WEBVTT
Kind: captions
Language: en
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In the last class we saw that, whenever we
make a measurement, we always take a sample
00:00:28.269 --> 00:00:37.680
of some quantity and then we obtain the mean
of that sample we get the sample mean and
00:00:37.680 --> 00:00:43.050
the sample standard deviation.
And from there we are trying to estimate the
00:00:43.050 --> 00:00:47.640
character of the population mean and the population
standard deviation.
00:00:47.640 --> 00:00:54.519
So, we have a smaller number data and we are
somehow trying to assess the character of
00:00:54.519 --> 00:01:02.140
the of the quantity ‘out there’.
And in doing so, we have seen that we had
00:01:02.140 --> 00:01:10.030
reversed the argument.
For example, if we have the measured quantity
00:01:10.030 --> 00:01:22.570
of x, sorry, x̿ which is the average from
the sample, and the the standard deviation
00:01:22.570 --> 00:01:33.850
obtained from the sample.
So, these are measured.
00:01:33.850 --> 00:01:42.050
And we are trying to find out.
So, we are trying to find out the population
00:01:42.050 --> 00:01:53.489
mean and the population standard deviation.
So, this is the target.
00:01:53.489 --> 00:02:04.300
And then the central limit theorem stated
that, if the number of data points is sufficiently
00:02:04.300 --> 00:02:17.530
large, then the means will be distributed
as a normal distribution something like this.
00:02:17.530 --> 00:02:25.060
And the mean of that normal distribution will
be the population mean 𝞵, and the standard
00:02:25.060 --> 00:02:33.920
deviation will be the population standard
deviation divided by this square root of the
00:02:33.920 --> 00:02:42.880
number of data points taken.
So, that is the claim of the central limit
00:02:42.880 --> 00:02:50.400
theorem.
Now, what we had done in the last class in
00:02:50.400 --> 00:02:58.260
an example that we worked out?
In that, we were given these and we were trying
00:02:58.260 --> 00:03:04.159
to figure out if we have the a particular
value, suppose this is the x scale and here
00:03:04.159 --> 00:03:14.310
is my x̿, we are trying to figure out if
they if I if I can define a range around x̿
00:03:14.310 --> 00:03:20.689
and can claim that the actual 𝞵 will lie
somewhere in this range.
00:03:20.689 --> 00:03:29.569
And in doing so we argued that, if say, 𝞵
is here, 𝞵 is here suppose,
00:03:29.569 --> 00:03:37.099
then the distance from x to the 𝞵 is the
same as the distance from the 𝞵 to the
00:03:37.099 --> 00:03:46.319
x.
And therefore, the probability that 𝞵 lies
00:03:46.319 --> 00:03:58.049
within — now here we had written it in the
form of a multiplier times the standard deviation
00:03:58.049 --> 00:04:14.680
— within z of z is the multiplier of x̿
and then we said that this is the same as
00:04:14.680 --> 00:04:34.740
P is equal to x̿ lies within the z of 𝞵,
sorry.
00:04:34.740 --> 00:04:52.180
And this z is a a a a a factor a factor times
the factor is z.
00:04:52.180 --> 00:05:01.730
So, that is the factor and this is the multiplier
of the standard deviation so standard deviation.
00:05:01.730 --> 00:05:12.830
Now, if we express it this way then it is
possible to obtain it because the value of
00:05:12.830 --> 00:05:19.870
z for each value of z each value of z for
example, if the z is here the the area under
00:05:19.870 --> 00:05:27.139
the curve to the left of z can be found from
the z table and from there we can solve the
00:05:27.139 --> 00:05:30.810
problem.
So, this z value is a factor.
00:05:30.810 --> 00:05:36.780
There is a multiplier of the standard deviation,
something times the standard deviation.
00:05:36.780 --> 00:05:49.160
So, x̿ then x̿ say x̿ is lying x̿ is lying
z of 𝞵.
00:05:49.160 --> 00:05:56.620
So, if if if if x̿ is at a distance z from
𝞵 then it is possible to find out what
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is the area under the curve to the left of
that value of z.
00:06:02.879 --> 00:06:11.509
And from that, we were able to to calculate
the probabilities, and hence the confidence
00:06:11.509 --> 00:06:19.120
with which we can state that 𝞵 will lie
within a certain range of x̿.
00:06:19.120 --> 00:06:26.220
You will notice one thing: what is the the
standard deviation here?
00:06:26.220 --> 00:06:33.669
It is the standard deviation of the distribution
of the means, which is this.
00:06:33.669 --> 00:06:39.750
And in that, we do not know 𝞂 because that
is a target.
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That is something that we do not know.
That is of the population out there.
00:06:43.789 --> 00:06:51.570
So, we argued that in the absence of the value
of 𝞂, we estimate it by the measured value
00:06:51.570 --> 00:07:08.849
of the standard deviation, which is s.
So, we argued that, we will use s as an estimator
00:07:08.849 --> 00:07:15.520
of 𝞂.
You might ask how logical will that be?
00:07:15.520 --> 00:07:21.000
Won’t that incur errors in the calculation,
in the measurement?
00:07:21.000 --> 00:07:27.169
Yes it will.
But there is a logic behind this substitution.
00:07:27.169 --> 00:07:38.430
The logic is that, if the number of samples
is reasonably large, then we have seen that
00:07:38.430 --> 00:07:47.069
the the central limit theorem claims that
the distribution of the means will be almost
00:07:47.069 --> 00:07:53.090
a normal distribution.
And if you increase the number of samples
00:07:53.090 --> 00:08:01.680
even more, it the it it it will not improve
the approximation any further.
00:08:01.680 --> 00:08:08.520
So, we know that around 25 is a good number
of readings to take.
00:08:08.520 --> 00:08:19.430
Now, if we take that minimum number of readings,
then the the theory in statistics—which
00:08:19.430 --> 00:08:28.580
will not go get into the details of that I
will not get into—shows that if the number
00:08:28.580 --> 00:08:41.159
of samples is reasonably large, then the difference
between s and 𝞂 will be really small.
00:08:41.159 --> 00:08:49.160
And since the actual value is 𝞂 divided
by square root of n, and n will be a reasonably
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large number, square root of that, therefore
the difference of 𝞂 by square root of n
00:08:55.930 --> 00:09:07.350
and s by square root of n will not be significant.
That is why this justifies this use of the
00:09:07.350 --> 00:09:13.380
s as an estimator of 𝞂.
But, one thing is clear: that we cannot do
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anything otherwise.
We do not have a handle on 𝞂.
00:09:18.029 --> 00:09:23.990
We only have s and therefore, we have to s
use s as a estimator of 𝞂.
00:09:23.990 --> 00:09:30.720
So, we have
we will then substitute in place of 𝞂 by
00:09:30.720 --> 00:09:34.769
s which is known,
n known and therefore,
00:09:34.769 --> 00:09:41.230
we have a a a handle on the standard deviation
of this curve.
00:09:41.230 --> 00:09:46.240
If we know that, then what factor needs to
be multiplied with the standard deviation
00:09:46.240 --> 00:09:52.190
to get the value, that is also known.
And therefore, we can then refer to the z
00:09:52.190 --> 00:10:06.680
table to extract the value of the probability.
Now, we know we have seen earlier that that
00:10:06.680 --> 00:10:23.519
if now I will plot that as a number here and
here is my I give a range.
00:10:23.519 --> 00:10:33.380
This is the value that I have I have measured,
which is x̿, and suppose here is x̿ minus
00:10:33.380 --> 00:10:44.889
the standard error of the mean and this is
x̿ plus the standard error of the mean.
00:10:44.889 --> 00:10:55.350
Then I know that in the curve that we had
already drawn, here I have this normal curve,
00:10:55.350 --> 00:11:06.910
and we are talking about a range which is...
this is my x̿ and this is the this value
00:11:06.910 --> 00:11:14.380
is here and this value is here.
So, it is basically within one standard deviation,
00:11:14.380 --> 00:11:30.339
and we know that the the area under this curve
here is 68.3 percent of the whole.
00:11:30.339 --> 00:11:38.550
What does that imply?
It implies that if I define a range which
00:11:38.550 --> 00:11:45.980
is x̿ minus the standard error (standard
error means the standard deviation divide
00:11:45.980 --> 00:11:50.420
by square root n) and x̿ plus the standard
error...
00:11:50.420 --> 00:11:58.260
If this range is defined, then I can be certain
that the actual mean will lie within this
00:11:58.260 --> 00:12:03.240
range, and I can state that with a confidence
of 68.3 percent.
00:12:03.240 --> 00:12:09.380
So, this is important.
This is important to understand what is the
00:12:09.380 --> 00:12:17.330
actual meaning of the statement.
And this, in many cases, is written as the
00:12:17.330 --> 00:12:23.589
error bar.
So, you we will see graphs something like
00:12:23.589 --> 00:12:43.650
this.
So, these are the data points.
00:12:43.650 --> 00:13:11.940
And if you add the error bars these will look
like... you normally put the extremities like
00:13:11.940 --> 00:13:22.290
this.
So, this is how the graphs are actually drawn.
00:13:22.290 --> 00:13:37.160
This means here is suppose a parameter or
an independent variable, I would rather rather
00:13:37.160 --> 00:13:58.490
say variable, and here is a dependent variable.
Now, for each independent variable, you measure
00:13:58.490 --> 00:14:02.019
the dependent variable and you may get these
values.
00:14:02.019 --> 00:14:06.700
Okay?
But you will also specify by an error bar.
00:14:06.700 --> 00:14:16.589
And what does that error bar signify?
It signifies that you are 68.3 percent confident
00:14:16.589 --> 00:14:22.700
that the actual value the actual value of
the dependent variable for that independent
00:14:22.700 --> 00:14:29.880
variable will lie within this range.
So, with 68 percent confidence you will state
00:14:29.880 --> 00:14:35.300
that.
That is the meaning of the error bar.
00:14:35.300 --> 00:14:45.120
Now, you might say that the 68.3 percent is
not a very large extent of confidence.
00:14:45.120 --> 00:14:53.430
Yes, surely, we will have to deal with that.
We would see how to to represent a higher
00:14:53.430 --> 00:14:59.440
level of confidence.
But in general, the the meaning of the error
00:14:59.440 --> 00:15:02.680
bar is that.
So, whenever you will read an error bar, you
00:15:02.680 --> 00:15:07.470
see an error bar in a paper, you have to interpret
that accordingly.
00:15:07.470 --> 00:15:11.720
It does not mean that the actual value will
be in this range.
00:15:11.720 --> 00:15:16.550
The actual value can be as well be outside,
because this this confidence level is not
00:15:16.550 --> 00:15:19.720
very large.
That means, essentially you are stating that
00:15:19.720 --> 00:15:27.870
there is 68 percent probability that the the
actual value the actual value that we are
00:15:27.870 --> 00:15:30.490
trying to measure,
which is out there, will lie somewhere in
00:15:30.490 --> 00:15:35.940
this range.
Okay?
00:15:35.940 --> 00:15:41.839
Sometimes we state the the value, for example,
suppose we have measured something and we
00:15:41.839 --> 00:15:52.330
have we have got x equal to, say, 3.56 centimeters.
But, then we will have to state it with an
00:15:52.330 --> 00:16:02.571
error which is plus minus, say, 0.03 centimeters.
So, we always state it like that, because
00:16:02.571 --> 00:16:08.209
we can never state the value exactly.
Because we know that it is subject to some
00:16:08.209 --> 00:16:12.529
random errors.
Okay?
00:16:12.529 --> 00:16:19.060
Sometimes these are also expressed a little
bit differently, as a percentage error.
00:16:19.060 --> 00:16:29.199
So, that can also be expressed as a percentage
error, something like this: I will write x
00:16:29.199 --> 00:16:37.290
as the average value that we have measured.
This is in centimeters may be.
00:16:37.290 --> 00:16:45.390
And then plus minus, now you have to put the
standard error, which is say standard error
00:16:45.390 --> 00:16:53.350
I I can write that it δx,
the change in x divided by your x̿, which
00:16:53.350 --> 00:17:01.339
is the extent of error in fraction into a
100 these many percentage.
00:17:01.339 --> 00:17:05.319
This is the percentage error calculation.
Okay?
00:17:05.319 --> 00:17:09.120
So, you might state the result in both these
ways.
00:17:09.120 --> 00:17:14.240
This is in absolute value and this is as a
percentage error.
00:17:14.240 --> 00:17:20.220
So, the point that I am making is that, whenever
you make an observation in measurement, you
00:17:20.220 --> 00:17:27.910
state the result of the measurement always
this way, and the extent of this error that
00:17:27.910 --> 00:17:34.830
you state is also objectively calculated.
Let me
00:17:34.830 --> 00:17:42.780
let me just illustrate that by means of an
example and then I hope that will be clearer.
00:17:42.780 --> 00:17:50.640
Suppose
suppose you have made a measurement of two
00:17:50.640 --> 00:18:00.760
variables x and y, and you have made a large
number of data points, say n equal to 25.
00:18:00.760 --> 00:18:08.280
25 data points you have calculated and then
from there you have you have found by calculating
00:18:08.280 --> 00:18:25.500
that x̿ is 5.018 and Ȳ is 3.335.
Okay?
00:18:25.500 --> 00:18:34.270
But, the data that you have calculated: this
25 data points for x, 25 data points for y,
00:18:34.270 --> 00:18:43.130
from there you can also calculate the s of
the x variable; s is the standard deviation
00:18:43.130 --> 00:18:46.100
of the x variable.
And suppose you know how to do that.
00:18:46.100 --> 00:18:48.830
you have... I have already said how to do
that.
00:18:48.830 --> 00:19:00.010
You have found that that to be 0.16.
And the s of y is, say, 0.21.
00:19:00.010 --> 00:19:13.500
Then then how do I specify these values?
We need to calculate the standard error.
00:19:13.500 --> 00:19:25.539
So, standard error in x will be 𝞂 by square
root of n.
00:19:25.539 --> 00:19:29.390
Now, 𝞂 we do not know, but we know the
s.
00:19:29.390 --> 00:19:37.860
Therefore, we substitute by that.
So, 0.160 divided by square root of n, is
00:19:37.860 --> 00:19:49.659
5, is equal to 0.032.
And the standard error in y is this 𝞂.
00:19:49.659 --> 00:19:57.110
This would be x and this would be y y root
over n.
00:19:57.110 --> 00:20:00.909
And we again substitute that by this standard
deviation in y.
00:20:00.909 --> 00:20:10.390
Major standard deviation in y.
So, 0.211 by, again, 5 because the number
00:20:10.390 --> 00:20:17.120
of data points were the same.
And then this comes to be 0.042.
00:20:17.120 --> 00:20:27.020
And therefore, having done this calculation,
we will state that I have measured x as here
00:20:27.020 --> 00:20:53.650
5.018 plus minus this 0.032 and I have measured
y as this 3.335 plus minus 0.042.
00:20:53.650 --> 00:21:01.400
Notice one thing, that after having these
values that you have actually measured, when
00:21:01.400 --> 00:21:08.929
you when you calculate the the mean, you could
have calculated up to a larger number of decimal
00:21:08.929 --> 00:21:11.990
points.
Similarly, for the standard deviation you
00:21:11.990 --> 00:21:14.559
could have calculated to a large number of
decimal points.
00:21:14.559 --> 00:21:18.620
If you have a calculator with 8 digits, you
can do that.
00:21:18.620 --> 00:21:24.700
But it will make no sense because the measurement
has been taken with some kind of apparatus
00:21:24.700 --> 00:21:31.010
which has a least count and it makes no sense
to specify something to a least count that
00:21:31.010 --> 00:21:38.370
is below that least count.
So, if the measuring apparatus has an accuracy
00:21:38.370 --> 00:21:44.260
which is meaningful to the third decimal place,
you should specify everything only up to third
00:21:44.260 --> 00:21:47.870
decimal place.
There is no point going beyond that.
00:21:47.870 --> 00:21:54.500
That is why, in this case I have expressed
everything up to the third decimal place.
00:21:54.500 --> 00:21:58.740
But remember to what decimal place you will
specify it.
00:21:58.740 --> 00:22:02.809
There is no pre-assigned prescription for
that.
00:22:02.809 --> 00:22:06.900
You have to do that depending on the instrument
that you use.
00:22:06.900 --> 00:22:12.360
Depending on the accuracy of that instrument.
Okay?
00:22:12.360 --> 00:22:20.161
Notice that everything hinges on the idea
that in the distribution of the means, the
00:22:20.161 --> 00:22:27.820
say, standard error of the mean or the standard
deviation of this graph is 𝞂 by square
00:22:27.820 --> 00:22:33.860
root of n.
And I have already told you that in the in
00:22:33.860 --> 00:22:39.510
the normal distribution curve you can easily
integrate the normal distribution curve and
00:22:39.510 --> 00:22:45.980
find out how much area is contained within
some specifically ranges... specific limits.
00:22:45.980 --> 00:22:53.370
And if it is up to the standard division of
this graph, then it is 68 the the area under
00:22:53.370 --> 00:23:09.350
the curve.
So, area within 𝞵 plus standard sorry 𝞵
00:23:09.350 --> 00:23:31.720
minus to 𝞵 plus standard error, 𝞵 plus
standard error, this range, is 68.3 percent.
00:23:31.720 --> 00:23:36.750
And we have also seen that the area if
if you can calculate this, then you can also
00:23:36.750 --> 00:23:44.360
calculate how much does it have to be taken
so that the area under the curve curve is
00:23:44.360 --> 00:23:58.049
95 percent, and that has been calculated.
Area within I want to have this at 95 percent
00:23:58.049 --> 00:24:13.640
𝞵 minus it is 1.96 1.96 standard error
and 𝞵 plus 1.96 standard error.
00:24:13.640 --> 00:24:35.830
This range is actually 95 percent.
And the area within sorry 𝞵 minus if you
00:24:35.830 --> 00:24:54.750
want to have 99 percent then 2.58 SE and 𝞵
plus 2.58 standard error.
00:24:54.750 --> 00:25:11.799
So, if you consider a larger range, which
is 1.96 standard error, this is a standard
00:25:11.799 --> 00:25:17.460
error,
then the area that is contained this area
00:25:17.460 --> 00:25:29.820
that is contained which is this area I will
I will patch it this area is 95 percent.
00:25:29.820 --> 00:25:40.190
This has great important...
In some fields the the confidence level of
00:25:40.190 --> 00:25:50.190
68.3 percent is considered too low and there
the demand is that you have to state it with
00:25:50.190 --> 00:25:58.710
a confidence level of 95 percent.
And if that is so, your error bar error bar
00:25:58.710 --> 00:26:15.140
will have: this is the mean value that you
have calculated and this is x̿ plus 1.96
00:26:15.140 --> 00:26:24.460
standard error and this is x̿ minus 1.96
standard error.
00:26:24.460 --> 00:26:32.360
In that case you have to specify that way
and this is actually true for most fields.
00:26:32.360 --> 00:26:42.120
So, but there are some fields that demand
even more level of confidence when you state
00:26:42.120 --> 00:26:52.080
the result, for example, 99 percent.
In that case you have to put 2.58 here.
00:26:52.080 --> 00:26:58.770
And there are some fields, especially something
like for example,
00:26:58.770 --> 00:27:06.970
the discovery of a new particle in particle
physics, the discovery of gravitational wave,
00:27:06.970 --> 00:27:14.220
is a discovery... in case of the discoveries
the demand is far larger.
00:27:14.220 --> 00:27:19.840
The demand is something that is stated as
5 𝞂.
00:27:19.840 --> 00:27:24.039
What does 5𝞂 mean?
So, in in in discoveries,
00:27:24.039 --> 00:27:38.169
when we will say something has been discovered,
in discoveries the demand is 5𝞂.
00:27:38.169 --> 00:27:41.899
What is the 5𝞂?
It is this number.
00:27:41.899 --> 00:27:50.330
This number will then become 5.
That means, almost the whole area is enclosed
00:27:50.330 --> 00:28:01.309
within 5𝞂, 5 times the standard error of
the mean and practically the whole area is
00:28:01.309 --> 00:28:03.730
enclosed.
What does this physically mean?
00:28:03.730 --> 00:28:10.460
It physically means, I will just write it;
it physically means that 5𝞂 will mean I
00:28:10.460 --> 00:28:46.220
will put the line here yeah this means that
only one in 3.5 million data points
00:28:46.220 --> 00:29:02.860
can lie outside this range.
Which means that if the result that is been
00:29:02.860 --> 00:29:11.779
obtained, for example, the discovery of Higgs
boson was a an observation an observation
00:29:11.779 --> 00:29:22.549
where there was there was a kink in a graph.
Now, if that kink happened due to random chance
00:29:22.549 --> 00:29:35.370
event, then the possibility of such a random
event being detected is one in 3.5 million.
00:29:35.370 --> 00:29:42.770
Only when we reach that level of confidence
we say that we have made a discovery.
00:29:42.770 --> 00:29:47.860
So, it is very very exacting demand for the
scientific community.
00:29:47.860 --> 00:29:53.660
They have to make measurements and repeat
it a large number of times and then only one
00:29:53.660 --> 00:30:02.250
can reach that kind of 5𝞂 level of confidence.
That means, if the observation that has been
00:30:02.250 --> 00:30:08.850
observed is caused by something that is
not what is intended to be,
00:30:08.850 --> 00:30:12.940
then the probability of that happening is
only one in 3.5 million.
00:30:12.940 --> 00:30:20.799
That means, the area under the curve within
5𝞂 is absolutely there
00:30:20.799 --> 00:30:26.470
the amount that is remaining is only this
much.
00:30:26.470 --> 00:30:31.650
So, that is a kind of demand in case of discoveries.