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; Analysis of Experimental Data; IntroductionIntroduction

Some of the types of errors that may cause uncertainty in an experimental data are:

1. Gross blunders in instruments construction.2. Fixed, systematic, or bias errors that cause the

same repeated readings to be mistaken. (Have unknown reasons!!!)

3. Random errors that may be caused by personal fluctuations, friction influences, electrical fluctuations,... This type usually follow a certain statistical distribution


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%


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 AnalysisUncertainty Analysis

Let R be the uncertainty in the result and 1,

2,…, n be the uncertainty in the independent variables, then:

For product functions:


R = [((R/x1)1)2 +((R/x2)2)2 +…+((R/x3)3)2 ]1/2

R = x1a1 x2

a2 …xnan

R/R = [(aixi/xi)2]1/2

For additive functions: R = aixi

Note that i has the same units of xi


R = [(aixi)2]1/2

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

±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.


C-1

Definitions: * Mean:

* Median: is the value that divides the data points into half * Standard deviation

* is called the variance

Statistical Analysis of Statistical Analysis of Experimental DataExperimental Data

n

iim x

nx

1

1

20,])(1

1[1

2 2/1

nxxn

n

imi

n

imi xx

n 1

2 2/1])(1[

2

Probability distributionsProbability 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 distributionsProbability distributions The probability that n events will succeed is

given as:

E.g. if a coined is flipped three times, calculate the probability of getting 0, 1, 2, or 3 heads in these tosses?

nNn ppnnN

Nnp

)1(!)!(

!)(

The Gaussian or normal error The Gaussian or normal error distributiondistribution

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:

22 2/)(

21)(

mxxexp


x

P(x) 1

2

1<2


The probability that a measurement will fall within a certain range x1 of the mean reading is

let =(x-xm)/, then P becomes

dxeP m

m

m

xxxx

xx

221

1

2/)(

21

11

2/ ,21 2

1

1

xdeP


Values for the integral of the Gaussian function are given in table 3.2.

Example

The Gaussian or normal error The Gaussian or normal error distribution-Confidence leveldistribution-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:

For small data samples; z is replaced by:

Using the Gaussian function integral values, the confidence level (error) in percent can be found. (Table 3.4)

zxx

nz

The Gaussian or normal error The Gaussian or normal error distribution-Confidence leveldistribution-Confidence level

The level of significance is: 1- the confidence level

See example 3.11

The Gaussian or normal error The Gaussian or normal error distribution-Chauvenet’s distribution-Chauvenet’s

Criterion 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(x-xm) 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 SquaresMethod 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:

This is accomplished by setting the derivatives with respect to a and b equal to zero.

n

i ii baxyS1

2)]([

Method of Least SquaresMethod of Least Squares Performing this, the results are:

22

2

22

)())(()( )(

)())((

xixnxyxxy

b

xixnyxyxn

a

i

iiiii

i

iiii

Method of Least SquaresMethod 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:

22

2)`(

nyyS ii

The Correlation CoefficientThe 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:

2/1

1

2

,

2/1

1

2

2/1

2,

2

2

)(

1

)(

1

n

yy

n

yy

where

r

n

iici

yx

n

imi

y

y

yx

analysis of experimental data; introduction

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