In today’s class we will talk about Error
Analysis.
While talking about performance characteristics
of instruments we have talked about static
error and dynamic error.
Now, since no measurement can be perfect and
may particular measurement will always been
error this error can be a various forms.
So, we should have some estimate of how much
error there can be in our measurement.
So, let us now talk about error analysis in
some detail Types of Error in Measurements.
The Types of Error in Measurement can be broadly
classified into 3 categories 1 the Gross error
or human error 2 Systematic or Determinate
Error and 3hree Random or Indeterminate Error.
So, all the errors that can be involved on
in our measurement can broadly be classified
intothis3 categories namely Gross error Systematic
error and Random error.
Now, Gross Error is caused by human mistake.
So, this is Wrong reading or wrong using of
instruments gross error which is caused by
human mistakes cannot be eliminated completely,
but it can be minimized by taking proper care
and appropriate uses of instruments.
Next let us talk about Systematic or Determinate
Error as the name suggests Systematic error
affect all the readings in a particular fashion.
This is caused due to shortcomings of the
instrument such as defective or worn parts
in the instrument improper design of instrument
loading effect etcetera.
However, systematic errors are usually recognizable.
So, when determined systematic error can be
corrected by careful calibration of the instrument.
So, let us repeat again systematic error are
those error which affect all the readings
in a particular fashion this is caused due
to shortcomings of the instrument such as
defective or worn parts improper design of
instrument loading effect etcetera.
When you can recognize a determined systematic
error this type of error can be corrected
by careful calibration of the instrument systematic
error May be of constant or proportional nature.
So, if we look at this figure is a float of
x variable versus y variable.
We can consider x variable to be the input
to the instrument and y variable as the output
of the instrument.
So, this can be considered as a calibration
curve.
So, if we look at the blue line let us say
blue line should be the correct characteristic
of the instrument.
If, we get the behavior of the x variable
and y variable that is the input and output
relationship is represented by a red line.
We see that there is an error associated with
the ready but, these error is a constant type
whereas, if the input output relationship
is characterized by the blue line will have
a Proportional error.
However, as the graph suggest that if, we
know how much of constant error or how much
of proportional error is associated within
instrument we can always introduce a suitable
correction factor.
So, it is rewriteable easy to handle systematic
error.
We can classify the Systematic errors under
three different categories: 1 Instrumental
error 2 Environmental error and 3 observational
error.
This can also be considered as source of systematic
errors.
So, systematic errors can be of Instrumental
error Environmental error or Observational
error.
Now, let us look at Instrumental error.
The Instrumental error is inherent with the
measuring instrument because, of their mechanical
structure like bearing friction, irregular
spring tension, stretching of spring, etcetera.
So, instrumental error is inherent with the
measuring device.
Instrumental error can be avoided by Careful
design of instrument and by careful selection
of the components of the instruments it can
also be avoided by calibrating the instrument
frequently against known standards.
So, Instrumental error is inherent with the
measuring instrument because, of their mechanical
structure, so just bearing fraction, irregular
spring tension, stretching of spring, etcetera.
And the best way to avoid Instrumental error
will be the Careful design of the instrument
and proper selection of the components of
the instrument.
Environmental error this is due to external
condition affecting the measurement such as
change in temperature, change in humidity,
change in atmospheric pressure, etcetera.
So, Environmental error is due to external
conditions effecting the measurement such
as change in temperature, change in pressure,
change in humidity, etcetera.
To avoid the environmental error we should
provide suitable control environment for the
instrument.
So, providing controlled environment for the
instrument is necessary to avoid the environmental
error.
Environmental error can also be avoided by
sealing certain components in the instruments.
Observational error is introduced by the observer
the most common observational error is parallax
error and estimation error while reading the
scale.
Next let us come to Random or Indeterminate
Error.
As the name suggests random errors affect
readings in a random way.
So, unlike systematic error which effects
all the readings in systematic fashion random
errors affect readings in a random way.
It cannot be determined easily we do not have
any control over random errors and these are
due to unknown causes.
So, causes for random error are not always
easily known.
Random errors cannot be eliminated completely
it is difficult to eliminate completely the
random errors because, the sources of random
error are usually not always known.
However, the random errors can be minimized
by increasing the number of readings that
means, multiple trials we take several readings
of the same value of the measuring variable.
So, we repeat the number of measurements and
then use statistical means to obtain best
approximation of the true value.
So, will take a particular reading several
times this is call multiple trials and then
we will use some statistical principles to
obtain best approximation of the true value.
This way we can minimize the random error
in our measurements, but we can never completely
eliminated.
Let us now, talk about some Basic Statistics
1st Average.
So, if we take a particular reading N times
and each time I have the major values in the
carried as xi.
So, I have values x1, x2, x3, up to xN, and
then some them up and then divide by the total
number of measurements.
So, this is known as average of n readings
or mean frequently represented as mu or x
bar.
Deviation is defined as difference between
a particular value and the mean.
So, we have to statistics as of now average
which is the sum of N readings divided by
N. It is frequently stated that, the mean
value is the most probable value of a set
of readings and that is why it is a very important
role in statistical error analysis the deviation
of the individual readings from the mean value
is obtain as di equal to xi minus mu where
xi is the i-th reading and mu is the mean.
We know one to have an idea about the deviation
that is why the individual readings are far
away from the mean value or not.
So, if we one to have some idea whether the
individual readings are far away from the
mean value or not can you use Mean of deviation.
So, if this is xi minus mu is mean.
So, this will represent Mean of deviations.
Unfortunately we cannot use the mean of deviation,
because if you follow this the mean of deviation
is always equal to 0.
So, instead we use Variance or mean squared
deviation 
which is defined as this.
So, Variance is mean squared deviation.
So, we find out the individual deviation square
sum of over N readings and divide by number
of readings N.
So, Variance or mean square deviation can
be used to find out how far away from the
a particular reading is how far away from
the mean value a closely related statistic
is Sample standard deviation which is square
root of variance.
We can have a biased estimate of standard
deviation or unbiased estimate of standard
deviation.
If, we want to have a biased estimate of standard
deviation we divide this by N if we want to
use unbiased estimate we should divide it
by N minus 1.
If, we have a large number of readings typically
more than 30 or so, it does not make much
difference whether we divide this by N or
N minus 1 the term standard deviation which
is square root of variance is often used as
a measure of uncertainty in a set of measurements.
So, we have Average or mean value then, the
individual Deviation which is difference between
a particular value and the mean then the Variance
or means square deviation and then standard
deviation which is square root of sigma square.
Now, let us talk about the probability distribution
function.
Suppose we have a sufficiently accurate Pressure
gauge, and where measuring a certain pressure
with this pressure gauge several times.
Let us say the given pressure gauge the given
pressure is say 10 kilopascal.
So, a measuring a 10 kilopascal pressure using
a pressure gauge and let us say for example,
the Data looks like this, you can have even
more number already let us say we end like
this.
So, we are measuring a 10 kilopascal pressure
with a pressure gauge which, can be consider
as a sufficiently accurate instrument and
I measure the same pressure several times
and I get the readings like this.
So, the Lowest is 9.80 and the Highest is
10.
45.
Now, let us Define 
an interval of let us say 0.05 kilopascal
and we want to determine, How many readings
fall in each interval of 0.05 kilopascal?
So, let us Define a quantity called Z which
is 
number of readings in each interval divided
by Total Number 
of readings divided by width of interval.
So, we have these data from repeated measurement
of a given pressure obtain kilopascal using
this Pressure gauge and now, if I plot the
values 
using an interval of 105 and calculate Z which
is define as follows Number of readings in
each interval by Total number of readings
divided by width of interval.
We can plot bar graph with high Z for each
interval which will be a histogram.
So, maybe I can get a histogram like this,
may be something like this 
where this is and this is Z.
So, here plot it x versus Z where x is the
readings scale readings.
Now, area of a particular bar is numerically
equal to the probability that is 
specific reading will fall in the associated
interval.
the area of the entire histogram has to be
1 or 100 percent because, there is definite
probability or there is a probability 1 that
a particular reading will fall anywhere, within
the histogram.
That is it will fall somewhere between the
lower value and the upper value that is somewhere
between 9.80 and 10.45 it this belongs to
if this is 10.
45.
So, it will lie between anywhere, between
here to here that you can say with probability
1.
Now, if I take infinite number of readings
here, I have taken a finite number of readings
each with an infinite number of significant
t digits we can make chosen intervals as small
as we want.
And still have each interval contain a finite
number of readings, in this particular plot
we have taken an interval width 
of 0.5.
Now, in this example as small as we want,
and still can expect to have definite number
of readings that will fall in each interval.
Now, if we do this then these steps in this
graph will be smaller and smaller and the
graph will approach a smooth curve in the
limiting case.
So, all we are saying is if we, increase the
number of readings that it will large value
if we repeat this experiments infinite number
of times.
We can make the intervals extremely small
and even then we can expect that some readings
will fall in each interval.
In that case the steps will be smaller and
smaller and this plot this plot will be like,
a smooth graph or smooth curve in the limiting
case.
Now, if we take this limiting abstract case
as a mathematical model for the real physical
situation then the function Z equal to fx
is call the probability density function.
So, if we take large number of readings this
histogram can be represented by smooth curve
and the functional relationship Z equal to
fx can be considered as the probability density
function.
So, From the definition of the Z, the Probability
of reading lying between a and b which can
be written as the Probability that x lies
between a and b, is the area under the curve
so.
So, the probability of reading lying between
a and b is the area under the curve.
So, this said region this probability information
is sometimes given in terms of Cumulative
Distribution Function 
the probability information can also be given
in terms of cumulative distribution function.
Which is defined as, the Cumulative Distribution
Function capital F x equal to probability
that reading is less than, any chosen value
of x.
So, this will be represented as, now most
random errors have a Gaussian distribution
or normal distribution.
Since, Normal Distribution or Gaussian distribution
is one of the most use full density function.
There are other probability density functions
as well.
In fact, there are many, but the normal distribution
function or the Gaussian distribution function,
is one of the most useful probability density
function.
The normal distribution function or the Gaussian
distribution function is define as this, 1
over sigma into square root of 2 pi into e
2 the power minus x minus mu square divided
by 2 sigma square.
Where mu is the mean value and sigma square
is the variance which we define previously.
So, Gaussian distribution can look like this,
the fact that most random errors may have
a Gaussian distribution is the consequence
of a very important theorem, call the central
limit theorem.
When you overlay many random distributions,
each with an arbitrary probability distribution
different mean value and a finite variance
the resulting distribution is Gaussian.
So, the Normal Distribution 
or Gaussian distribution which is defined
as, where mu is mean and sigma square is variance.
So mu can be obtained as, and the variance.
So, given the probability density function
P x mean is, defined as follows and the variances
is defined as this equation.
So, if you look at the Gaussian or Normal
Distribution function it has two characteristic
parameters.
One is sigma which is variance; another is
mu which is mean.
So, the shape of the Gaussian or normal distribution
function is determine by mu and sigma.
Now, we can have a Gaussian Distribution like
this we can also have a Gaussian Distribution
like this.
Obviously, both has same mean but, different
variance or different standard deviation.
This distribution has small standard deviation
whereas, this has large standard deviation
both has same mean.
So, when the standard deviation is small it
indicates that, there is a high probability
a reading will be close to mu or mean.
So, we have a distribution with small standard
deviation it indicates that, there is a high
probability that the reading will be close
to mean value mu, while if, the standard deviation
is large, the readings will be more scattered
along the mean.
It may be pointed out here, that if you look
at this mathematical definition of the Gaussian
Distribution it says that, there is a small
probability that even a very large reading
will also occur.
But, a real distribution will always have
their tails cut off.
So, instead of 
this a real distribution will have will be,
where their tails will be cut off.
However, most of the or many measurements
the errors the random errors associated with
many measurements can be represented by a
Normal Distribution Function.
Now, having defined some of the basic statistics
let us, now see.
How the error can be propagated in a particular
measurements?
A particular instrument can have various components.
So, a particular instrument can be chain of
components, the question we ask is if we know
the error or uncertainty associated with each
component.
What will be the overall error or what will
be the overall uncertainty measurement?
You can also think of a situation where, a
particular measurement involves various other
measurements.
So, particular experiment can involve various
measurements.
And we calculate a value of a particular variable,
from the measure from these measure values.
Now, if I know the uncertainty or error associated
with each measurement.
Can I compute the overall error or uncertainty
measurements.
So, let us now see how we do that?
So, the question you ask is, if an experiment
involves several measurements.
Say, using several instruments and if, I know
the individually inaccuracy or individual
error uncertainty.
How do I combine this individual uncertainties
or errors to get an estimate of overall accuracy?
So, now, consider the problem of Computing
a variable Y 
which is known to be a function of n independent
variables.
So, we want to compute Y which is a function
of n independent variables x1, x2 up to xn.
So, we may are measuring x1 x2 up to xn.
If we know the errors or uncertainties associated
with x1 x2 up to xn we want to compute the
overall accuracy in Y.
So, we say Y is a function of x1 up to xn.
For a small change in the independent variable
x1 x2 up to xn from a given say operating
point.
A Taylor series expansion will give an good
approximation for the corresponding change
in Y.
So, which can be represented as where you
can only the First-order terms.
So, if there are change is small change is
in the independent variables x1 x2 up to xn
around a given operating point.
The corresponding change in Y can be obtain
from a Taylor series expansion of these function,
which is written as this where we are written
only the First-order terms.
We can think of these partial derivatives,
as the sensitivity of Y to changes in the
particular x.
So, these are sensitivity of Y to changes
in particular x.
So, this is sensitivity to change sensitivity
of Y to change in x1 sensitivity of Y to change
in x2 and.
So, on and so 4th.
If partial derivative is large we say Y is
very sensitive to that particular x.
Now, Consider delta Y to be the uncertainty
in measured value Y.
Which we represent as let us, say similarly
delta x1 to be uncertainty in measured value
x1 which we represented as u x1.
Similarly, delta xn is to be considered as
uncertainty in measured value xn which will
represent as U xn.
Then we can write as, Uy is 
we have just, use this definition in this
equation.
Now, the maximum value of uncertainty will
be when all the uncertainties happen to have
the same sign.
So, the maximum value of uncertainty can be
taken as, would be obtained when all the uncertainties
happen to have the same sign.
And this would be the worst possible case;
however, the probability of such an occurrence
is generally very small.
Therefore, the more realistic way is to square
both sides to give equal weight age to both
positive and negative values of uncertainties.
So, more realistic way of representing uncertainty
will be, to square the terms, here we have
Neglected terms like cross terms like.
So, this will be More realistic uncertainty.
So, we can now take overall uncertainty as,
where N is this x1 up to x2 up to xn, an number
of independent variables.
So, if you have an estimate of the individual
uncertainties, we can determine the overall
uncertainty.
But, is the inverse problem which is more
important.
What do you mean by inverse problem is as
follows?
When you originally plan an experiment we
must decide, how much accurate our measurement
will be.
So, we set our goal that our measurement must
meet this much of accuracy, and accordingly
we need to know what will be the allowable
uncertainties in individual measurements.
Now, this problem is not.
So, easy to solve because there can be various
combinations of these uncertainties which
will give the same over all accuracy.
So, we have to get started somewhere.
So, at this stage we can make use of something
called Method of equal effects, where we assume
that all the instruments or all the individual
components contribute equally to the overall
error.
In other words the individual components contribute
equally to the overall error.
So, if we look at this equation, under the
assumption of Method of equal effects.
We can now, write 
or 
so we can now, have an estimate of; what should
be the accuracy of individual component to
meet the requirement of overall accuracy?
If you can find instruments which meet all
these nets we have at least one solution to
a problem.
If one or more requirements cannot be meet
we should check if some of our instruments
are better than that the above equation requires.
If, that is the case, then you can relax the
requirement that we cannot meet ultimately
the overall accuracy must be meet.
