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Transcript Thermodynamics-review
Analysis of Experimental Data;
Introduction
Some form of analysis must be performed on all
experimental data.
In this chapter, we will consider the analysis of data to
determine errors and uncertainty, precession, and
general validity of experimental data.
Real errors are those factors which are vague to some
extent and carry some uncertainty. Our task is to
determine just how uncertain a particular observation
may be and to device a consistent way specifying the
uncertainty in analytical form.
Analysis of Experimental Data;
Introduction
1.
2.
3.
Some of the types of errors that may cause
uncertainty in an experimental data are:
Gross blunders in instruments construction.
Fixed, systematic, or bias errors that cause the same
repeated readings to be mistaken. (Have unknown
reasons!!!)
Random errors that may be caused by personal
fluctuations, friction influences, electrical
fluctuations,... This type usually follow a certain
statistical distribution
Analysis of Experimental Data;
Introduction
Errors can be broadly analyzed depending on the
“common sense” or some “rules of thumb”.
e.g. Assume the calculation of electrical power from
P = EI, where E and I are measured as
E = 100V 2V, I = 10A 0.2A
the nominal value of P is 10010=1000W
the worst possible variations:
Pmax=(100+2)(10+0.2) = 1040.4W
& Pmin= (100-2)(10-0.2) = 964.4W
i.e., the uncertainty in the power is +4.04% & -3.96%
Uncertainty Analysis
Yet, a more precise methods for estimating uncertainty
are needed.
Consider the following
“Suppose a set of measurements are made. Each
measurement may be expressed with the same odds.
These measurements are then used to calculate some
desired result of the experiment. We like to estimate
the uncertainty in the calculated result. The result R is
a given function of the independent variables
x1,x2,…,xn. Thus: R = R(x1,x2,x3,…xn)”
Uncertainty Analysis
Let
R be the uncertainty in the result and 1, 2,…,
n be the uncertainty in the independent variables,
then:
R = [((R/x1)1)2 +((R/x2)2)2 +…+((R/x3)3)2 ]1/2
For product functions:
R/R = [(aixi/xi)2]1/2
R = x1a1 x2a2 …xnan
Uncertainty Analysis
For additive functions: R = aixi
R = [(aixi)2]1/2
Note that
i has the same units of xi
Uncertainty Analysis
E.g. The resistance for a certain copper wire is given
by:
R = Ro[1 + (T-20)]
Where Ro = 6±0.3% at 20ºC, = 0.004 C-1 ±1%,
and T of the wire is T =30±1ºC. Calculate the
resistance of the wire and its uncertainty?
Sol
……
Read examples 3.2 and 3.3.
Statistical Analysis of Experimental
Data
Definitions:
* Mean:
xm
1 n
xi
n i 1
* Median: is the value that divides the data points into half
* Standard deviation
1 n
2
[
n
1/ 2
(
x
x
)
]
i m
i 1
1 n
2 1/ 2
[
(
x
x
)
i
m ] , n 20
n 1 i 1
* 2 is called the variance
Probability distributions
It shows how the probability of success, p(x), in a
certain event is distributed over the distance x.
Two main categories; district and continuous.
The binomial distribution is an example of a district
probability distribution. It gives the number of
successes n out of N possible independent events
when each event has a probability of success p.
Probability distributions
The probability that n events will succeed is given as:
N!
p ( n)
p n (1 p) N n
( N n)! n!
E.g. if a coined is flipped three times, calculate the
probability of getting 0, 1, 2, or 3 heads in these
tosses?
The Gaussian or normal error
distribution
It is a continuous probability distribution type.
The most common type.
If the measurement is designated by x, the Gaussian
distribution gives the probability that the measurement
will lie between x and x+dx, as:
1
( x x m ) 2 / 2 2
p ( x)
e
2
The Gaussian or normal error
distribution
1
P(x)
2
x
1<2
The Gaussian or normal error
distribution
The probability that a measurement will fall within a
certain range x1 of the mean reading is
x m x1
1
( x x m ) 2 / 2 2
P
e
dx
x m x1 2
let =(x-xm)/, then P becomes
1
P
2
1
1
e
2 / 2
d ,
1
x1
The Gaussian or normal error
distribution
Values for the integral of the Gaussian function are
given in table 3.2.
Example
The Gaussian or normal error
distribution-Confidence level
The confidence interval expresses the probability that
the mean value will lie within a certain number of
values. The z symbol is used to represent it. Thus:
x x z
For small data samples; z is replaced by:
z
n
Using the Gaussian function integral values, the
confidence level (error) in percent can be found. (Table
3.4)
The Gaussian or normal error
distribution-Confidence level
The level of significance is: 1- the confidence level
See example 3.11
The Gaussian or normal error
distribution-Chauvenet’s Criterion
It is a way to eliminate dubious
data points.
The Chauvenet’s criterion
specifies that a reading may be
rejected if the probability of
obtaining the particular deviation
from the mean is less than 1/2n.
The attached table list values of
the ratio of deviation (d=abs(xxm) to standard deviation for
various n according to this
criterion
n
dmax/
3
1.38
4
1.54
5
1.65
6
1.73
7
1.80
10
1.96
15
2.13
25
2.33
50
2.57
100
2.81
300
3.14
500
3.29
1000
3.48
Method of Least Squares
We seek an equation of the form:
y = ax + b
where y is a dependent variable and x is an
independent variable.
The idea is to minimize the quantity:
S i 1[ yi (axi b)]2
n
This is accomplished by setting the derivatives with
respect to a and b equal to zero.
Method of Least Squares
Performing this, the results are:
a
n xi yi ( xi )( yi )
b
( yi ) ( xi ) ( xi yi )( xi )
n xi ( xi) 2
2
2
n xi ( xi) 2
2
Method of Least Squares
Designating the computed value of y as y`, we have
y` = ax + b
And the standard error of estimate y for the data is:
( yi y`i )
S
n
2
2
2
The Correlation Coefficient
After building the y-x correlation, we want to know
how good this correlation is. This is done by the
correlation coefficient r which is defined as:
x, y
r 1 2
y
where
2
1/ 2
2
(
y
y
)
m
i
i 1
y
n 1
n
1/ 2
2
(
y
y
)
ic
i
i 1
n2
n
x, y
1/ 2