Hello, everybody.
In this particular session, we will be discussing about a new law, which is yet another model of another
testing method used in statistics.
Now know what is similar to other testing methods, but the only difference is that there the other
testing methods like.
Guys, where they help us compare one or two variables, ANOVA helps us in deal with multiple values,
that there's multiple variables.
Now, what is an over and over is analysis obedience, which is taking the in from analysis and or from
all the wrong variance analysis of variance.
Now, it is a technique from statistical inference that allows us to deal with several populations.
So as I said, in the case of other tests, we used to have either one or two population and we were
trying to find out if these populations are similar or not.
Now, in case I have to compare not just two populations, but if I have to compare five, six or hundred
populations.
In that scenario, I will have to conduct a lot of this to come to a conclusion, but I know what helps
me to reduce the problem and do just one thing the best.
So instead of conducting hundred faced.
I can find out the conclusion by conducting just one single test, which is unknowable.
That is why I know what is extensively used when we have a large number of populations, when we have
to conduct this for multiple populations.
So let us get further about it.
So what do we have for that is so let us say we have a similar example.
OK, now we have the same data set where we have a lot of Eminem's.
So we want to connect the world with them now for this, when we want to know what.
We will test a different null hypothesis.
So usually what the null hypothesis used to be was that is one is equal to.
For another test, it will be Mutu equal to military, for another test, it will be military will do
food and there will be more number of tests.
So if I was conducting would be test, I will have to create so many null hypothesis.
But in case of a Nova, I have just one null hypothesis, and the null hypothesis will be it's not that
this new one is equal to the military is equal to Muifa.
So I am combining all the formal hypothesis into just one null hypothesis here.
So this state that there is no difference between the mean waves of red, blue and green Eminem's,
there is no difference between these four populations.
And the alternate hypothesis will be that there is at least one population, which is different.
Right.
That is at least one population which is having a different date.
It could be either red, the blue, green or orange.
And any one of these, if they have a different way, then it will be an alternate hypothesis.
Now, how this is a combination of those several hypotheses, so let's say I was doing this, then these
would be the different hypotheses which I will have.
What are these?
One is that the mean little population of Red Gandy's is not equal to the mean weight of the population
of the blue.
Gandy's right.
This is one alternate hypothesis.
What is the one that the mean weight of the population of blue candies is not equal to the weight of
the green can be similarly green compared with orange and the green compared with red and blue, compared
with orange and blue compared with red.
So all the combinations which we have.
So we are combining all of those.
I'm creating just one list here.
So instead of conducting so many tests, we are combining all of those details in just one single test.
This is what I know, what helps says in achieving.
So now the P value, which we will be using in probability distribution for ANOVA is known as if that's
OK, as we know that for every test we have a different P value rate based on the problem than the solution.
The distribution is different.
Hence, we have a different P value for each and every test.
The bewailing for the test is calculated differently, the distance differently.
So.
Guys, when four guys split again, the statistics this guy did differently, so similarly, in case
of another, we have the P-value as if distribution.
Now, the calculation which we will be doing in case of ANOVA is a little more complex in comparison
to the other calculations which we have done because again, solving a very difficult problem.
So that is why we won't usually conduct this using by home.
So we will be using some computer to solve this problem.
OK, similarly for other guests, also, usually what we do is we solve these problems by using a system
I'm not by.
So all the examples which I have shown you now, we have solved the problems by hand so that you actually
know what is happening behind.
But in actual life, you will be solving those problems using a system, using a computer.
OK, so I'll show you how you would be doing that.
But now let us again try to solve this by hand.
OK?
So what are the steps which are involved in the ANOVA test, so let us go through these steps once and
then we will go through each step one by one.
So the first thing that we do is we calculate the sample mean for each of these sample as well as we
mean for all the same for OK, so we will be calculating through means.
One thing will be for each and every population, for each and every sample and another we would be
for the all of the samples which we have from the data, all of these.
OK, the next is calculating the sum of squared error.
OK, now with each sample we square the division of each do the data value from the sample.
We and the sum of all the squared deviation is the sum of the square in an abbreviated sample sweater.
So basically what we are doing is we are just finding out the mean and we are comparing the mean with
each and every value in the sample and finding out the.
It is from that submissives attracting the expected value from the actual value, the value which we
have in hand, that the mean which we have and the values in the sample, and then we square those and
we square those what each and every sample we add those values and see how we do that.
But for now, this is just somebody offered.
Next, we calculate the sum of squares of three twin.
OK, so what does a Mosquera Friedman Square division of each sample mean from the overall mean?
So here what we were doing is initially we were finding out the sample squared error for each and every
sample.
OK, so the squared was a sample specific thing, but now we are finding out the sum of squared error
specific to all the data, so we will be calculating an overall sum of spread error here.
OK.
Then we find out the degree of freedom again, there will be two types of degrees of freedom, one degree
of freedom will be the respective to the Sam Phillips in, for example, we will be calculating the
degree of freedom and another degree of freedom will be for the entire treatment for the entire removalist.
So we will be calculating for all the samples which we have.
So based on all the samples, we will be calculating the other degree of freedom.
OK, then we will be calculating the mean squared of error and mean squared off frequent.
And later, we will get the stylistics, which is equal to the.
Mean squared, we mean squared off treatment, divided by the mean squared off at this particular value,
divided by this particular value.
OK, so let's go ahead and lay this out.
It will be a little difficult for you to understand because it is a very complex process, but once
you see the example, it will be a little easier to understand.
So for us, we will find out the date in time for me.
OK, so what do we have?
We have four independent population.
OK, so for those four independent population, the null hypothesis will be No one is equal to me is
equal to Mutiny's equal.
And what will be the alternate hypothesis?
The alternate hypothesis will be that either one of these.
Is not equal to the others, so it will be like at least one of these population mean is different from
other samples.
OK.
Now, for this example, we will be using a sample size of three so that it is simply not the way to
solve this problem.
You will be having a large dataset.
So it will be a little complex in nature.
But for example, I'm thinking small number of samples, so then it is easier to this time.
So what we do here is so this is my first population.
Plus, population contains three values, 12, nine and 12, second population contains seven then and
for the third population contains five, eight and 11.
Next we have five, eight and eight.
Now, what is the sample mean, the sample mean will be the meaning of these three values for the first
population, it comes out to be 11.
For the second population, the mean of these three values comes out to weepin.
For the third population, the meaning of these sea values come out to be eight, and for the population,
the mean of these three values turns out to be seven.
So now what we have done.
We have calculated the sample mean for each of our sample as well as the mean for all of this.
So this is the mean for these samples.
Now we need to find out.
The mean for all of the details will mean of all of the data we can give.
You mean of all these four values?
OK, so it comes out to be nine, so we have two types of means.
No.
One is the entire sample means no.
I mean, of all the samples and these are means specific to each and every sample.
OK, next, what we are doing here, this.
Next, we will include the sum of squared deviation from each sample mean.
OK, so what do we have here?
The meaning of the first sample sample from the first population is 11.
And these are the values 12, nine and 12.
So what I do is I subtribe this mean value from each of these values in the sample.
I where those values.
So I get the.
So a sum of squared off error for each population like this, so this is what the population does,
what the second population, third population and fourth population.
Again, how did you how did I do it?
The values are 17 and 13 for the second population, minus 10.
So what will be the values the most, whether it will be seven, minus 10 will square plus 10, minus
10 will squared plus 13, minus 10, Woodsworth, what is it, seven, minus 10.
Will 20 plus then minus 10.
Looks with plus 13 minus.
Then we'll square.
It is equal to 80.
Right.
So this is the sum of squarer for this population.
Similarly here, the menas e so I subtract these values from each of the population values.
I'm square those values to get the sample mean some squared header here.
Right now, this is the sum of squared it up for.
Each and every population, and now we want to find out the summer of sweat and for the entire population,
for all the values which we have so that we will get by adding six plus 18 plus 18 plus six.
So it comes out to be forty eight.
OK, what have we done?
Now we have calculated the sample mean for sample and the mean of all the samples.
Then we calculated the sum of sweat and what each and every population.
Then we calculated the sum of squared error for each and every value in the.
Some of all the squared deviation, right?
So this is the sum of squared, Ayda.
Next, we want to find out the sum of square off treatment of gay.
So now we want to find out the sum of squared off three minutes.
How do we find that out?
So we square the division of the sample mean from the overall sample, me and the sum of all of these
is multiplied by one less than the number of samples.
So what we do here is.
We have the.
Value here, what is the sum of squared error in this 40?
Now we want to find out the sample mean.
What is the sample mean of all data comes out to be nine.
I mean, for example, 11, then eight and seven.
So we have 11, ten, eight, seven, and this is the mean for all the data for the oil.
So how do we find out the population mean for the first sample, minus the mean for all?
The whole square plus population mean for the second sample, minus the mean for all the data is with.
Plus, I mean, for example, minus all the mean for all the double squared plus I mean for the fourth
sample minus the overall, you mean both with like and multiplying this with what, B and multiplying
this with the number of samples minus one.
So what is the number of samples that are one, two, three, four samples.
So we multiply that with.
So this is multiplied by three.
So this is the sum of squared off.
So that is how we calculate the sum of squared off treatment, right.
So what have we done?
Now, we found out the sample means we found out the overall mean.
We found this of squared of some of square Edda.
For so basically, it's almost better for each and every population and and all of these.
For all the populations.
Then we find this of squared off treatment by multiplying by the number of population minus one and
finding out the sum of squared off error with respect to each and every population, me with the entire
mean of the all the data.
So this is what we have until.
Now, what do we have done, we have done this particular save, this is all you've done, this is also
done now.
We want to find out the degree of freedom.
Now, what are the degrees of freedom, the degrees of freedom are now there are two big data values.
One for Sampas.
So the dignity of freedom for the treatment will be number of samples, minus one, that is number of
populations, minus one.
So there are the whole population.
So the degree of freedom of the treatment will be four minus one.
That is three.
You can say this.
What is the treatment, which I have been talking about?
Maybe you are confused about the treatment ward, which I have been using.
So this treatment is nothing but legacy.
I'm talking about say there are different types of of that and that these doctors are being fed by different
types of food.
They're being fed by different types of food.
So these are four different food items which I am feeding to the characters.
And I'm comparing these four food items and finding out if they have if these four are equivalent oil
for food works well on these this or there is one or there is any specific food which works extremely.
But then all of the food in the way of the causes war for any one particular specific item of what,
a specific food item.
So this treatment is nothing but what I am doing to make these populations different.
That is one of the treatment.
This makes a number of degrees of freedom of the industry.
That is, instead of having the full food items.
I am so out the food food items when they are there.
So the of freedom of choosing the food item will be three.
Right now, the degree of freedom of error is basically the degree of freedom of what we can select
from the populations.
So there are potential values.
There are two, three values, so from this sample, I can pick up randomly the values from each sample,
I can go through values randomly.
So gross food comes out to the EP.
So this is the big deal, freedom.
Of Ed.
Now, the means squared off treatment and means squared off of it has to be found so mean squared off
treatment will be cut by three and means word of error of will be fourteen by eight.
What is this the.
And what is this for?
Be it.
This is some sort of treatment.
So some of square of reedman divided by the degree of freedom of treatment.
Gives the means where it gives the means of treatment.
And when we divide.
The.
Sum of squared off error by the degree of error of a degree of any degree of freedom of error, because
if so, this is the mean squared off error means we're afforded, OK?
So these are the values that we have will be.
Then on six.
Now I can simply find out the statistics by dividing them by six, so it comes out one point six six.
OK, so you can go to this again in case you want to let me take you through this again.
OK, now what do we have?
We have these samples.
So out of these samples, we calculate the means for each and every population and we find out the mean
of all these populations, which comes out to be nine.
OK.
This is the mean with respect to each and every population.
This is the mean with respect to the entire population.
Then we find out the mean minus something mean minus sample mean, which is.
11.
So what is this?
This is night.
So we are subtracting 11, minus nine, then minus nine, eight minus nine and seven minus nine, which
comes out of the four one one four.
And we are thinking of this.
Some of these.
Now we are finding out the sum of squared error, so what is the sum of squared error in this?
Twelve minus 11 will squared plus nine, minus 11 squared, plus 12, minus 11 was when it comes out
to be six.
This is some of square and what, each and every population.
So what are we doing again, seven minus 10 or 12 and minus 10 squared, 13 minus then will square and
the sum of all of these comes out.
We begin with just one more square in a row population.
So similarly, we calculate this what each and every population, some of square.
And when we add these some of squared with respect to each and every population together, we get some
of the complete some of squared added.
Right, and what is the sum of squared of treatment, the sum of squared off?
This is the value which we call the 48 value, which we got, and this is the sum of squared of treatment.
So what is the most frequent examples where treatment is in the sample mean?
So what does mean mine sample mean, so this is the mean.
Minus sample mean.
This value.
I mean, minus sample mean.
These values, so what is the sum of these values, these some of these values is then.
Multiplied by the 320 degree of freedom.
So what is the degree of freedom when degree of freedom is the number of populations minus one?
So this comes out to be three.
So three then comes out with 30.
Now, this is the sum of squared off treatment.
This is the sum of squared error, what else we need to find?
We need to find the degree of freedom for each degree of freedom for tourism industry, which we have
already found no degree of freedom for the entire population comes out to be to.
Plus two, plus two plus two, which comes out to be eight.
How do we find this out?
We find this out by the sum of all of these values, the total number of values which we have in the
population minus.
The number of populations, so it will be 12, minus four comes out to be.
So this is what we calculated, and lastly, what we do is we find out the means of treatment, that
means of means of treatment will be 30 by three and means we're available before the break.
Thirty by three and fourteen by eight, which is dividing by their own degree of freedom, basically
a degree of the sum of squared off divided by the degree of freedom of religion.
And some of squared off and divided by any degree of freedom of ÉDITH.
And we simply divide the sum of squared of treatment divided by degrees of divided by this square of
error, divided by negative.
And so we get down by six with just one point six and seven.
So this is the statistics value.
Now, next thing is to find out what is the meaning of this F statistic value, which we have just found
out.
So we go to the distribution table.
And in this distribution, they will compare different degrees of freedom.
So what is the degree of freedom that we have?
We have the first degree of freedom as eighth and the second degree of freedom.
So the value corresponding to this would point zero six six.
Now, four point zero success is with respect to the Alpha zero point zero fight.
Make sure that you check the alpha value related papers and distribution will have different deals for
different values.
And because Alpha zero point zero five is of a particular value, which is.
Considered to be a standard, so we will consider this if value for now, so it comes out to be four
point zero six six.
Now, what do we do on this zero four point zero six six?
The actual value is one point six, six, seven.
Now, the statistical value is really less than the value which we have got from the distribution deal,
which means that this there is no significant difference between these populations.
These populations are similar to each other.
That is, there is no impact of these.
The men on the.
So this is what we find out from this particular test.