Transcript Document

Welcome to 125:315
Measurements and Analysis Laboratory
Instructor:
Head TA:
Yves Chabal
Mehmet Kaya
BME 101
BME 118
Lecture in B-120 (Eng.)
Labs: Weeks 1, 2, 3, 4, 10,12,13: BME 104
Week 5, 6, 7: Engineering
Weeks 8, 9, 11: Engineering A209
315: Measurements and Analysis Laboratory
Monday
Wednesday Thursday
Lab
#5
Lab
#1
Lab
#3
3
4
Lab
#2
5
Friday
Lecture
1
2
Tuesday
Lab
#4
Purpose of the course
1. Consolidation of experimental methodology
 Mandatory lab report for each lab (55% of grade)
2. Exposure to BME topics and faculty members
 Mandatory attendance of lectures (15% of grade)
Final exam will test knowledge and experimental methodology
(30% of grade) learned over the whole semester
Schedule
Week 1 (9/6/05)
Introduction
Y. Chabal
Week 2 (9/13/05)
ECG
J. Semmlow
Week 3 (9/20/05)
Evoked Potentials
J. Semmlow
Week 4 (9/27/05)
Respiratory Physiology
J. Semmlow
Week 5 (10/4/05)
Biomechanical Testing
N. Langrana
Week 6 (10/11/05)
Biomedical Fracture
A. Mann
Week 7 (10/18/05)
Soft Tissue Mechanics
D. Shreiber
Week 8 (10/25/05)
Protein Analysis
C. Roth
Week 9 (11/1/05)
Cell Growth and Imaging
P. Moghe
Week 10 (11/8/05)
Renal Physiology
T. Shinbrot
Week 11 (11/15/05)
Gene Transfer
C. Roth
Week 12 (11/29/05)
Cardiac Physiology
S. Dunn
Week 13 (12/6/05)
Neuro Physiology
W. Craelius
Exam (12/21/05)
Y. Chabal
Expectations
Prior to new lecture/lab (i.e. before Tuesday morning):
Download from website (http://coewww.rutgers.edu/classes/bme/bme315/ ):
•Background material on the particular lecture
•Lecture notes
•Lab write up
Read background material and lab write up
Copy lecture notes and lab write up (to take to lecture and lab)
Be ready for a short quiz at beginning of each lab
Perform lab completely (consider extra credit at least one time)
Write lab report independently following the required format
(turn within one week of your lab period)
Lab report
Note: important that each lab report represents 100% personal effort (wording, description,
analysis). Each students must go through complete exercise, even when working in pairs.
Critical thinking and creativity will be rewarded. The mandatory format of the report is:
• Introduction: Motivate lab in your own words (personal assessment of topic
importance); Give justification for use of method used in the lab
• Description of measurements:
Describe experimental protocol
Method and care taken to perform lab
• Results:
Data (Objective summary of results)
Focus on organization of data
• Analysis
Significance of results, how statistical analysis can help?
What conclusions can or cannot be derived and why?
• Summary & conclusions
•References (quote all sources used, including web sources, no plagiarism)
Exam
The final exam will include:
• Questions on big picture and associated basic knowledge
(back of the envelope calculations, quantities and units, ….)
Questions drawn from first two slides of each lectures
• Topical questions from main points of each lecture
To test knowledge of subject matter in more detail
• Lab questions to test understanding of measurements
e.g. why and how measurements are done, derivation of cause-effect,
possibly some details on analysis
• Test of analytical skills
Give set of data (observations), derive meaningful conclusions
• Design of experiment (extra credit)
Pose problem and ask to design an experiment to solve it
Lab #1: Measurements and Error Analysis
Experimental measurements only have a meaning with an error bar
• to test a theory (e.g. light deflection by sun)
• to answer a practical question (e.g. materials problem)
Types of errors:
• random errors unpredictable statistical fluctuations (temp., voltage, etc)
equally likely to be positive or negative
can be made smaller by taking more data points
************* precision: small random errror *****************
• systematic errors
always present during measurements, with a definite sign
due to something different from what is assumed (e.g. slow timer)
cannot be made smaller by increasing the number of data points
************* accuracy: small systematic error *****************
Statistical treatment of data
Given a set of measurements x1, x2, x3, ….., xN
the mean (average) is:
1
x 
N
N
x
i 1
i
To estimate the error in the mean, we need to introduce the idea of a distribution,
or histogram:
# readings in interval 0.02
4.615
4.638
4.597
4.634
4.613
4.623
4.659
4.623
5
0
4.6
4.7
Distribution function
Definition: Curve that would be obtained for a very large (infinite)
number of measurements (interval is very small)
The distribution function, f(x), is such that:
f(x) dx is the fraction of N readings that lie in the interval x to x+dx
i.e., f(x) is the probability that a single measurement taken at random
will lie in the interval x to x+dx

By definition,

f ( x)dx  1

And the mean (the average over all measurements) is:
X   x 

 xf ( x)dx  1

Error
The error in a measurement with value x is:
x - <x> or x - X
And the standard deviation of the distribution is s where:

s  e  x 
2
2
2
(
x

X
)
f ( x)dx


1
S 
( N  1)
2
or for a finite set of measurements:
 x  x
N
i 1
The root-mean-square, or standard deviation is therefore:
S 
1/ 2
2
1 N
xi  x 


N  1  i 1



i
2
Standard error in the mean
The standard deviation of a distribution of the means pf sets of
measurements, each set containing the same number N o single
measurements is:
Sm 
S
N
Where S represents the error in a single measurement and Sm the error in
the mean of N measurements.
The Gaussian Distribution
For a number of reasons, the function
f ( x) 
1 1
exp  ( x  X ) 2 / 2 2 
2 
has most of properties of a distribution of measurements with random
errors:
1. It is symmetric about X
2. It has its maximum value at X
3. It tends rapidly to zero as x- X  becomes large compare with .
Note: it is normalized, i.e. its integral = 1
its standard deviation is 
the points of inflexion occur at x= .
Probability: 1  (68%), 2  (95%)
Propagation of Errors
Important to estimate the error in a quantity that is a function of one or
several measurable quantities, each with some error or uncertainty
Example A.
Volume of Cylinder measure r, h
V = r2h
V + V =  (r + r)2 (h + h)
All terms with 2 are small, so….
V + V = r2h + 2rhr + r2h + …
V = 2rhr + 2h
V =
 V +

 r

r


 V 

 h

h


where
 V 

  2 rh

r


and
 V 
2

  r
 h 
Propagation of errors
In general, for independent random errors:
if q = x +… +Z – (u+…+w), then
q = ( x)2  ....  ( z )2  ( u )2  ...  ( w)2
if q 
x  ...  z
u  ...  w
and if q =
xn
then
q
q
, then
q
q
 (
n
x
x
)2  ....  (
z
z
x
x
For a function of a single variable, q=q(x)
then
q 
q
x
x
)2  (
u
u
)2  ...  (
w
w
)2
Laboratory this week: Measurement and uncertainty
Measure your reaction time to a stimulus (set of data)
Plot a histogram
Produce a distribution
Analyze using a Gaussian distribution approximation
Derive meaningful conclusions about the measurements and
the ability to distinguish reaction times
References:
“Practical Physics”, G.L. Squires, Cambridge University Press
(1968, 4th edition, 2001)
“An introduction to Error Analysis”, John R. Taylor, University
Science Books (1982, 2nd edition 1997)