Lucas Parra, CCNY
Cit y College of New Yor k
BME 2200: Biostatistics and
Research Methods
Lecture 9: Measurement error and error
propagation
Lucas C. Parra
Biomedical Engineering Department
City College of New York
CCNY
[email protected]
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Lucas Parra, CCNY
Cit y College of New Yor k
Content, Schedule
1. Scientific literature:
 Literature search
 Structure biomedical papers, engineering papers, technical reports
 Experimental design, correlation, causality.
2. Presentation skills:
 Report – Written report on literature search (individual)
 Talk – Oral presentation on biomedical implant (individual and group)
3. Graphical representation of data:
 Introduction to MATLAB
 Plot formats: line, scatter, polar, surface, contour, bar-graph, error bars. etc.
 Labeling: title, label, grid, legend, etc.
 Statistics: histogram, percentile, mean, variance, standard error, box plot
4. Biostatistics:
 Basics of probability
 t-Test, ANOVA
 Linear regression, Least-squares curve fit
 Error analysis
 Test power, sensitivity, specificity, ROC analysis
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Lucas Parra, CCNY
Cit y College of New Yor k
Measurement Error*
There are two types of errors:
Systematic error: Result of a mis-calibrated device, or a
measuring technique which always makes the measured value
larger (or smaller) than the "true" value. Careful design of an
experiment should eliminate or correct for systematic errors.
Example: Using a steel ruler at liquid nitrogen temperature to
measure the length of a rod. The ruler will contract at low
temperatures and therefore overestimate the true length.
Random error: Even when systematic errors are eliminated one
may obtain different measurements every time the experiment is
repeated.
Here we will deal with random errors only.
*Slides 3-7 follow www.rit.edu/~uphysics/uncertainties/
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Lucas Parra, CCNY
Cit y College of New Yor k
Measurement error
When giving a measured quantity always quote
the accuracy!
x± x
This range gives a 67% confidence interval: We expect 67% of
repeated measurements to fall within that range.
Example: Normal distributed measurements have  x=s x
 x=1.2 cm
5.5±1.2cm
x=5.5 cm
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Lucas Parra, CCNY
Cit y College of New Yor k
Instrument Limit Error and Least Count
Least count is the smallest division that is marked on the instrument.
Thus a meter stick will have a least count of 1.0 mm, a digital stop
watch might have a least count of 0.01 sec. In an instrument with
digital display it will be the smallest (steady) digit on the display.
Instrument limit of error (ILE) is the precision to which a
measuring device can be read, and is always equal to or smaller than
the least count (some fraction of the least count). Depending on the
space between the scale divisions you may be comfortable in
estimating to 1/5, 1/10 or 1/2 of the least count.
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Lucas Parra, CCNY
Cit y College of New Yor k
Repeated Measurements
The instrument error is a good start, but the best way to estimate the
random measurement error is to repeat the measurement several
times and present the standard deviation:
 x=s x
Example: Time measurements with a stop watch (in seconds)
Time, t
7.4
8.1
7.9
7.1
〈 t 〉 =7.625
t− 〈 t 〉
-0.2
0.5
0.3
-0.7
〈 t− 〈 t 〉 〉 =0.0
 t− 〈 t 〉 
∣t− 〈 t 〉∣
0.2
0.5
0.3
0.6
〈∣t− 〈 t 〉∣〉 =0.375
〈
2
0.04
0.25
0.09
0.36
 t− 〈 t 2∣ =0.1569
〉
Standard deviation: st =std t =0.4573
The result are always shown rounded t=7.6±0.5 s
to the least significant digit!!
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Lucas Parra, CCNY
Cit y College of New Yor k
Absolute vs relative error
To put an error in perspective it is customary to quote the relative
error (often in percentage)
Example
Absolute error
x
Relative error
x
x
x± x
x± x / x %
t=7.6±0.5 s
t=7.6 s±7 %
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Lucas Parra, CCNY
Cit y College of New Yor k
Function of random variables
You may have measured a few random variables and now you have
to compute some quantity that depends on these variables.
What is the error of the derived quantity?
Example: You want to create a salt solution with a salt
concentration of 0.5 g/ml by mixing 5g of salt with 10 ml of water.
The pipet to measure the volume of water has an accuracy of 0.5ml
and the scale to measure weight has accuracy of 0.1g. How accurate
is the value 0.5 g/ml of salt concentration?
5.0±0.1 g
g
c=
=0.5±? ?
ml
10.0±0.5ml
Note that here concentration is a function of two random variables,
mass m, and water volume v
m
c= f m , v =
v
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Lucas Parra, CCNY
Cit y College of New Yor k
STD of a function of random variables
More generally, say we measure N samples of two variables, ui and
vi, with standard deviations su and sv. Assume that the variables are
independent so that the correlation vanishes, suv = 0. We can then
compute the standard deviation of the resulting variable x
x= f u , v 
If we take the Taylor expansion around the mean to first order we
get
 
 
df
df
 x i −〈 x 〉=u i −〈u 〉
v i −〈 v 〉
...
du
dv
Square this equation, sum over the N samples, divide by N-1,
invoke the definition of the standard deviations, and we get (up to
second order):
2
2
   
df
2 df
s =s
s v
du
dv
2
x
2
u
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Lucas Parra, CCNY
Cit y College of New Yor k
STD of a ratio
If the function is a ratio (as in our example):
u
x= f u , v =
v
The formula simplifies to
  
2
2
u
2
2
2
v
2
s 2 s 2
1
2 u
s =s
s v 2 = x  x
v
v
u
v
2
x
2
u
Or with the conventional choice
 x=s x
     
2
2
x
u
v
=

x
u
v
2
In words: For a ratio the relative errors squared are additive.
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Lucas Parra, CCNY
Cit y College of New Yor k
STD of a ratio
For the concentration example we have
5.0±0.1 g
g
c=
=0.5±? ?
ml
10.0±0.5 ml
   
2
2
2
c
0.1
0.5
=

=0.0029
c
5
10
g
 c=0.0269
ml
g
c=0.5±0.03
ml
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Lucas Parra, CCNY
Cit y College of New Yor k
STD of a product
Surprisingly we obtain the same result for a product:
x= f u , v =u v
The formula simplifies to
s =s  v  s  u  =
2
x
2
u
2
2
v
Or with the conventional choice
2
s
2
u
2
u
2
x
s
v
2
v
2
x
2
 x=s x
     
2
2
x
u
v
=

x
u
v
2
In words: For a product relative errors squared are additive.
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Lucas Parra, CCNY
Cit y College of New Yor k
STD of a addition (subtraction)
If the function is a sum
x= f u , v =uv
The formula simplifies to (and the same for the subtraction)
s =s  1  s  1  =s s
2
x
2
u
2
Or with the conventional choice
2
v
2
2
v
 x=s x
  x  =  u    v 
2
2
u
2
2
In words: For the addition the absolute errors squared are additive.
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Lucas Parra, CCNY
Cit y College of New Yor k
Error of the mean
If a measurement devise has a large measurement error  x
(compared to the magnitude x that is to be measure) it is good
practice to repeat the measurement several times and report the
mean value:
N
m x =〈 x 〉=∑
i=1
xi
N
Since m is again a function of random numbers m=f(x1,..., xN) we
can use the same rule again and get for the error of the mean:
     
 x1
 xN
 x 
x
  m  = N ... N = N N = N
2
2
2
2
2
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Lucas Parra, CCNY
Cit y College of New Yor k
Standard error vs STD
The accuracy of the mean of N measurements is the standard error.
x
 m=
N
We find therefore that by repeating the measurement of a variable x
many times the mean m x =〈 x 〉 becomes more and more accurate.
Sometimes one may quote the error of the mean instead of the
standard deviation.
For the previous example one may write
std t /   N =0.2287
t=7.6±0.2 s , with N =4
However, if not otherwise stated you should assume that the standard
deviation is given and not the standard error. To avoid a false sense
of accuracy I suggest you mostly use the standard deviation.
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Lucas Parra, CCNY
Cit y College of New Yor k
Assignment
Assignment 9:

Measure your pulse rate and give the accuracy of your measurement.

How can you improve the accuracy of you measurement?
Should you quote the standard error or standard deviation of
multiple measurements? And why?

Submit your result and discussion in writing (No programming
needed).
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