Well, now whenever we have a situation where I...
I described a few kinds of situations. Situation number one: we are trying to measure
the the mass of an electron.
We know from theory that all the electrons have the same mass and therefore, we are trying
to measure a constant. When we are trying to measure a constant,
the variability comes from the random changes that we have, random variation of the results
that we can, we can have. But there are situations where the thing that
we are trying to measure has a inherent variability, for example, the weight of a bird.
It has an inherent variability. Or if a geologist is trying to find out the
the average density of the Earth’s crust, then, obviously,
he has to, he or she has to take samples from different parts and one has to be very careful
that it is a representative sample of different types of densities.
So, whenever the thing that we are trying to measure has a variability, then we have
to be very careful how we take the samples. So far we are talking about, we have a population
out there and we are trying to take samples from there.
And if it is a population with some variability, there is additional consideration that,
when we take samples, when we draw the samples, we have to deliberately
draw the samples keeping in mind a variability. And therefore, we have to make sure that,
we are... if there is a variation, some chunk is at the higher level, some chunk at the
lower level, then we have to draw samples from all of them.
And in every field there are very well laid off prescription for doing that.
In every field. In order to to illustrate, let me, let me
give you the example of how samples are collected from the soil.
I mean soil quality measurement is a very standard thing that, I mean agricultural scientists
have to do all the time, and they have the same problem.
Suppose they have a large field in hand and from that large field, they have to to find
out what is the character of the soil of that field.
But the character of the soil in this part of the field,
it may not be the same of the character of the soil in that part of the field.
Character of soil in the top layer may not be the same as the character of the soil,
in the, you know, one feet below. So, we have to then take that into consideration.
How is it actually done? How it is actually done is...
How it is actually done is, suppose I have a field,
suppose I have a square field like this. Suppose it is about this much.
Okay? And I have been asked to find out the character
of that soil. How do I do that?
Because the character of the soil here will not be the same
with the character of soil here here here, it will be different, and as I said, at different
layers it will be different. So, what is done?
What is done is that, we make holes about in in every 20 meter distance.
So, like this, every 20 meter distance, holes will be made.
Okay? So, this holes will be like this
and about 20 meter distance and soil will be collected from the top,
from the middle, and from the bottom. Okay? And then this soil that has been collected
will be some... I mean, if this is done from all the holes
all around, it will be huge amount of soil.
Now, that soil will be dried and will be spread over a hard surface, maybe cemented surface.
And let me now use that this as the cemented surface.
So, the sample is spread over the whole thing and then this is divided into four parts by
doing like this. And then this part is, this part and this
part are retained and the rest thrown off. Again the soil is mixed, again it is spread
over the the the square area, again the same process is repeated.
That means, you you mix up everything, spread it,
and then you retain the main diagonals and discard the off-diagonals, and so on so forth.
Ultimately you carry on this process until you have got something like 2 to 3 kgs remaining.
When you get the 2 to 3 kgs, remember then it has a reliable mixture,
it is a reliable mixture of soil collected from all parts of the field, as well as from
the top, middle and bottom, and therefore it is a reliable sample from
the field. But after this 2 to 3 kgs is obtained, it
is further ground and it is seived through a, you know, 2 millimetre sieve.
And finally, what is collected, that is, what is tested.
Notice, what we are doing, the the... this is the standard laid-out procedure
for the soil scientists. Similarly, for every field there would be
similar procedures laid out, very well laid out procedures, and using that we collect
the samples. If you, if you have to collect the samples
from a population of humans, we know the humans can be old, middle aged, children, babies,
male, female. So, we have to collect from all the the,
We have to make sure to collect samples from all the parts of the population.
So, this is very important in sampling. In any measurement there are two types of
errors. The first is random error and the second is
systematic error. Systematic errors are those, which make the
results that you obtain systematically differ from the actual population mean.
There is a value out there, the actual population mean, and you are trying to obtain an estimate
of that by obtaining the sample mean. Now, if there is a systematic error,
there would be a systematic deviation from of the sample mean from the population mean.
That can happen due to various reasons. But this is something that we have to somehow
figure out and eliminate. Because, if a systematic error remains, then
the results will definitely be wrong. But random errors generally cannot be avoided.
Random errors happen because of, you know, thermal fluctuations and effects like that,
which are not within our control. Therefore, a scientist has to learn how to
sort of, take stock of the random errors. So, systematic errors can be eliminated, but
random errors normally cannot be eliminated. So, a scientist has to learn to live with
random errors. Systematic errors will depend on the area
in which you are working. The systematic error for a physics experiment
may not be the same as a systematic error for a Earth science experiment, for example.
So, different area different fields will have different instruments
used, and different systematic errors.
But nevertheless, scientists have two general prescriptions of getting rid of systematic
errors. The first is calibration.
Wrong calibration is the most common source of systematic error.
And therefore, it is necessary to periodically recalibrate all the equipment that you use,
because with age the calibration goes bad. With, you know, environmental condition, the
calibration may change. So, recalibration every now and then
is necessary for every equipment. We scientists sometimes do rely on whatever
a big equipment, the data that a big equipment churns out, without bothering whether it is
properly calibrated or not. So, it is very important to check the calibration.
Second is that, whenever possible measure the same thing by two or more different ways.
So, measure the same quantity by ways. The advantage is that, it is very unlikely
that the same systematic error will creep in, in two different instruments.
Two different methodology means, you will use two different equipment, and
it is very unlikely that the same systematic error will creep in.
So, if you get the same, you know, results within some tolerance,
then you know that your result is reliable. But if you you...
if you measure by two different, two or more different methods and measurement, and you
get substantially different results, you know that there is a systematic error somewhere
and you have to take care of that. Okay?
So, generally, in order to avoid systematic errors, these are the two general prescriptions.
But for every field the source of systematic error will be something and you need to figure
out depending on the character of that particular field.
Ok, there are a few types of systematic error. Random errors is something that you have to
learn how to live with that. Now as I said,
always we take, we set up the equipment, set up the experiment,
such a way for every measurement, we can take a large number of measurements
and can take an average. This is a fundamental prescription.
Nobody can avoid that. And sometimes people, when using big equipments,
people sort of miss out this particular point, because a a one crore worth equipment and
you have fed in a sample, it is giving rise, giving some data and you
are simply trusting that. And you are believing that.
I have put in the sample, it is giving the data.
No, what it has actually done is that, this equipment itself has within it, the built-in
possibility of making multiple measurements, and averaging out.
What it is churning out is essentially the average of multiple measurements.
So, always remember that without multiple measurements and taking the average, you cannot
get rid-of the the random errors. So, when we, we, for example,
measure the resistance of a wire by applying different voltages, by measuring different
values of current and then tabulating them, dividing the voltage by current, by getting
thereby we get a large number of values of x: x₁, x₂, x₃...
Each will have a random error in it, but the random errors in some cases will be positive,
in some cases will be negative. And therefore, when you actually do the act
of averaging, then the positives will cancel out the negatives and ultimately the the the
effect of the random error will cancel off. That is what the naive belief is.
Okay? I mean: common-sense logic will say that.
That, if you have positive errors as well as negative errors,
then obviously the positives will cancel out the negatives and ultimately you will get
a reliable mean value. But the moment I say that, a few questions
immediately arise. And now before we end today’s class, let
me enumerate the question that immediately arise.
For example, how many observation should we take in order for the result that we get to
be reliable? The common sense does not immediately tell
us any any estimate of how many readings should we take.
Okay? We actually get a large number of values x₁,
x₂, x₃ ... each value is different.
Out of that, which one should we state in a paper?
Okay? That is, as as a measured value how how how
do we state it? How reliable will be that measured value?
Can we state some kind of a degree of confidence that we have in that measured value?
And how do we state it, how do we state it, in the sense that, an experiment has to be
repeatable? Repeatable anywhere in the world.
Therefore, I have to state it in a form, so that anybody repeating the experiment anywhere
in the world should get the same result. And how do I state it?
If I state a number, another person doing the experiment somewhere else will not get
the same number. So, how is it repeatable then?
There has, there has to be some way of stating the number that we measure in such a way that
it is repeatable. So, how do we state it?
Should we declare it as a range? If so, how big is the range?
And so, you know that we always use something called ‘error bar’.
How big will be the error bar? How do we decide, how big the error bar will
be? And when we state it with the error bar, our
statement is of the form that ‘I am fairly confident that the value lies within this
range’. What do you mean really, by saying that ‘I
am fairly confident’? Can you state a number as a degree of confidence?
So, all these questions arise. Natural questions.
We will come with this questions... We will deal with this questions in the next
class.