In this session, we will discuss about that school, also known as kind of school.
A comparison of the relative standard like what the mean and the standard deviation can we need using
Z score or standard scooter?
So let us see, we have a particular dataset.
So a particular data set will have some distribution as.
Now, let us see in that particular data set or in that particular sample, I want to find out the location
of a particular data point or I want to find out how far that data point should be present in terms
of the standard deviations or in terms of the distance, in terms of mean and standard deviation.
So that is something which is provided by zig zag.
Score is just a value which is calculated and the value of the Z score is four, then calculated in
the Z.
So we have a Z score, which we calculate from the plot.
And based on that Z school, we get value from the Z table, which gives the probability value.
So this probability value will actually give me that what is the probability of this value being present
in the plot?
And in what region of the world should the value be available?
So let us try to understand this more clearly and we will begin a lot about this value.
So what is it?
The number of standard deviations are data value is above or below the mean for a specific distribution
of the value.
So let's say they have a particular distribution.
Let us say we have this distribution and I want to find out the location of the zettl pointed in terms
of the mean of this distribution, then I can find it on the basis of the Z Squaw Valley.
So that is a very useful statistic because it allows us to calculate the probability of a score occurring
within our normal distribution.
So based on the Z score, I can actually find out what is the probability of a particular value occurring
in this distribution, in this normal distribution.
So I can find that opio.
Now, as it's called.
Enables us to compare to schools that are from different normal distribution, so like I say, I have
two different normal distributions and I want to compare these two schools belong to the same population
or the other.
Where do they actually stand?
In front of each other in a particular sample.
So these things, this kind of comparison can actually be made using the zip code.
Now, the value of the Z score will be from zero to one.
So if a Z score is zero, it will indicate that the data point score is identical to the mean score.
If the Z score value, this is the means for.
The mean score will have the value as Zettl now if the Z score value is towards one, it is anything
greater than zero then it is toward the point is actually lying towards the right hand side of the knee.
If a point has a Z score less than zero, it means that the point this line towards the left hand side
of the knee in their distribution and it is always calculated as a part of the normal distribution.
So when we are calculating the score, we are assuming that the distribution is actually a normal distribution.
And because the distribution is normal and it is symmetric in nature, so if we have let us see a value
of minus zero point seven five a Z value of minus zero point seven five, then we will look at the probability
in the Z.
So let's see, we have the value of Z as my minus point seven five and we are looking up in the Z table.
Then in the Z, the value of the probability, which will be their legacy, it is willing to do so.
It means that the area below this point in the normal distribution has 22 percent of the data.
It means that there is a twenty two point six percent probability of the point being here.
So there is a twenty two point six six percent data below this.
And again, if we have the value as point seven five, this is negative.
This is positive.
Then this becomes again twenty two point six percent.
So the percentage of area is also equal here.
So if let's say we want to find out something here, then the probability of the area which is covered
in this region will be equal to the area covered in this region.
So we can simply calculate the value of.
One.
So if we have a point minus zero point eight.
On minus zero point eight one, then we can calculate zero point eight one and it will have seen the
on the board.
To see for the.
As one would indicate the value that is one standard deviation from the mean.
So if your value is minus zero point seven five, it means that it is zero point seventy five in one
standard deviation of the mean.
If it is at one, this means that it is minus one standard deviation from the mean.
If it is minus two, then it means that it is two standard deviation from the mean.
If it is minus three, it means that it is three standard deviations from them.
And you remember from the standard deviations, we have this particular plot here, which means that
if something is one standard deviation away from the mean, then the area is thirty four point one three
percent.
If something is to a standard deviation from the mean, then it is 95 percent of the data.
If something is three standard deviations from the mean, then it contains ninety nine point seven seven
percent of the.
So we can derive this also from the Z value.
The formula for Z being calculated for a particular value is so let's say we have a value zero point
or fifteen, then the Z value will be fifty minus the mean of the plot divided by the standard deviation
of the plot.
So this will be X minus one divided by sampling error in case of sample and X minus population mean
divided by the population standard deviation in the population.
So this is what the formula Z is.
Now using the Z.
We can actually standardize the date also.
So what is that school standardization, Zelikow's scores, standardization is actually used to scale
the data so that all the variables have the same range.
So let's say we have a column each and a column Silalahi.
And the range of age and salary is very different here.
Age is from 20 to 52, and here the salary is from twenty thousand forty five.
So the scale is a lot different.
So in case I want to find out that what is the probability of having this age with respect to the salary,
then we can only find that out because the scheme is a lot different.
So what we can do is we can scale this age in salary and then we scale these using the Z score to the
standard normal distribution will be change in such a way that the mean will be at zero and the standard
deviation will be one.
For this both of these columns and the Z value will be X minus will divided by Sigma, which is actually
the Z value from this particular formula, which we are very.
X minus moderated by Sigma.
So similarly, we can scale the data and we will get the scale.
Now here you can see the values are ranging from minus one point to nine, minus one point one point
six nine two one point four.
So here we can kind of compare these values.
This is about that school and let us look for the now legacy now we have understood this, but we need
to have more clarity on this.
So let us look for the legacy.
We want to find out the probability, which is the area under the curve.
Remember, the probability is idiomatically for a normally distributed variable by transforming it into
a standard normal variable by using the formula.
So we are using this particular formula and we are trying to find out the probability associated.
So let us see a few examples related to this.
So this is the first example.
So what we are doing here is we are seeing that of the average or the mean of a particular value, so
legacy, we are talking about the students study.
So let us say a student is studying three point one hours every day.
So this is the mean average of all the students in the world study.
And we say that the standard deviation is zero point five.
We say there is a standard deviation of zero point five in the average or the mean value of three point
one, as well as where the students study.
And we want to find out the percentage of students studying less than three point five hours.
So we are seeing the average student studies for three point one dollars and the standard deviation
is of zero point five.
And we just want to find out the percentage of students who are studying less than three point five
hours.
So if we have this particular block.
So from this particular plot, let's try to.
Point out.
So this would be the mean value.
This is the mean value would just.
Three point one in this particular case.
This is the mean value and the standard deviation is zero point five, so the zero five zero point five
is a standard deviation.
So zero point five would be somewhere here.
This is zero point five zero point five.
So if we think about it based on the values of let us look at it on the basis of the Belko.
So this is the Belko.
And this is one standard deviation.
We are saying that this is one standard deviation and this point, this zero point five, and we are
asking about zero point eight, which is somewhat higher.
So we are trying to ask the question, we're asking a question like, what is the probability of student
study less than zero point eight hours, less than three point daters.
Less than three point five hours, less than three point five hours.
So three point one is here.
OK, this is three point one.
And the standard deviation is zero point five, so this is if this is three point five, this is the
standard deviation to zero point five, then one standard deviation will be three point six.
One standard deviation will be three point six.
This means.
That this is actually one standard deviation and this value is.
Three point.
We want to find out what is the percentage of people spending less than three point five hours and we
think that this is three point five.
So give me the percentage of people selling for less than three point five hours.
So if we think about it theoretically and just on the basis of Belko, so let us look about it just
on the basis of Valko.
So this is our Belko, and this is one standard deviation, which is three point six.
This is three point six and this is three point five.
So what will be the percentage?
So the percentage will be 2.8.
Plus thirteen point five.
Plus thirty four point five.
Plus, some part of this thirty four point five, not completely this complete thirty four point five,
but some part of it.
OK, let's say to something like that.
So the sum of these values will be.
Do point to wake.
Plus, thirteen point five nine plus thirty four point one three, which is 50, so the Swatch area
is 50, which is already understandable because the idea is 50 percent of the block.
So this area will have 50 percent of the data.
And on top of it, there will be some more percentage, which we have to calculate.
So how do we calculate that?
So we calculate that using this formula.
So what is this formula now?
Our new value is three point five.
New Orleans, three point five, the value of mine is three point one, the standard deviation is zero
point five.
So what is the value of this?
The value of this is zero point eight.
This is the Z School.
This is the Z score, which we have found.
Now we just found out the Z score, this is the Z School and this is one standard deviation.
So one standard deviation will be one.
And this isn't code for zero point eight.
This is a policy to point.
This is a school for one standard deviation is one that's for one standard deviation is one school for
mean a zero.
So let us find out the percentage here.
So how do we find out this percentage?
We go to the vet, they will know.
So now look at the Z.
So this is the Z table.
So what does this table represent, this table value represents that area to the left side of the Z
score.
So if we have a Z score of zero point eight, we have the Z score of zero point B, so four zero point
eight.
That's all we want to find out the area below zero point.
So towards the left hand side of zero point eight.
And that is exactly what we want to find.
So we will go to this table.
We will look at zero point eight.
And what is the next value, zero point eight point zero zero, because there is nothing else in the
decimal point that is nothing else in the decimal point.
So we will look at zero point zero.
So what is the value?
The value is zero point seven eight eight one.
So this is the percentage of.
Students who are study less than three point five hours.
So what is the percentage, the percentage is seventy eight point one percent.
So they stand for less than fifty point five hours.
So that is how we calculate this.
So it is very simple.
All we need is the mean value.
We need the standard deviation value, and we want to find out with respect to what we actually want
to find out the percentage.
So based on this percentage value, we will find out the Z score, the Z score will actually give us
where in the plot we want to find out the probability.
And from the Z score, once we find out the Z score, we go to the Z value and from the Z table, we
look at the value here.
So if the value was zero point eight to one, then what will happen, zero point eight and then we'll
get to zero point eight two and the percentage would have changed to seventy nine point three.
It was.
So this is how we find out the percentage fun.
Now, look at the next problem in hand.
This is the second problem.
Now, what is the problem here?
The problem is that the mean is 2010.
So now the mean value is 2012 and the standard deviation is to increase.
Mean is to when the standard deviation is 12, so what is the mean value?
Mean value is.
Twenty eight at Zettl.
Zero, the mean value is.
Wendy.
It.
And we want to find out what will be the percentage between 27 and 31, what is the value between 27
and 31?
So simply we will apply the formula.
So we want to find out between 27 and 31.
So first we will find out the area.
Above twenty seven, so we will find out what is the area at 27, what is the area at 31, how much
it is there below 31.
And then we will surprise the area below.
Twenty seven from the area below 31.
And this will give us this particular area.
So let us find out the area below twenty seven.
So below twenty seven will be twenty seven, minus 20 divided by the standard deviation we just do so.
Twenty seven minus twenty eight divided by two gives us minus.
Zero point five.
Next, we will find out the area, the Z value at 31 valued 31 will be X minus MU, which is thirty
one minus 20, divided by two, which is one point five.
Now, what do we want to find out?
We want to find out the area between 27 and 31.
So what will that be?
That will be.
This entire area.
Minus only this Majida.
We want to minus only this small area from this entire area so that we get this shaded portion, which
we actually need.
So what do we do?
We find out the area of the probability of xai.
At one point five.
And we subtract the probability of that minus one point five from it.
So what are these values?
So we want to find out the probability at one point five, so what is eleven point five?
We have one point five and one point five with this ninety three point thirty one.
What is the probability at minus zero, point five?
So we will look at the negative Vados, this is minus zero point five and it is.
Zero point three zero eight.
So we will subtracted from this one, so we get 60.
Zero point six two four seven.
So this shaded area is 62 percent of.
The entire block.
So the area between 27 to 31 is sixty two point four, seven percent.
Now, let us look at the next problem, the next problem is finding specific data values for a given
percentage using the standard normal distribution.
So now the question is entirely different.
So it is the opposite of what we were doing earlier.
Earlier, what we were doing is we had certain values and based on these values, we had to find out
the percentage.
But now what the problem is.
We are given a percentage and we want to find out the standard distribution.
So what we do is.
We are living in the top 10 percent.
So the top 10 percent of data is given.
So in the top 10 person, the meanest two hundred.
And the standard deviation is 20.
Find the lowest possible school to quality.
So we want to find out the lowest possible.
So what is this 10 person?
We want to find out the Z score for this straight, so how do we find out we have this 10 percent value?
This 10 percent value implies that the value is actually zero point one zero.
OK, this is zero point one zero one method, what we could have done is because this is zero point
one zero.
And this plot is symmetry.
So if I plot another line here.
And look at 10 percent here.
Then if I find out this area.
Then the value will be just in the negative of this.
If we have been blessed in this region and we find them in this region, then the Z value will be negative
in here and here, it will be positive that so we can simply find out what is zero point one zero.
We can simply find out this is 10 percent.
So, again, we can find out what this 10 percent in this region and then we can find out the Z value.
So let us look at that post.
So in the negative region.
The Z value is.
Then so this is the zero point one zero zero.
So this is 10 percent.
This is 10 percent.
And what is this?
What this is four minus one point.
So we can simply say that here the Z value is one point to begin to validate the values of one point
to the Z value will come out to be one point.
So how we can calculate this now, similarly from the Z table, you can look at that then and you can
see that this value is zero point zero three.
And on the basis of that, you can find out the minus one point to it.
So you can see it simply see the Z value is one point to it, which is exactly what we just did.
So this is one point two and it is at the eighth column Zettl.
Three.
Five, six, seven.
And so this is one point two eight.
So their value is one point, do it.
We have the mean value, we have the standard deviation so we can find the value here by X is equal
to Z Endou.
God plus.
So it comes out to 26, so we can say the minimum value has to be 226.
We have simply find out that the lower value actually has the same percentage because the plot is symmetrical.
So based on symmetry, we have found out the value of Z to be does the negative of this value.
So we found out it is one point, do it.
And from one point to it, once we put it in the formulative, we get the value to 26.
So the value, which is the actual numeric value here, will be to 26 and the mean will be two hundred
at this particular point.
Now we have another problem.
Now, what this problem is that we want to select the middle 60 percent of the population.
And we say that the mean is 120.
And the standard deviation is eight.
Find the upper and then lower values.
So let us give you the bell curve again.
OK, let us get to the Belfour.
This is the Belko.
And from this vehicle, you can see that one standard deviation is almost 60 percent.
One standard deviation is almost sixty eight percent.
So let us get back.
Now we want to find out in the middle 60 percent or so, because this much area.
Is actually sixty eight percent, one standard deviation is 68 percent.
So 60 percent will anyhow be between these one standard deviations.
So and what is one standard deviation, one standard deviation is eight.
So these the values.
These raised values will actually be.
120 plus it, that is 128, this value will be.
One.
Twenty eight.
Mean my writing here and this value will be 120 minus eight, which is one hundred twelve.
So this much idea is 68 percent.
So 60 percent will be between these two values.
So how do we find that out?
So we can simply kind of see that the values will be between 100 and twelve and 120.
Now we have to find out the actual values.
So how do we find out these?
So we can find that out by simply thinking that this is 60 percent middle 60 percent means that this
area is.
The middle line.
So here we have some 30 percent.
And here also we will have other people of the area which we want to have now, because here we want
to have a 30 person.
This means that this 30 person will again be.
A part of this area.
So we have 30 percent here.
And we have another 20 percent you.
Similarly, we will have a 30 percent year and we have another 20 percent this this that is in totality
will make the one hundred percent right.
This is the end of the goal.
The area under the curve is hundred percent.
So if we have 30 percent, if we have the middle 60 percent, then from the middle line, that is the
mean.
That is the median from this middle line.
This much area will be the 30 percent vaidya.
And this area will again be another 30 percent media, which will totally add to 60 percent.
And further, there will be a 20 percent.
And here also we will have another 20 percent.
So how do we see that?
We can state that in a form that the probability of area between minus area of the probability with
E is minus E is less than say.
Minuses less than, say, and Zeller's, less than a.
Well, that is the mean value.
We have certain value, any value could be there, and that value has to be greater than mine.
MASEY and less than eight.
Then the area between this and will be 60 percent, which is zero point sixty.
Now, because this is symmetric in nature, then we can see because this is symmetric.
So instead of seeing all these infarcts, we can simply see a Z of less than minus.
In total.
Area of Z, less than minus eight.
What is it?
Less than minus eight, this 20 percent.
This 20 percent is nothing but.
This Bizzell.
This feasible minus Z, less than minus eight is simply the probability of area being less than minus
eight, which is this portion.
And when we do the case of this, then it becomes 40 percent and we're approaching it from one.
This means that we are getting the 60 percent of the.
Because this is symmetric, right, so we can equate this, we can equate this entire thing with this
probability of A minus A less than Z, less than a.
Which is this?
And then we are just swapping it and solving the equation.
So what do we get the probability of that less than is equal to one minus probability of this entire
thing, which is 60 percent divided by two.
So when we solve this entire thing, we get one minus zero point zero, which is probability of this
just what we have found out here and divided by two.
So one minus 60 percent this.
Forty percent, but just zero point zero four divided by to give zero point zero to.
And we just want to point out this zero point zero two percent.
So now when we take zero point zero two percent in the table, that is the value.
What is the Z value for the person?
Zero point zero.
So zero point zero, actually, Lezak minus zero point eight.
One zero, one, two, three, four.
So 20 percent lies and zero point eight four minus zero point eight.
This means that this minus eight.
This minus eight is that zero point eight for.
So from the zero point eight four, if we want to find out the value of X1 x2, how do we find then
we simply say Z in the new plus new.
So it will be easy to find each four in do eight, which is a standard deviation.
Plus the mean value is equal to twelve point seven two.
And similarly, we will apply the same formula for finding out X2.
So it comes out to be plus zero point eight four.
So it gives us twelve point seven and one hundred and thirteen point two it.
So these are the values which we get we get one hundred and thirteen point week and one twenty six point
seven, which is the value of minus and plus.
This value is incorrect.
This value has to be one twenty six point seventy.
So please get the.
So this is the solution to this problem.
Now, one last thing which we need to discuss here is that the probabilities of the distribution of
sample mean.
The mean of the sampling will be same as the population.
Secondly, the standard error of the mean will be.
Sigma by a Rueben, which we have derived from these and the.
Also, from the same theorem we can get that they will do X minus MU.
Divided by Sigma.
So, for example, because the sigma, for example, will be.
Sigma upon in so we can replace it, so we can see on the road in X minus mu divided by.
Sigma.
So this will be the Z value, for example.
So this is what we have the I for the sample.
So when we are walking with samples, that will be the.
X minus mu divided by the population standard deviation.