PERCENTILES DISCRETA Y CONTINUA

Now we are going to see what is "percentiles" It is not something complicated, but in general it is complicated for students So what I recommend is that you look at this class very concentrated because here comes the complicated part First we are going to work with the following phrases / words When we are told "minor"

I can't be talking, let's assume that the number is 7, with values ​​less than 7, right? What are they going to do? The 6, the 5, the 4, the 3, the 2, the 0, right? As I said, when I say as a maximum, again, lower values, but here it has the same, less than 7, the 6, the 5, the 4, and the 7 included, right? There would be a little shortcut.

When I say the sum, the same: lower or equal values. That is, values ​​less than 7, 6, 5, 4, 3, 0, -1.

and equal values to 7, so 7 is also included. In this part we work on the left side, the lower values we could also say. For example here, when it says "mayor" as "minimum" to "minimum" we will work on the right side. Here I have 7, I'm going to go to this side. "Mayor" to 7 will be 8, 9, 10, 100, 1000,

"como mínimo" is going to be "mayor" to 7, 8, 9, 10, 1000, but it is also going to be the same, that is, the 7 is included. "Menos" is equal to "como mínimo", it is going to be "mayor" to 7 and the 7 is also included. And finally we are going to have "cuando nos diga alguno". "Cuando diga alguno" the exercise refers to one or more than one, yes? It will always be the number one or more than one.

These are words that will appear in the exercises that we are going to perform, which are exam exercises. But before we get to what exam exercises are, we are going to see percentages with a small example, simple, to be able to interpret.

Well, here I have a variable that is clearly a discrete variable. Remember that we work with discrete variable or continuous variable in this class. Let's start with a discrete one, where the discrete variable represents the age of people. How many people did I ask their age? And we can see that I asked 1, 2, 3, 4, 5, 6, 7 people I asked their age. That number 7 is called sample size.

There are 7 data because there are 7 people who ask their age. We are going to have the age variable here, right? I'm going to ask you how many people are 23 years old, how many people are less than 23 years old, how many people are at most 24 years old, how many people are at least 26 years old, etc. Let's do some to see, let's kill it. How many?

Well, that's one. How many people are at least 26 years old? Well, we can see that it says "at least", right? So, as it says "at least", we know that "at least" refers to more,

26 or 26 inclusive. I have to include those of 26. So let's see: How many people are older than 26? And this person. And how many people are 26? And this person. How many people are at least 26 years old? Two people. So there we are counting the number of people, right? There are two people. The issue is that the issue of the number of people I can also work in percentages.

which is what will happen generally in the exam. So you say: "Well, instead of 4 people, we are going to work it in percentages." You will say: "What percentage of people is at least 26 years old?" You say: "Well, those who are at least 26 years old are these two, but they ask me in percentages. I know it in quantity, in quantity they are two, in percentage it is not two.

How am I going to calculate the percentage in a discrete way? It will be with the rule of three simple. We know that seven people represents 100%, that is, the total of the sample. Now, how many people are at least 26 years old? Two people.

So, to get that percentage, if 7 people represent 100, I want to know how many people represent 2 people. So here we are going to do the simple 3 rule, which if you remember is 2 divided by 100 divided by 7. So we do 2 divided by 100 divided by 7. And that will give us 200 in the numerator divided by

7 in the denominator and that little account gives us 28.57 28.57 percent, it means that 28.57 percent is at least 26 years old and these two little people that we are going to write like this these two people

represent 28.57% of people. So the answer was either two people or 28.57% of people. That is, 28.57% of people would be the total number of people. That is the percentage that is occupying, that meets the condition that at least they are 26 years old.

In general, in the exam they ask what percentage, not what number of people. But the number of people will help me to get to the percentage with simple rule of three. Now, let's go with another question. Let's see. What percentage of people has as a maximum 23?

It says as a maximum. So we said that when it says as a maximum, the lower ones will be the same. Let's put the same color. The lower ones are the same. That is, those who are less than 23 years old and those who also have 23, because they have the same. So I'm going to look at the sample here and say, well, these have less than 23 and these have 23. They are going to be all of these.

That is, 4 people. There are 4 people who are at most 23 years old. The issue here is that they do not ask me the number of people, they ask me what percentage of people. So I know that the number of people is 4, but I have to take that to percentage. Let's go again. We know that 7 people represent 100% of the sample.

Now, 4 people, what percentage will they represent? And we are going to do the simple rule of 3: 4x100/7. We do 4x100/7. Well, that will give us 400 in numerator, divided by 7 in denominator. We do the math and that gives us 57.14.

57,14% So, 4 people represent 57,14% of the samples. So, this result is giving us 57,14% of the samples. You see that sometimes I start to calculate percentages. Here we have one percentage and here we have another percentage.

But sometimes the percentage is based on the people who are at the end of the sample. For example, these two people represent 28.57%. But in reality, these two people also represent 28.57%. These two people will also represent 28.57%. And here, four people represent 57.14%. The same would happen if I talk about these four people.

these four people here, right? Because we are talking about the number of people expressed in percentages. Now, to differentiate those percentages from when a percentage is calculated based on the end of the sample or when a percentage is calculated at the beginning of the sample, we are going to call that K-percent. So look, this one I got here, we are going to call it K-percent.

that these will always come out of this, from the minors, minors or equals, yes? From here we will get k% I will write it up here. These that are going to be calculated from the final part of the sample, yes? We are going to call it k% *. So here we are going to write, I'm going to erase it, k% *. So there

we will have percentages that will be counted from the beginning of the operation and others from the end of the operation. You say: and this changes me if the answer is going to be the same? It will not change anything in what is discrete variable. Now when we go to continuous variable they will see why the declaration of k and k*. Here in discrete the truth will not bother us,

You can call it, if you don't know if it's K or K, it won't change, you can identify it or not. But you will see that when we see "continue" it will be very important. Now we are going to have the number of people, which in this case is 7 people. And we are going to have the number of people expressed in percentage. We said that one is obtained through the other by simple rule of three.

So I say, well, here, what could be unknown? And that percentage could be unknown. As the teacher just asked me, what percentage... What percentage... has as a maximum... 23 years? Yes? See that here what he was asking us... Ah, it was with blue, sorry, again.

What I was asking is: What percentage is 23 years old? So here we are working with what is K% Because as it says, as a maximum, it enters what is K% This percentage that I was asking is K% Which is what we calculated a while ago, that gave us 58 and a bit

There may be another unknown. Now, he can ask us: What age does 20% of people have at least? See? Here what he asks us is: What percentage? And he gives us the age. Now here he asks us: What age? And he gives us the percentage.

It says "as a minimum" and I know that I am here and it is "k%*" This is going to be "k%*" As it says "as a minimum" it is "minor" and "igual" The percentage that is throwing me is

As it says here, the percentage will be K. I have to be careful. As I told you, here the exercise changed, the percentage does not ask me, it asks me the age. It says that the age has at least 20% of people. So I would have to be able to identify what is the 20% of people. I know that here I have one person, two people, three.

4, 5, 6, 7 people but I have it in quantity, in percentage. How much would 20% of people represent? To know how many I have to grab. Yes, because I know that all are 100%, but I don't know 20%. So, let's do the following: Let's say that 100% is 7 people, right?

But I don't want 100% of the people, I only want 20% I want to know how many people are 20% Simple 3 rule: 20 divided by 7 is 100 Remember that simple 3 rule was this divided by this and that will give me the number of people

represent 20%. So we do 140, because 20 divided by 7 is 140, and we divide it by 100. This gives us 1.4 people. Yes, we did 140 divided by 100, and that gives us 1.4 people. Now, we said that there was a K and there was a K*. What was the difference? It was that K* started from here.

Instead, "ka*" started from here. One starts counting from this side and the other starts counting from this side. In this exercise, it will be important to know what is the "k" that is asking me. Sorry, what is the "k" that is giving me. As it says "como mínimo", I understand that it gives me "ka*". Remember, "como mínimo", "ka*".

So, if it is a "Casterisco" we will have to start counting from the end. See that it gave me 1.4 people? That is, it gave me between 1 and 2 people, right? 1.4 people would be between 1 and 2 people. So here I am going to start counting the number of units that I have to take. In this case it gives me 1.4, so I'm going to count: 1

And see, here it would already be 2, so I went over. So it's going to be 1.4, it would be like in the middle of these two, in the middle between 26 and 30. Let's get out of there. So it means that 1.4 people are going to represent 20% of the people. What does it mean? That here we have that 20%.

And now the question is: What age do 20% of people have at least? So here we have 20% and where would we have to look at age? And the age is here, the ages are these, remember? This was 20 years, 22 years, 23 years, 24 years, 26 years, 30 years. There we could say that 20%

of people have at least, as I was there, I will say that at least they are between 26 and 30 years old or I could take directly the 26, that is more than anything the criterion of the teacher, but hey, we could take an average of those two, we are going to put there, what age does the 20% of people have at least?

26 years old. When we talk about 20% of people we are talking about a person and a half. Well, it doesn't exist, right? We have one or two people. But well, what I want to know is where that 20% leaves me to be able to look at the age. And I fell in 26 years. I know that in this sample 20% of people are between 26 and more years old at least.

Well, this is something very important. Sometimes you can ask me the percentage and other times you can ask me the value of the variable. This part is quite complicated for the student.

I think it's practical, you have to practice. But well, review this, so we can move forward in what is continuous. Now you will see how this would be in continuous. Let's do one more so it's a little clearer. Well, question: what age do they have? In the sum, do you remember that in the sum it was less or equal? In 50% of people.

Well, since it is "a lo sumo" I know it is "K%" the one that is giving me. Well, it asks us about age, so age is going to be one of these. It has "a lo sumo" the 50% of people. There the people are giving it to me in percentage. So I say, well, I know that 100% are 7 people. The 50%

Well, half, right? We do this by this, divided by this. We know that this will give us 3.5 people. But now, it is making me work with K%, how do I know? Because it said the sum. Remember that in the previous one it said as a minimum and then it made us work with K'.

Here, as it makes me work with K%, I will start counting from here. Remember that with K* I started counting from here, but here we have K%. How many people am I going to count? 3.5 people. So let's count. Here we have 1, here we have 2, here we have 3.

And here we have 3.5, we would be here, almost in position 4 or in the middle of these two we could also say. But well, the average of these two is 23. So, what age does 50% of people have? Let's say he is 23 years old.

See that I didn't reach position 4, but I'm in the middle. See? 1, 2, 3 and here this would be position 4. It would be there. But see that both are 23, so it takes 23.

That's going to be the teacher's criterion. There are other teachers who say: "Well, take this value." Others say: "No, take this value." And others say: "No, take the semi-sum of the two." That is, to get an average, add them and divide them by two. Yes? Good. Good. Now we are going to calculate presentives with a continuous variable. A while ago we did it with a discrete variable. Now we are going to do it with a continuous variable. To calculate presentives with a continuous variable,

we use a formula that is probably very well known to you because it is the same formula that we use in quartiles. We are going to use this formula. This formula, very important, has two variables. Here it will work with the variable

In this case, the variable that is being analyzed is the age of the people. And in this part where "K" appears, it will work with the percentage.

Let's do an exercise, I'm going to write it down here. I don't remember which one we did in the previous one, but I think it was... what percentage? 20% of people. Yes? Well, let's see here. Did you see that it says "as a maximum"?

So, as a maximum, it is equal to or less than and represents a percent. This percentage that is giving me here is a percent. As a maximum, it is always equal to or less than. And the question is: what does it give? If I go to my formula, I see that the formula has what the variable is and has what the percentage is. I have to see what it gives me to be able to load that formula.

Does it give me the age and ask me the percentage? Or does it give me the percentage and what it asks me is the age? It will always give me one and the other will be the unknown. In this case we can talk about what the percentage gives us, see? It gives us 20%. It tells us that the percentage K is going to be 20%. And the unknown is the age. It is what it is asking me, what it wants us to calculate.

Well, how did we use this formula? What we did was to identify the range with which we were going to work. For that we have to use the percentage and look at the accumulated frequency. In this case it is 20, see here I have a 12, I have to look at the accumulated, I have a 12, I did not reach 20. Here at 44 I already reached 20. So this is the range with which I am going to work.

From this row I am asked for the lower limit, which is simple, plus the amplitude, which is the difference between 10 and 5. What will it give us? 10 minus 5 is 5. The 20%, which is k%, over the simple percentage frequency, which is 32%.

minus the previous one, which is 12%. And that gives us the value of x, which represents the variable that is being analyzed in the exercise, and in this exercise the variable is being analyzed. We are going to solve that little math, be careful with the calculator, do the math to see how much it gives you, and if everything is fine, it would have to give you

6.25 And if everything is fine, that value 6.25, well, we know that it represents age, it will be years, right? The age in years.

it will represent 6.25 years and if it is all ok, that number has to be in the interval that we are analyzing. You see that the interval goes from 5 to 10, then the 6.25 is indeed within the interval. Well, then, you saw that the variable, that age, which goes from 0 to 5, from 5 to 10, from 10 to 15,

and from 15 to 20. What this exercise asked us is: "How old are 20% of people?" As I said, I was looking for the same minors. The minors equal to 20%. That 20% told me that I can take from here to here, for example. Or take to here, or take to here. What it is telling me

I'll take from here to here, which is 6.25. This would be our K% and as it says as a maximum, it is less or equal, that's why it's all this strip. That is, from 0 to 6.25 are the years that it covers. If it said as a minimum, it would be from here to there.

So, what age does 20% of people have at least? And at least it has 6.25 years. Then, within that 20% of people, there are others who will have 5 years, another who will have 3 years, another who will have 1 year, but the maximum of that 20% is 6.25 years. Well, having said that, now we are going to change the exercise. Instead of as a maximum,

we are going to put "como mínimo" As it says "como mínimo" we are going to change here. And this is going to change everything. "Como mínimo" as we can see represents an equal major that represents a casterisk. So the percentage as it says "como mínimo" the percentage that it is giving me is not "k" is casterisk. So

Well, now we are going to calculate the exercise again, but considering that K is K*, because it says "as a minimum". Well, the unknown again is the "da", so this is going to be unknown. And what is the percentage giving me? The percentage tells me that it is 20%, but when I go to the formula, the formula says that it does not accept K*, it only accepts K. See that the formula says K, it does not say K*.

So I already have a problem. I say: "Well, could I load this 20% in that formula?" No, it would be wrong, because what it is going to give us is something else, it is going to give us this. And it is not asking me this, it is asking me the final 20%. So, here we are going to work with a conceptual issue, yes? That says that the sum of k + k* is 100%.

Yes? The sum of k and k* will always be 100. But we just said that k* is worth 20. But I can't load it there, because I need k. But if I have k*, I can calculate the value of k. Well, let's calculate the value of k. How much is k? And if k* is worth 20 and with k they add up to 100, necessarily k% is worth 80.

is 80% here we will replace it with 80% If I get a "k" the formula only admits "k" The formula once it has "k" loaded tells us to identify the row with which we are going to work in this case the row with which we are going to work the one that takes the 80% is this

because here I didn't reach 80, I just reached here, right here. So we are going to work with that third row. Well, the lower limit of that third row is 10, plus the amplitude, 15 minus 10 is 5, times k%, minus the previous accumulated frequency, which is 44%, over the simple percentage frequency, which is 36%. And there, with that, we will be able to calculate

How much is x? We are going to do 10 + 10 + 5 times 80 - 44 over 36 and that gives us 15. Remember that x represented age and age is represented by years.

What was this exercise asking us? What age do at least 20% of people have? I talked about 20% again, but not this first 20%. I was talking about the final 20%. And he said to me: "Well, what age do you have at least?" In this case I stayed here, in 15. Right in this 15, it's just a limit.

What you are telling me is that at least 20% of people are 15 years or older.

At least it is 15 or more. It will be 16, 17, 18, 19, 20 years old, but at least it is 15 and then it is more. On the other hand, in this one it was as a maximum, it is 6.25 or less than 6.25 years. That is the big difference of this exercise with the previous one.

Let's do another one. I already said it several times, but I will say it again. Something very important is that the formula only works with k. It does not work with k. If it appears as a k-data, that will help me to calculate k.

In the formula, K is always loaded and with K is the row with which we are going to work in the formula. You see that you say: "Well, but which one do I take? The 20% or the 80% row?" And notice that you loaded in the formula. If you loaded an 80% in the formula, you have to work with the row that accumulates the 80%, yes? Not because here it says 20%, you are going to work with the row that says 20%.

because we always rely on the formula to detect the range. Another very important thing that I want to say is that the sum of k and k* will always be 100. So one is a complement to the other. If one is 30, the other is 70. If one is 40, the other is 60. And something very, very, very important is that

Did you see that here it talks about "ca" and "casterisco"? In fact, in statistics, it only talks about "ca". It is understood that the other, which would be "casterisco", is the complement, but by giving it a name and guiding us in the class, I call it "casterisco", because in class the teacher is not going to talk about "casterisco", she is going to tell you the complement of "ca".

I gave it that name, "Casterisco", so that we can guide ourselves when we are talking about one thing and when we are talking about another thing. But I mention it more than anything, that in the class the teacher does not mention it, and you say: "Prof, Casterisco" and I say: "Who is Casterisco?" It's something of mine so that we can identify ourselves, that it represents the complement of the K percent. Yes, that. Let's go for another exercise.

What percentage is 17 years old? As you can see, we have a minus and a k percentage. So, minus equals... and the percentage is k percentage. Now, another data tells us that it is 17 years old. Let's go to our formula, which is the one that relates variables with percentages.

Well, in this formula the variable can be unknown or the percentage can be unknown. If the percentage asks me, it will give me the variable and if the variable asks me, it will ask me about the percentage. We see that now the unknown is percentage. It is asking me how much k% is worth. So we know that now k% is the unknown.

And what does it give me? It gives me the variable, it tells me that it is 17 years old. Here we are going to complete with 17. Well, the subject is the following: in what range am I going to work? Do you remember that before, to select the range, we looked at the percentage accumulated frequency, but now we don't have the percentage. How do you think

to detect which is the row of the frequency table with which we have to work. Perhaps some of you have noticed. As it says 17 years, I have to look where the 17 years would be.

we see that the age of 17 enters the interval that goes from 15 to 20. So what we are going to do is work with the last interval, we are going to work with this interval. Because the 17 years are in that interval. Well, what is the lower limit of the interval? It is 15. What is the amplitude? 20 minus 15 gives 5.

by k% minus previous accumulated frequency. Well, the previous accumulated frequency is 80% and the simple frequency is 20%. And that would be the equation that now I'm going to have to solve where my incognita is k. Here we are going to solve the equation. I separate in terms.

And the terms without the incognito will go on the other side. The 15 that is adding will go subtracting. Right? And 17 minus 15 will give us 2. The 5 that is multiplying will go dividing. Well, we continue here. The 20 that is dividing will go on the other side multiplying.

We will have k minus 80 k. If you want, in that part we can solve it. 2/5 times 20 will give me 40/5. The minus 80 will pass on the other side. As it is subtracting, it will pass adding up. And that will give us how much k is worth.

If we do the math, 4 divided by 5 is 8, plus 80 is 88. Now the million question: Is 88 years or percentage? And K represents a percentage, so that's 88%. K always represents a percentage.

So, what percentage has a maximum of 17 years? And the percentage is going to be 88% of the students have a maximum of 17 years, that is, there are quite a few who are under 17 years old. Now, let's go for another one. Now we are going to do this exercise. It says: what percentage has at least, remember that at least,

was greater or equal and represented a k*. So the percentage that is asking me, as I said, as a minimum, which is greater or equal, the percentage now is a k*, it is not a k. And it tells me, at least, 17 years. Let's go to the formula and let's load the data that it is giving me. X is the variable in the age that gives it to me, it tells me that it is 17 years.

It says: "What percentage is at least 17 years old?" Here the question is: "How much is Kasterisco?" And with this formula I can only calculate K%, as we did a while ago. Remember that we calculated K% and it gave us 88%. So, with the formula I can only calculate K%, but here it asks me Kasterisco.

Would it be useful for me to do the formula and calculate k% if what I want is k*?

Yes, it would be useful for me, since if I have K*, it is a way to know how much K% is worth. Why? Because of this relationship, remember? That they were complementary. So if K* gives me 20, I automatically know how much K% is worth. K* is worth 80.

The formula, although it does not say "k*", will help us because knowing "k" we will know "k*". Well, how are we going to identify the range with the age? We are going to work again with this range, because we are 17 years old and 17 years old is in this range. Well, the lower limit is 15, the amplitude is 5, k% minus the previous accumulated frequency,

over a simple frequency of 20. We are going to solve this and we are going to get the K% value. And that will give us 88%. Because it is the same exercise we did a while ago. Right? Remember that we had separated in terms, what was adding was subtracting, what was multiplying was dividing, and it had given us this result. The point is that I know how much K is worth.

but here it asks me how much is "Casterisco" To calculate "Casterisco" I know that both add 100 So I say, well, 100% I subtract it and that will give me the result of how much "Casterisco" is worth

So the result will give us 22%. That's the percentage that this exercise was asking me. What percentage has at least 17 years? So we already saw the four cases that can be presented to me when using the percentage formula in continuous variable.

As you can see, calculating percentages in continuous variables is very different from how it is done in discrete variables. You saw that in discrete we do not use any formula and in continuous we are using formulas. That is the great confusion that is made to the student, that in continuous sometimes

forgets that it has the formula and tries to do something else and in discrete sometimes wants to use the formula that does not correspond because it is only for continuous, those are the confusions. What they would have to do is review, review what I saw in this section of the class because this is a very important part and that gives many confusions. Well, to close this

Did you see that here we calculate the 88%? I'm going to look at it from this side.

A while ago it gave us 88% here and then here it gave us 22% This gave us K and this gave us K* As you can see, K starts counting from the beginning, from 0 and in this case we reached number 17

from 0 to 17, instead K% started counting from 17 to 20, what we had mentioned before, K%*, look at the end of the sample, instead K% starts counting from the beginning of the sample.

The problem is that this formula will only work with k% Then we always charge k% Then we calculate the complement because we are looking for k*