92 0.0109
Mexico 72 6.6 0.1515
Morocco 64.5 21 0.0476
Pakistan 56.5 73 0.0137
Russia 69 3.2 0.3125
South Africa 64 11 0.0909
SriLanka 71.5 28 0.0357
Uganda 51 191 0.0052
United Kingdom 76 3 0.3333
United States 75.5 1.3 0.7692
Vietnam 65 29 0.0345
Yemen 50 38 0.0263
58% of life expectancy is associated with TV’s.
Would we save lives by sending TV’s to Uganda?
Excel does not automatically
provide  estimates!

31. Linearize it!
Observant scientists are adept at seeing curvature. Train
your eye by looking for defects in wallpaper, door trim,
lumber bought at Home Depot, etc. And try to straighten
out your data, rather than let the computer fit a nonlinear form,
which it is quite happy to do!
Chart Title y = -0.1156x + 70.717
R2
= 0.6461
0
10
20
30
40
50
60
70
80
90
0 50 100 150 200 250
People per TV
Life
Expectancy
Linearity is
improved by
plotting Life vs.
people per TV
rather than TV’s
per people.

32. These 4 plots all have the
Same slopes, intercepts and
r values!
Plots are pictures of
science, worth
thousands of words
in boring tables.

33. From whence do those lines come?
Least squares fitting.
“Linear Fits”
the fitted coefficients
appear in linear part
expression. e.g..
y =a+bx+cx2+dx3
An analytical “best fit” exists!
“Nonlinear fits”
At least some of the fitted coefficients
appear in transcendental
arguments. e.g.,
y =a+be-cx+dcos(ex)
Best fit found by trial & error.
Beware false solutions! Try
several initial guesses!

34. All data points are not created equal.
Since that one point has
so much error (or noise) should
we really worry about minimizing
its square? No.




n
i i
fit
i y
y
1
2
2
2 )
(


We should minimize “chisquared.”
n is the # of degrees of freedom
n  n-# of parameters fitted
Goodness of fit parameter that should
be unity for a “fit within error”




n
i i
fit
i
reduced
y
y
1
2
2
2 )
(
1

n


35. 2 caveats
•Chi-square lower than unity is meaningless…if you
trust your 2 estimates in the first place.
•Fitting too many parameters will lower 2 but this may
be just doing a better and better job of fitting the noise!
•A fit should go smoothly THROUGH the noise, not
follow it!
•There is such a thing as enforcing a “parsimonious” fit
by minimizing a quantity a bit more complicated than 2.
This is done when you have a-priori information that the
fitted line must be “smooth”.

36. Achtung! Warning!
This lecture is an example of a very dangerous
phenomenon: “what you need to know.” Before you were
born, I took a statistics course somewhere in undergraduate
school. Most of this stuff I learned from experience….um…
experiments. A proper math course, or a course from LSU’s
Department of Experimental Statistics would firm up your
knowledge greatly.
AND BUY THOSE BOOKS! YOU WILL NEED THEM!

37. Cool Excel/Origin Demo