In the last class we saw that, whenever we make a measurement, we always take a sample
of some quantity and then we obtain the mean of that sample we get the sample mean and
the sample standard deviation. And from there we are trying to estimate the
character of the population mean and the population standard deviation.
So, we have a smaller number data and we are somehow trying to assess the character of
the of the quantity ‘out there’. And in doing so, we have seen that we had
reversed the argument. For example, if we have the measured quantity
of x, sorry, x̿ which is the average from the sample, and the the standard deviation
obtained from the sample. So, these are measured.
And we are trying to find out. So, we are trying to find out the population
mean and the population standard deviation. So, this is the target.
And then the central limit theorem stated that, if the number of data points is sufficiently
large, then the means will be distributed as a normal distribution something like this.
And the mean of that normal distribution will be the population mean 𝞵, and the standard
deviation will be the population standard deviation divided by this square root of the
number of data points taken. So, that is the claim of the central limit
theorem. Now, what we had done in the last class in
an example that we worked out? In that, we were given these and we were trying
to figure out if we have the a particular value, suppose this is the x scale and here
is my x̿, we are trying to figure out if they if I if I can define a range around x̿
and can claim that the actual 𝞵 will lie somewhere in this range.
And in doing so we argued that, if say, 𝞵 is here, 𝞵 is here suppose,
then the distance from x to the 𝞵 is the same as the distance from the 𝞵 to the
x. And therefore, the probability that 𝞵 lies
within — now here we had written it in the form of a multiplier times the standard deviation
— within z of z is the multiplier of x̿ and then we said that this is the same as
P is equal to x̿ lies within the z of 𝞵, sorry.
And this z is a a a a a factor a factor times the factor is z.
So, that is the factor and this is the multiplier of the standard deviation so standard deviation.
Now, if we express it this way then it is possible to obtain it because the value of
z for each value of z each value of z for example, if the z is here the the area under
the curve to the left of z can be found from the z table and from there we can solve the
problem. So, this z value is a factor.
There is a multiplier of the standard deviation, something times the standard deviation.
So, x̿ then x̿ say x̿ is lying x̿ is lying z of 𝞵.
So, if if if if x̿ is at a distance z from 𝞵 then it is possible to find out what
is the area under the curve to the left of that value of z.
And from that, we were able to to calculate the probabilities, and hence the confidence
with which we can state that 𝞵 will lie within a certain range of x̿.
You will notice one thing: what is the the standard deviation here?
It is the standard deviation of the distribution of the means, which is this.
And in that, we do not know 𝞂 because that is a target.
That is something that we do not know. That is of the population out there.
So, we argued that in the absence of the value of 𝞂, we estimate it by the measured value
of the standard deviation, which is s. So, we argued that, we will use s as an estimator
of 𝞂. You might ask how logical will that be?
Won’t that incur errors in the calculation, in the measurement?
Yes it will. But there is a logic behind this substitution.
The logic is that, if the number of samples is reasonably large, then we have seen that
the the central limit theorem claims that the distribution of the means will be almost
a normal distribution. And if you increase the number of samples
even more, it the it it it will not improve the approximation any further.
So, we know that around 25 is a good number of readings to take.
Now, if we take that minimum number of readings, then the the theory in statistics—which
will not go get into the details of that I will not get into—shows that if the number
of samples is reasonably large, then the difference between s and 𝞂 will be really small.
And since the actual value is 𝞂 divided by square root of n, and n will be a reasonably
large number, square root of that, therefore the difference of 𝞂 by square root of n
and s by square root of n will not be significant. That is why this justifies this use of the
s as an estimator of 𝞂. But, one thing is clear: that we cannot do
anything otherwise. We do not have a handle on 𝞂.
We only have s and therefore, we have to s use s as a estimator of 𝞂.
So, we have we will then substitute in place of 𝞂 by
s which is known, n known and therefore,
we have a a a handle on the standard deviation of this curve.
If we know that, then what factor needs to be multiplied with the standard deviation
to get the value, that is also known. And therefore, we can then refer to the z
table to extract the value of the probability. Now, we know we have seen earlier that that
if now I will plot that as a number here and here is my I give a range.
This is the value that I have I have measured, which is x̿, and suppose here is x̿ minus
the standard error of the mean and this is x̿ plus the standard error of the mean.
Then I know that in the curve that we had already drawn, here I have this normal curve,
and we are talking about a range which is... this is my x̿ and this is the this value
is here and this value is here. So, it is basically within one standard deviation,
and we know that the the area under this curve here is 68.3 percent of the whole.
What does that imply? It implies that if I define a range which
is x̿ minus the standard error (standard error means the standard deviation divide
by square root n) and x̿ plus the standard error...
If this range is defined, then I can be certain that the actual mean will lie within this
range, and I can state that with a confidence of 68.3 percent.
So, this is important. This is important to understand what is the
actual meaning of the statement. And this, in many cases, is written as the
error bar. So, you we will see graphs something like
this. So, these are the data points.
And if you add the error bars these will look like... you normally put the extremities like
this. So, this is how the graphs are actually drawn.
This means here is suppose a parameter or an independent variable, I would rather rather
say variable, and here is a dependent variable. Now, for each independent variable, you measure
the dependent variable and you may get these values.
Okay? But you will also specify by an error bar.
And what does that error bar signify? It signifies that you are 68.3 percent confident
that the actual value the actual value of the dependent variable for that independent
variable will lie within this range. So, with 68 percent confidence you will state
that. That is the meaning of the error bar.
Now, you might say that the 68.3 percent is not a very large extent of confidence.
Yes, surely, we will have to deal with that. We would see how to to represent a higher
level of confidence. But in general, the the meaning of the error
bar is that. So, whenever you will read an error bar, you
see an error bar in a paper, you have to interpret that accordingly.
It does not mean that the actual value will be in this range.
The actual value can be as well be outside, because this this confidence level is not
very large. That means, essentially you are stating that
there is 68 percent probability that the the actual value the actual value that we are
trying to measure, which is out there, will lie somewhere in
this range. Okay?
Sometimes we state the the value, for example, suppose we have measured something and we
have we have got x equal to, say, 3.56 centimeters. But, then we will have to state it with an
error which is plus minus, say, 0.03 centimeters. So, we always state it like that, because
we can never state the value exactly. Because we know that it is subject to some
random errors. Okay?
Sometimes these are also expressed a little bit differently, as a percentage error.
So, that can also be expressed as a percentage error, something like this: I will write x
as the average value that we have measured. This is in centimeters may be.
And then plus minus, now you have to put the standard error, which is say standard error
I I can write that it δx, the change in x divided by your x̿, which
is the extent of error in fraction into a 100 these many percentage.
This is the percentage error calculation. Okay?
So, you might state the result in both these ways.
This is in absolute value and this is as a percentage error.
So, the point that I am making is that, whenever you make an observation in measurement, you
state the result of the measurement always this way, and the extent of this error that
you state is also objectively calculated. Let me
let me just illustrate that by means of an example and then I hope that will be clearer.
Suppose suppose you have made a measurement of two
variables x and y, and you have made a large number of data points, say n equal to 25.
25 data points you have calculated and then from there you have you have found by calculating
that x̿ is 5.018 and Ȳ is 3.335. Okay?
But, the data that you have calculated: this 25 data points for x, 25 data points for y,
from there you can also calculate the s of the x variable; s is the standard deviation
of the x variable. And suppose you know how to do that.
you have... I have already said how to do that.
You have found that that to be 0.16. And the s of y is, say, 0.21.
Then then how do I specify these values? We need to calculate the standard error.
So, standard error in x will be 𝞂 by square root of n.
Now, 𝞂 we do not know, but we know the s.
Therefore, we substitute by that. So, 0.160 divided by square root of n, is
5, is equal to 0.032. And the standard error in y is this 𝞂.
This would be x and this would be y y root over n.
And we again substitute that by this standard deviation in y.
Major standard deviation in y. So, 0.211 by, again, 5 because the number
of data points were the same. And then this comes to be 0.042.
And therefore, having done this calculation, we will state that I have measured x as here
5.018 plus minus this 0.032 and I have measured y as this 3.335 plus minus 0.042.
Notice one thing, that after having these values that you have actually measured, when
you when you calculate the the mean, you could have calculated up to a larger number of decimal
points. Similarly, for the standard deviation you
could have calculated to a large number of decimal points.
If you have a calculator with 8 digits, you can do that.
But it will make no sense because the measurement has been taken with some kind of apparatus
which has a least count and it makes no sense to specify something to a least count that
is below that least count. So, if the measuring apparatus has an accuracy
which is meaningful to the third decimal place, you should specify everything only up to third
decimal place. There is no point going beyond that.
That is why, in this case I have expressed everything up to the third decimal place.
But remember to what decimal place you will specify it.
There is no pre-assigned prescription for that.
You have to do that depending on the instrument that you use.
Depending on the accuracy of that instrument. Okay?
Notice that everything hinges on the idea that in the distribution of the means, the
say, standard error of the mean or the standard deviation of this graph is 𝞂 by square
root of n. And I have already told you that in the in
the normal distribution curve you can easily integrate the normal distribution curve and
find out how much area is contained within some specifically ranges... specific limits.
And if it is up to the standard division of this graph, then it is 68 the the area under
the curve. So, area within 𝞵 plus standard sorry 𝞵
minus to 𝞵 plus standard error, 𝞵 plus standard error, this range, is 68.3 percent.
And we have also seen that the area if if you can calculate this, then you can also
calculate how much does it have to be taken so that the area under the curve curve is
95 percent, and that has been calculated. Area within I want to have this at 95 percent
𝞵 minus it is 1.96 1.96 standard error and 𝞵 plus 1.96 standard error.
This range is actually 95 percent. And the area within sorry 𝞵 minus if you
want to have 99 percent then 2.58 SE and 𝞵 plus 2.58 standard error.
So, if you consider a larger range, which is 1.96 standard error, this is a standard
error, then the area that is contained this area
that is contained which is this area I will I will patch it this area is 95 percent.
This has great important... In some fields the the confidence level of
68.3 percent is considered too low and there the demand is that you have to state it with
a confidence level of 95 percent. And if that is so, your error bar error bar
will have: this is the mean value that you have calculated and this is x̿ plus 1.96
standard error and this is x̿ minus 1.96 standard error.
In that case you have to specify that way and this is actually true for most fields.
So, but there are some fields that demand even more level of confidence when you state
the result, for example, 99 percent. In that case you have to put 2.58 here.
And there are some fields, especially something like for example,
the discovery of a new particle in particle physics, the discovery of gravitational wave,
is a discovery... in case of the discoveries the demand is far larger.
The demand is something that is stated as 5 𝞂.
What does 5𝞂 mean? So, in in in discoveries,
when we will say something has been discovered, in discoveries the demand is 5𝞂.
What is the 5𝞂? It is this number.
This number will then become 5. That means, almost the whole area is enclosed
within 5𝞂, 5 times the standard error of the mean and practically the whole area is
enclosed. What does this physically mean?
It physically means, I will just write it; it physically means that 5𝞂 will mean I
will put the line here yeah this means that only one in 3.5 million data points
can lie outside this range. Which means that if the result that is been
obtained, for example, the discovery of Higgs boson was a an observation an observation
where there was there was a kink in a graph. Now, if that kink happened due to random chance
event, then the possibility of such a random event being detected is one in 3.5 million.
Only when we reach that level of confidence we say that we have made a discovery.
So, it is very very exacting demand for the scientific community.
They have to make measurements and repeat it a large number of times and then only one
can reach that kind of 5𝞂 level of confidence. That means, if the observation that has been
observed is caused by something that is not what is intended to be,
then the probability of that happening is only one in 3.5 million.
That means, the area under the curve within 5𝞂 is absolutely there
the amount that is remaining is only this much.
So, that is a kind of demand in case of discoveries.