In this session, we will discuss a potential.
So, first of all, let us discuss about the frequency distribution.
What is the probability distribution?
So the frequency distribution gives the exact frequency or the number of times a data point occurs,
while the probability distribution gives the probability of occurrence of the given data point, when
the first number of the best cases are large, the frequency distribution and probability distribution
are similar in shape.
So let us see what is the difference between frequency distribution and probability distribution.
So let's say we have these data points.
Now, these data points range from 10 to AP.
And the frequency of these data point is two for them, three elevens, five to six turbines for four
beings, zettl, 15 values, 14, 16 values, then 17 values and six 18 values.
So if we want to find out and the total number of these values is 50.
So there are in total 15 values and this is the number of times this particular value occurs.
So 14 times 16 is occurring in the data.
So 16 is the mode of the data.
Similarly, we can find out the values and how often they occur and calculate the mean and median.
When you want to find out the relative frequency related frequency, you will give the probability of
the number occurring.
So let us see.
We want to find out the relative frequency of then occurring.
So the relative frequency of then occurring will be two out of 50.
So two times out of 50 then will be happening, three out of 50 times 11 will be a.
Five times out of 50 will be the opponents of 12, similarly, 14 times out of the 16 will be occurring.
So these are the frequencies it related to each other.
So these are the number of frequencies where these data will be happening and the complete some of these
relative frequencies comes out to be one.
And when we find out when we divide these values, we actually get the probability of the value occurring.
So the frequency actually helps us in driving the probability of the data occurring.
So the probability of occurring is zero point zero, which means that there is a four percent chance
of then occurring.
When we randomly pick out the value from this particular population, similarly, there is a 12 percent
chance of 13 occurring.
Here there is a 28 percent chance of 16 occurring when we take a sample out of this data.
So this is what the probability is and this is what the frequency is, and we can plot these together,
frequency distribution and the probability distribution.
So when we plot the frequency distribution, the area and the frequency distribution actually provides
the probability of the value occurring.
So here you can see this is the frequency distribution plot where the frequency of every value is given
here, the frequency is almost Siedel here.
The frequency used in frequency is 20 year, frequency is almost 22.
And out of this, the probability distribution plot is created, which is this line of code which is
given.
And this will give the probability of a particular value of three.
So if you want to find out the probability of one occurring, then we can find out the area below this
particular area, below this particular area, and it will give the probability of one knackering.
And similarly, we want to find out the probability of value occurring below 1.0.
Then we can find out the area occurring below this particular area of Dakhil.
Then it will give the probability of one point to work.
Similarly, if you want to find out the probability of value occurring between and B, then we can find
out the area below the Gulf between A and B, and it will give the probability of values B this between
this and B.
So this is what the distribution gulf will provide us.
Now, what is the distribution of sample means now we are getting close to the central limit.
Now we know what the frequency distribution is.
We know what probability distribution is.
So let's see what is the distribution of sample means.
So the distribution of sample means is the collection of sample means for all the possible random samples
of a particular size and.
So what we are doing is we are creating a collection of some means, so we are collecting.
We are creating a lot of samples.
Let us say we create hundred samples from those hundred samples.
We calculate different means.
We take out different means, and then we block these means together.
So we collect the sample means for all the possible random samples of a particular size.
And so the samples which we will be taking from the population will have a common size.
So the size will be seen and because the samples will be taken randomly, then the values in the samples
will be different.
So because the values of the samples will be different, the mean value will also be different for all
the samples.
So we can obtain these samples and the means from the population and then we applaud them together.
Now it is not a distribution of scores, but the distribution of statistics because they are not taking
the mean of the entire population.
We are not taking values from the entire population.
We are only plotting the needs of all this data means of different samples.
Now, when we do this, what will happen is the distribution will be kind of in a normal form, the
distribution will be normal in nature, life will be normal in nature, because when we take out samples,
the samples will be ranging from, let's say we have a population and the values of the population range
from one two hundred.
So now when we take all the samples, the sample values will be randomly distributed from one to hundred.
And they are taking the mean of these samples now, the mean of the sample will never be always the
mean of the samples and things would be fafi.
Sometimes it will be 51, sometimes it will be 52.
Sometimes it will be 48.
Sometimes it will be 42.
To now, based on the sample and how the random sample is generated, those values will be more number
of samples will have values closer to the center values, which is 50, and they will be very few samples
for which somehow the mean value will be near the edges.
So this is the reason why the gold which will be created from these machines will be in normal nature.
So it will be almost perfectly normal if either the population from which the sample is drawn is normal
or the number of sample is relatively large.
That is be almost basically the size of the sample is large, so if the size of the sample is large,
then the distribution has a mean that is equal to the population.
So let's look at some properties of normal, Goebels.
So what is the properties of the namiko?
A normal distribution of esveld sheep, the mean median and mode will be equal and will be located in
the center of the distribution.
Then a normal distribution is unimportant, which is it will have a single peak.
It will be symmetrical nature.
Then the Gulf is above the mean, which is equivalent to saying that if this ship is same on the both
side of the Gulf.
So if we have this line passing, so the shape of the Gulf towards the left will be the same as the
shape of the globe on the right.
Then the goal was continuous, that is, the values will range from some value to another value, but
there will not be any gaps in between.
Then the Gulf will never touch the axis here.
You can see this going endlessly beyond, but it is never touching the x axis here.
So theoretically, no matter how far in either direction the globe extends, it never needs access.
The total area under the normal distribution code is equal to one or two hundred, so the area and this
entire gulf, which is the probability actually, so the probability of a value occurring under this
Gulf will be.
So it will be either it will range from zero to 100 or zero.
The one will be the probability and the probability percentage will be zero to 100 percent.
Right.
So this is about the normal gulf and the area of the Gulf.
Give the probability of the point falling under the sea.
So if we have any point.
Then the area under this particular point bin line will give the probability of that value occurring.
Now, there are a few things about the Namiko, which is the area under the part of the normal Gulf
that lies within one standard deviation.
Now, this is the mean value.
This is the mean median mode value, the central value.
Now, this is one standard deviation away in the region and one standard deviation of it in the positive
direction.
So this is basically one standard deviation area of one standard deviation.
This one standard deviation area contains approximately sixty eight percent of the.
This one standard deviation bill contains approximately 60 percent of the data.
Then the two standard deviation.
This is the standard deviation area, it will contain almost 95 percent of the data and the three standard
deviation will contain almost ninety nine point seven percent.
This is the rule which is already defined for the normal golf or the baker, which is this is the percentage
distribution.
So how can we get help from this now?
I think we have already discussed this example, but again, talking about.
So we have this kind of distribution in the nonlegal.
So what will happen is that let's say we want to find out the thought of the two percent people out
of two point two percent, people who should be given a lease or should be given extra rewards for their
hard work.
So what will we do?
We will find out later, see the dolphin things.
And from the top ratings, we will find out the top two point, the weight person, people who are operating,
and then we will be giving them a reward.
So how we can do it, we can simply find out the mean value.
Because we cannot really find this value, right?
How will we know what is going to it?
So what we can do is we can simplify the mean value and we can find out the standard deviation.
And based on the mean value plus the standard deviation, we can declare that whoever is getting the
rating above mean the standard deviation.
We will give them whatever.
Or let's say someone who has everything to you mean minus two standard deviation, we will give them
additional training so that they perform better next quarter.
They actually can do something like the.
So this is the population distribution.
So this is the normal population distribution curve, and for this particular article, the mean is
zero point one, the meanness zero point one.
This is the distribution and the mean value is zero point one to.
The standard deviation is twenty one 06, this is the standard deviation for this block, we have mean
as zero point one two and standard deviation as twenty point zero six.
Now, what we are doing is we are taking out several samples, so we will take, I would say, 50,
100 or 200 or 500 samples out of this of this population data.
Out of this entire data, we will take out different samples.
Now we are taking out samples so the values will anyhow range from this minus two plus hundred.
OK, but we need to find out some sample values out of it.
So what we are doing is we are taking out a few samples.
So the values of the samples, you can see the distributions are different.
See this, there are different distributions for the sampas, they're not exactly the same samples.
So what happens is the standard deviation is also slightly different for the samples here.
The standard deviation is nineteen point something.
Here, the standard deviation is eighteen point something and so on.
So different samples will have different standard deviations.
Similarly, these samples which we have generated will have some mean value.
Now, this means that you will also be different here, the mean value is minus zero point one eight
one here, the mean value is minus one point seventy seven.
Here, the money will mean value is zero point three.
Here, the mean values, minus zero point zero four.
So these mean values are also different.
So what happens is when we take five hundred or two hundred or any number of such samples, then we
create the.
We create the sampling distribution out of it, which is the distribution of the sample means so here
what we have done is we have created a block of all the means.
Of the samples that we have obtained, so here we have certain mean values, so we have only plotted
these export values in this particular judge you.
OK, so now when we have plotted this.
Sample means here.
Now, this is where central to them comes in.
So what we have done enough, we had one population out of this population, we have taken out different
samples, the sample size, we are expecting to be more than 30 or at least 30 so that it is a good
representative of the population.
And they have created several samples from these samples, we have calculated the mean value of each
and every something so mean value of each sample comes out to the buttocks.
But we have, let's say, one hundred or one, two, one hundred samples.
And we have the economics blood from these samples.
Now, from the samples, we have generated a lot.
And this is actually a lot of the mean values, which is the X values from these samples.
So this is a lot of different values from different samples of what we created.
We needed a distribution means one 60 random samples.
We have taken 160 random samples and taken out the mean value and plotted them here.
Now each consists of one hundred and three observations.
So the sample size here is one hundred and three.
That is the end value.
The end value here is one hundred and three.
And the number of samples is 160.
Now, from this sampling distribution, this distribution of means.
The distribution of means is also already sampling distribution.
So from this sampling distribution, the mean is zero point zero eight.
The calculated mean of this sampling distribution is zero point zero eight letters compared with the
population mean.
Here, the mean is zero point one to.
And the the standard deviation is twenty point zero six.
Here, the standard error is one point seven four.
Now let us see.
So.
The central limit theorem states that.
The distribution of sample mean.
The sampling distribution should be nearly normal.
But just through the sampling distribution, which we have created, which is the distribution of sample
means, is normal.
And the mean of the sampling distribution should be approximately equal to the population.
That is the mean of the sampling.
Distribution should be approximately equal to the population me, which is zero point one.
So what is it?
The mean of the sampling distribution is zero point zero it.
And the mean of the population is zero point one, which is almost similar.
The mean is almost close to each other.
Now, the standard error is the standard deviation of the sampling me, the standard error which has
been calculated, is actually the standard, the standard deviation of the sampling distribution.
Standard error is the standard deviation of the sampling distribution.
So what does that signify?
This standard error is actually equal to the standard deviation of the population divided by the square
root of the size of the sample.
So twenty point zero six, which is the.
Standard deviation of the population.
Twenty point zero six, Bob, standard deviation of the population.
And here we have the square root of one zero three, which is the.
One hundred and three observations, so Vera.
110.
Today is the end, which is the size of each sample.
So this gives us one point nineteen.
Which is.
One point seven four, so this is also equivalent, so this is what we are doing from this.
Send a to.
So we can find out the characteristics of the population.
From the sampling distribution, from the sampling distribution, we can create different sampling distribution
and find out the characteristics of the population here.
So this is what central imitative is, which is that the distribution of the sample mean should be nearly
normal.
In the mean of the sampling, distribution should be approximately equal to the population mean.
That is the mean of the sampling distribution of this sampling distribution will be almost equal to
the mean of the population distribution.
And the standard error, which is the standard deviation of the sampling distribution, will be equivalent
to the.
Standard deviation of the population distribution divided by the square root of the sample size, the
number of observations in each sample.
So this is what we have learned from the central limit.