The normal distribution is a symmetric, bell-shaped continuous probability distribution characterized by its mean (μ) and standard deviation (σ). Key properties include: (1) symmetry about the mean, where mean equals median equals mode; (2) the total area under the curve equals 1; (3) approximately 68% of data falls within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ. To solve probability problems, any normal distribution can be standardized to a standard normal distribution (Z-distribution) with mean 0 and standard deviation 1 using the formula Z = (X - μ)/σ, allowing the use of a standard normal table for probability calculations.
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Normal Distribution || Mean, Standard Deviation and Symmetric PropertiesAdded:
Yeah. What's up everyone? Good evening.
How you doing?
Uh I hope you had a beautiful Sunday.
Uh you're welcome to this class.
uh our very first class on normal distribution.
So uh we have a couple of things to talk about and um see that this is the very first class uh you need to understand that you have to be very attentive and focused and don't shoot yourself in the leg.
Make sure you uh you get calm. You're not going to have the whole knowledge once. It's possible you may not get it the very first time we uh we mention it but you can just stay put and uh in some few minutes everything will get very clear. So please stay very focused. I really want to delve into this very interesting topic.
This is uh a very interesting yet consequential topic in major exams and uh I really want to clarify every bit of intricacies around it. It's very simple yet if you don't have the real basics you would have issues. It's just that simple. Uh guys please make sure you stay as attentive as you can because this is uh the very first class on normal distribution.
All right.
First of all, let's start from a different name. You may see this being called another name. This could be called is the goshian distribution.
All right.
Goshian distribution.
And um as I said earlier in some of our classes before now I told you guys that when we talk of discrete distribution functions we talk about Benoli binomial we talk about poison we talk about geometric and some others.
Of course the four is what we we limited the classes we have done to why I said when we talk about the continuous distribution functions one of the most extremely important continuous probability distribution is the one we're considering in tonight's class and that's the normal distribution.
It's a very extremely important one because it mirrors and it models the daytoday our day-to-day living. So it's one of the most important extremely important distribution we study in statistics.
Now how do you relate with a normal distribution? Number one, one major description of a normal division is it is symmetric.
A normal distribution is symmetric.
And not just symmetric, but it's symmetric about the mean. And I will explain to you what that means.
A normal distribution is symmetric about the mean. All right, let me explain to you. Number one, let me try to remind you of the word symmetric if you have forgotten.
I told you guys in some some times back that I gave you distribution shapes and one of the shapes I drew for you if you can recollect is a symmetric shape. Can anybody in the comment section give me another name I called the symmetric shape? Anybody? There's another name we can call uh a distriution that has this shape. If we don't want to use the word symmetric, anybody can anybody give me another name for instead of saying symmetric is there something else I can say?
Nobody. All right. Uh symmetric or bell-shaped? I told you guys bell-shaped.
You check your notes, you will see this there.
All right. So you can say symmetric you say bell shape. Now what's the real uh understanding of the word symmetric symmetricity of uh a distribution comes with one important value.
We said symmetric about the mean.
Meaning that this uh point right here which you can see divides the shape into two equal parts. You can see that this point right here is the mean.
All right. This is the mean.
So that's why we say it's symmetric about the mean because the the midpoint which I'm sure as you heard me say the word midpoint I'm sure you agree with me that if this is the midpoint if I was running this uh in a different uh if I'm talking about this in a different way I can also call this what the median I'm sure you recognize that because if This is the midpoint of a distribution.
This obviously will be the median. But it's the mean also and that's why we say it's symmetric about the mean. These are just some properties of normal division that you have to take note of. Number one, the mean is equals to the median.
But not just that, can you guys think about something else that is also this point is also represented by this point right here of course which I highlighted this point that divides the shape into two equal parts. What do you feel this place can also be that I have not mentioned? Is there anybody there?
What do you feel this place can also be?
Anybody?
Of course. This is also the mode.
So you can see that one of the important property of a symmetric distribution is that the mean is equals to the mode is equals to the median. You can see that's very important feature that we have the mean, the mode and the median as the same. This is a very perfect objective question. You would see where they will tell you to distinguish between different uh expression and which one they will ask you which one represent a symmetric distribution. Of course the option I told you that the mean is equals to media equals to the mode is the perfect one cuz that is an important feature of of a distribution being symmetric. All right that's that about that. So this is when you hear the word being symmetric about the mean that's telling you that the midpoint the mean is going to be a dividing line between uh the two different equal parts of the distribution and also it will also represent the median as well as the mode.
All right so this is just uh of course we're introducing so be very attentive as I said. Now the next thing we need to talk about because you recollect first of all don't don't forget I told you guys that this is very essential in our day-to-day living you will see several questions we'll be solving and how much they represent real life experiences now with everything we have been doing when we talked about binomial benoli poison and uh geometric there's one thing that we did throughout if you guys can recollect we always stated their PMF All right, we stated their PMF meaning that if we're talking about normal distribution, we have to state the PDF.
Don't forget I told you guys that discrete probability distribution function, we tend to use the word PMF, probability mass function. Whereas when we're talking about continuous distribution function like the normal distribution we say PDF that's probability density function. So we say probability density function when dealing with continuous.
So the PDF of a normal distribution is quite very unique and I want you guys to be you don't need to get scared of course it's uh what it is. So we have it as 1 / sigma square t of 2 pi e ra to power - 1 / 2 sigma square let me erase is because I believe you have taken that note 1 / 2 sigma square then x - mu all² so right here is the pdf of a normal distribution so you can see it right here very simple and beautiful to see 1 / sigma Huh?
2<unk>i e^ - 1 / 2 sigma square x - mu everything all squared.
All right.
Now I need to introduce to you. Of course there's something I'm missing out here which you guys should always be attentive to. Whenever you state a PMF or a PDF, you must always give uh the band for your X. Your X cannot just take any value. So for this PDF of course your x this is the value for your x and don't forget uh the value for your sorry the values for your x ranges between minus infinity we can say for x being between minus infinity to infinity. Now let me give you a quick uh check back on some of the things we learned earlier. I told you guys that one of the precise difference between discrete and continuous is that discrete has to do with countable data set whereas continuous has to do with measured data set. So when we talking about discrete we talk about whole numbers 0 1 2 3 4 5 and so on could be minus 1 - 2 - 3 but we never talk about decimal but you see when you're dealing with normal distribution you have to always highlight it in bands all right so you see what we have right here as I always told you guys this is just representative of a model you don't have to minus im just represents that x can take any value on a positive side and you can take any value on the negative side. So it's not restricted by anything. This is different for some other p from for some other distribution we have talked about in the past. For instance, if you recolct when we're dealing with geometric, our x value starts from one. All right? 1 2 and so on. We cannot have zero in geometric distribution. Of course, if you go back to the video on that class, you'll be able to understand what I'm saying because of the popularity of the distribution and its properties. Now, if you look at that of binomial, we said X ranges from zero up until N. The bandwidth for X will be from zero till N. It terminates at N. The number of trials uh that of Benoli X is always zero or one, one of the two. It cannot be more than one because we only have one trial in Benoli. For poison, we have X being between 0 1 2 and so on. Because we don't have number of trials in poison, we don't talk about an end point being N or whatsoever. It all depends on whatever the nature of the question is.
All right. Now for your normal distribution, we have our X being between whatever value you have on the left hand side and whatever value you have on your right hand side. That's the meaning of the minus infinity to infinity. Now another thing we need to talk about is first of all what is the representation of this sigma we have here. I'm sure you guys should be very much aware of what sigma represents.
Sigma represents the standard deviation.
All right. And sigma square is what the variance.
All right. So standard deviation sigma sigma square is obviously uh is obviously standard deviation and then sigma square is variance. Your sigma here takes value that must be greater than zero. That's the that's the uniqueness of this distribution. The sigma is greater than zero.
All right.
Now, um for your sigma square, your sigma square, of course, if the sigma is greater than zero, you know that if you square whatever value is greater than zero, the sigma square will also equally be greater than zero. So these are some of the things you need to note to understand to make sure you are in line and to have proper understanding of distribution. And as always you know I don't always like to leave any stone unturn especially at the beginning of any explanation to ensure that you are very s in whatever you are doing.
Now I want you to understand you can see right here that we have a mu value. I want to explain every datas everything we have here. Mu that's the mean. All right. It has its uniqueness is that it takes values between infinity to minus infinity to let me put this in a box so that you don't mix it up. Let me put all let me put this these are the let me put it in a box so you don't mix them up. So your mu is between minus infinity to infinity. The meaning of that essentially is just saying uh it all depends on the nature of your question. your mean will be take any value on the left hand side and on the right hand side. All right, don't get scared. Uh you will not be using too much of this formula at this your level. But of course any other person watching this video not doing the A level GP should understand that this is a formula you would use often. But there is a limitation a restriction to the coverage under the Jupb syllabus. So obviously this is just for knowing sake.
You will not be using more of this at this your level. But of course you need to note it because we're going to be doing more of distributions in the future. You'll be needing to understand the application. Now of course the pi that is here is the normal pi. Of course you know what pi is. I've talked about sigma. E is obviously exponential. What you're going to press on a calculator.
Sigma square is what we have talked about already. That's this variance. New being the mean X is what I defined already earlier. Yeah. So good. This is great. So we are done with this quick introduction to this aspect. Now let's talk about something else. Some other properties of the normal distribution.
It's very important you understand uh some key features. So I'll be rubbing off this uh PDF. I want to focus on something else.
So uh let's talk about something else.
I'm sure you guys can you paid attention to what we just said. Uh the key to understanding any topic is being very um patient and ensuring that you follow through on everything we're saying. Now let's talk about some some feature of this uh curve we have right here.
Now um just pay attention to something I'm going to do right here. Not this.
Let's add Okay.
And uh let's have another one here.
Okay.
Then let's have one here.
Okay. Now these are different divisions I made. They are very essential for something. Now look at something. We already defined this midpoint as mu. That's the mean.
Believe you all followed at that point.
This is mu.
Now I already talked to you guys about standard deviation.
I want you to understand that you see this point here from here to here from this point you seen right here that's the mu till this first red line will be one standard deviation. So from here to here is one standard deviation. All right. That means of course if if from here to here is one stand deviation that means this point this point right here if you are if you can follow through will be mu plus this point right here will be mu plus one standard deviation.
You can understand what I just did.
uh if this point is mu I told you guys guys that from here to here is one standard deviation that means it's more like saying if I said if I said this point is five and I said this point from here to here is one of course that means this place will be 5 + 1 that's just what's happening here so if this place is mu and from here to here is one stand deviation that means this place will be mu plus one stand deviation all right and if you Look at that.
This place to this place also will be of course one standard deviation. That means this place now if you guys can guess will be mu - one standard deviation. I'm sure you agree with that.
It's more like if I said this place to this place is five is one. That means this place now will be 5 - 1 that will be four. It's just simple uh logic.
If this place is five and we say from here to here is one of course from here to here will be one also. That means this place will be four. This place will be six. That's what we just identifying here. Good.
Now that you understand that uh of course what we say right here is that mu minus one mu minus one that's this point right here which you can all see mu minus one that identifies with one standard deviation below the mean. This is the mean. So if this is mu minus one that's one standard deviation below the mean. Whereas mu plus sigma mu minus sigma is one standard deviation below the mean. Mu plus sigma is one standard deviation above the mean. I told you guys already that from is one standation. That means this is one stand deviation above bigger than the mean.
Simple. And from here will be one standation lesser than the mean. That's why we say one standation below the mean. Very simple. Good.
Now uh let's try and use uh dots uh to shade this point everywhere we just talked about.
You can see we just identified we identified uh one standation above and one standation below. Now this thing I want to talk about next is very essential because you'll be needing this understanding very soon not probably for this particular thing we saying but to understand the whole concept around normal distribution now you need to know that don't forget guys that of course we have done this before I told you guys that let me see if you guys can recollect this what's going to be the total area under this curve guys can anybody shoot the answer to me quickly what's the total area under this curve. Anybody?
What's the total area under the curve?
I'm waiting for answers.
What's the total area under the curve?
Somebody said two.
What's the total area under the curve?
Zero. Really?
We talked about this when we're dealing with continuous distribution. Zero.
People are saying zero. One. What's going on?
The total area under a curve is one. I told you guys that already. Yeah, that's the reason why we said when you people are writing zero total area under the curve really really I told you guys that when dealing with continuous distribution of course a function brought about the cough this cough did not just happen a function made it happen if you have f of x being something we then said for you to get uh different probabilities we go on to if you want to get the probab gravity from maybe year to year. We said you will integrate uh the function with respect to that band whatever the band was. Then I told you guys that if you want to get of course if the band if the total band was let's say uh minus infinity to infinity let's just put that as a model. If you integrate the function with respect to the total band that mean the beginning the last point here and the last point here. If you integrate that function, you must get the value of one. Told you guys that already. That's quite very basic. The area under the C is always one. And that's the reason why we said if something is a continuous random variable, the integral of the function with respect to the band, the limit, the limit you have must be one. It's a basic knowledge. All right. Now what I I said that to say this if the whole area under the cough all all everywhere I don't if we could shade everywhere if the whole area under the curve is equals to one so I want to talk to you about this particular area what it will always be so now that means if you ask a question any objective question and you are told to talk about the area uh between the area that lies between one standard deviation above and below the mean that's this point I just shaded all this point I just shaded now this area this area is going to be 0.68 note that standard this by standard that means I mean the whole area this whole area I shaded in black below and above what's meation below and above this area now so if this whole place is one this particular place I shaded is 0.68 68.
Note that very very important is a very important thing to note. That means that area area in between mu minus sigma and mu plus sigma is approximately equals to 85% sorry 65 68% of the area. All right.
So this uh this uh mu minus sigma and mu plus sigma the area it takes is always 68% of the whole curve. All right note that now let's talk about the next one. Of course if this is always one one so that means this place now particular one will be mu - 2 2 sigma. So if if you understood that from here to here was one standard deviation that means from here to here will be two standard deviation from here to here one stand from here and that standard deviation.
So that means from if this place is mu this place will be mu minus 2 standard deviation more like just understand it like this let me use this other one if I have 5 - 1 and all these place are one one of course you expect that this place should be what 5 - 2 if this was five this is four this will be three so that's what is just happening here if this is five this is one star let's say this was mu this one stand elevation this two star elevation so this place will be 5 minus two standard deviation. That's how we got this. And this other place will be mu plus two standard deviation. All right, that means this particular spot is two standard deviation above the mean. And this particular spot is two standard deviation below the mean. This is the mean. This is below it. This is two standard deviation above it. All right. So let's talk quickly. Let's try to shade that whole point. Let me use uh let me use this marker now. So everywhere now including that black spot everywhere I'm shading now in blue everywhere I'm shading of course the place that encloses the place that mu minus two standation and mu plus two standation encloses what's going to be that area now so I've talked about this other area so let's try and talk about this area so this area that is enclosed by mu minus 2 standation and mu plus two standation is 95% of the whole area 0.95.
So imagine let's try and ignore this.
I've spoken about that already. So let's say we are dealing with this whole area now. All right. This will be 95% of the whole. So if this whole place is one, this place will be 0.9. This place that we have shaded is 0.95 of the whole curve. All right?
0.95 of the whole curve.
Now uh 0.95 of the whole cough note that so that if you are asked in an objective question and MCQ that what's the area um between two standard deviation below and above the mean that area is 0.95 95% of the curve. Now that's that about let's see for the last one of course can anybody tell me what this place will be based on what I've been explaining. What do you think this place will be? This will be mu plus what? Tell me what this will be. Mu plus what?
And this place will be mu - what?
Can anybody give me an answer there?
Yeah, this will be mu + 3 standard deviation.
mu - 3 standard deviation.
All right, that's just uh the old basic. Now, let me talk about the area of course. Let me uh take this out now and let's assume we are going to be shading every of this now.
every of this now I'm talking of everywhere but don't forget we have not shaded everything there's still a small thing left so this area obviously as you can guess will be almost one but not exactly one that will be 0.997 so this whole area shaded everything from here to here everything is 0.997 that's obviously 99.7% of the area so that means 9.7% of the area lies within three standard deviation of the mean. That's just the conclusion.
99.9 sorry 99.7% of the area lies within three standard deviation of the mean. So this is a very important property uh that I want you guys to note. All I'm trying to explain here is an important property that comes with normal distribution. Of course I told you already that one of the basics of normal is that it's symmetric about the mean.
Meaning that this midpoint which is always going to be the median and the mode is the mean. But we are going to focus our attention more on it being the mean. So and I told you guys that from here to here is one standard deviation.
Of course if you can look at this same measurement this same place this will be one standard deviation. Of course, if this is mean, this this particular spot will be mu plus one standard deviation while this particular spot will be mean minus one standard deviation. Just basic. I try to explain it using a number line here. If this was five, of course, if this point represented one, that means this place be 5 + 1 and this place be 5 - 1. Of of course, that's no brainer. This was five, you expect this place to be four, you expect this place to be six. If the uh if the dimension was in one was counting in ones. All right.
Now uh now uh if you're talking about the other side that means two standard deviation above the mean, two standard deviation below the mean, three standard deviation above the mean, three standard deviation above below the mean. So that's just uh that's just basic. Let's uh this is I just want you to you know as I told you guys already uh this is all uh introductory and uh all these properties are things you don't say how do we get like for instance you trying to say how do we get this this is just it's just normal understanding you don't need to crack your mind around this it's something I'm telling you that is happening that if you have three elevation above and below the mean the whole area that will be uh covered by that band. All this place now that is shaded is 99.7% of the whole area. It's not something you need to work on. What how did you get it? It's something you need to just know that that's just it. I told you that the one one standation above and below is 68% 0.68. That's standard. You're not trying to get that.
It's something that I'm trying to tell you works. All right? It's not something that you are going to learn how to get it. You can you can get it if you can go uh with trying to prove it. You can it's not it's provable especially if you have a computer to do that. If this whole area is one this the one standation above and below will occupy 68% of the old area. It's just basic but you don't need to wrap your head around that. All right. But the one for two standard deviation above and below I told you it occupies a space of 95%.
That's 0.95.
And uh the other one uh 0.997 is 3 sigma below and uh above the mean. So it's all basic. It's all basic. These are things you just need to understand. This is the this is the first uh class. So you don't expect that everything comes very um of course you've not known anything about this. So these are things you just note on your left hand. I'm trying to just make sure we talk about this so that at least there is no confusion in the future. So this is all just start getting familiar with this C. It's a very interesting one.
Very interesting one. All right.
Now uh let's get on to something very important.
Uh we have All right. So we have uh we always talk about we always talk about uh we said a random variable X follows a Benoli and I told you guys that the parameter there is parameter P. All right.
And uh we said a ben sorry a random variable x follows a binomial and I told you guys that this will be n comma p.
These things are very important I'm writing because sometimes you're not always going to be seeing it as uh sometimes they will just say uh x follows a binomial and it wrote 10 comma 0.01 something like that. So you need to know what all these things mean.
All right. And uh we have another the next one follows a poison and we have the parameter there as lambda. So whenever you see something and we have poison and probably they just wrote three. So you need to understand that the word three there the parameter the three there is representing the parameter lambda. All right. And uh we have uh that for geometry the parameter in for geometry is p probability of success. All right. So let me touch uh that for normal of course note all these things very very important.
You could you don't always get familiar with the use of the word priority of success or this. They may just write it in this form and expect you uh to bring out the values yourself. So for that of a normal distribution it follows a normal distribution. This is how you write that of a normal distribution. The parameter is mu and sigma square.
This is the way you write that of a normal distribution.
All right.
So when you have when you have a question on normal just note that the way it parameter is written is this in this way. So that means if you see something like this X follows a normal distribution and you see for instance let's say they wrote five comma 1.5 of course you need to understand that the five there will be representing the mu that's the mean and 1.5 representing the variance. Now I need you to understand that sometimes we have different forms. Some people tend to sometimes want to write the form as this mu and sigma. So uh it's all going to be indicated in whatever you're dealing with. Uh if you're using a table and the table may tell you that this is the format we are following. So I tend to follow this format but whichever one uh this is still okay. This is okay. But just note that you could see any of these two. Don't get confused. All right.
Now there's something very important I need to talk about next. This is now the meat of the whole thing. Everything we have been saying is just basic property and um you need to really really understand the next thing I want to talk about. This is now where you tend to see uh questions and this is where you will be really tested in your understanding of what I want to say next.
All right. No.
When we were dealing with continuous random variable, I told you guys that when you have a cough, if you're told to find for instance, the probability of let's say we have let's say this is 5 10 15 20 25 for instance.
If you are told to find the probability that the random is between 10 and between 25, you are told to find this.
You are told to find this. I told you guys in that class that what essentially they asking you to find is the area under the cough. That's what I told you guys when we were dealing. If you could find the area under this cough, of course, between the 10 and 20. If it was 15 that means you'll be finding area from 15 to 25. So the whole understanding is when you see a question like this under continuous random variable or continuous distribution function understand that you're trying to find the area between that band.
So in this case now if it was 10 to 25 you're trying to find the area between 10 to 25. Now I told you guys also in that same uh class that you don't tend to uh have the opportunity or the ability to find the area under this curve generally. So most of the time what we do is to go on to integrate the function. So we integrate whatever function brought about this curve setting the limit between the band we are trying to find.
That was how we used to evaluate these questions. All right, we used integration. Integrating using the band we were given to find a solution. That's the way we used to solve questions like this. I'm sure if you have been following you can really understand what I'm saying. Now, but there's another way we can solve this question and that's the use of a table.
All right, the use of a table. I sent you a piece of the table uh some moments before this class. Uh I'm sure you just uh I'm sure you you of course you will not be using it so much this night. But if you see the table you would see it has different uh numbers in it. So those numbers represent different probabilities.
the answer to different probabilities which I'll be explaining to you not in this class but some other classes. All right. So another way of finding all I want to say here is another way of finding this question of solving this question without having to integrate which is what you have been doing in the past is by using a normal table and that's what we want to be delving into in uh subsequent classes. But what I want to do majorly in this class is to teach you a very important thing you must know how to do if you're going to be able to solve questions like this.
All right. Now, this is one of this is a very big uh problem that used to arise when talking about normal distribution. Just pay attention now. You would understand what I'm saying. Now I told you guys already now that a normal distribution follows we have the mu and let me even just let's use standard deviation as that second parameter.
Now take a look at something. Now I just said now that if you have followed what I've been saying so far you would see that I said for us to find this probability we can decide not to move through the route of making integration of the function with respect to the limit based on the bands that was given in your probability.
Now you have this you can do it by using integration if you have the opportunity to but another way of solving this is by using tables all right using tables to solve it now but there's a problem that can arise from what I just said and the question is how do we find area under a normal cough now look at something this is a possibility.
Guys, pay attention to what I want to say next. You would I will try and draw as many diagrams as I need to to get this very clear.
Let me make this space available.
Now look at uh this particular what we have right here.
Let's say we have another one with 10 and let's say 3.5.
Let me see let me write the third one that we have a normalization with uh let's say 12 and uh 3.8 just an example. All right.
Now this is the problem that arise from us trying to use a normal table.
If you check your normal table, of course, that table I sent to you is what we call a standard normal table because that's a way standard normal table. I will explain to you the reason why we have that.
The reason why we have a standard normal table is because of this problem that comes with a normal distribution. Don't forget I said standard normal table. I did not say normal table. Let's assume we don't have a normal distribution.
What would have needed to have done would have been that for every of these kind of question now the mean is five the standard deviation is 1.5.
We will need a separate table just like what I sent to you. Now we need a table for when the mean was five and the standard deviation was 1.5. There will be a table for that. Then that table is now what you will use to find answers to whatever question arise from a situation like this. But look at the the problem that will come as a result of that.
What about when you're not dealing with that parameter again? Now you're dealing with a new parameter of 10 and 3.5. You see the table I sent to you, you will see they drew a cough, a normal cough there.
But imagine you have a different parameter. That means you'll be needing a different table for when mean was 10 and standation is 3.5. And let's say now we have to go on to they say okay that question we're done with that question.
Let's say you just had a question to solve it. Then they give you another question. They say okay in this case now the mean is 12 the standard deviation is 3.8. That means you will need a different table that will be solving that question. Now this is very uh almost impossible to handle.
All right. That's the reason why we talk about something called a standard normal table. So the question is what is a standard normal distribution? Of course, when we say a standard normal table, that's obviously a table that mirrors the properties of a standard normal distribution. So that means from what I just said now, we have a normal distribution and we have a standard standard normal distribution.
And there is something you need to note there. These two distribution they are similar but they are different.
All right.
So what's the difference between a normal distribution and a standard normal distribution? They both normal distribution but a normal distribution has a peculiarity.
This is a normal distribution. What I just wrote here, uh, X follows a normal distribution mean.
All right. For that of a standard normal distribution, we have Z following a normal distribution.
All right. But in this case now the mean is zero, the standard deviation is one.
Now if you look at what I just said right here, this is a normal distribution which I've mentioned before. We've talked about this. This right here is a standard normal distribution. That means you can clearly define a standard normal distribution without wasting time because you can see that what I wrote there indicates something. That means a standard normal deviation is a normal distribution with mean of zero and standard deviation of one.
Let me repeat that a standard normal division is a normal distribution. It's a normal distribution. You can see I said zed follows a normal distriution.
It's a normal distribution but the normal division has a mean of zero and a standard deviation of one.
All right, that's the pecularity of a standard normal division. The question that you should ask me is what then is the excense?
Why do we need to talk about a standard normal division? Now I just explained a very big problem that can arise from just discussing a normal distribution is that for each different scenarios of course these are just a few I just you can go on to write a th00and or 10,000 or a million possibility of course mean can be five staration can be 0.1 m can be five staration can be 0.2 mean can be five staration can be 1.2 to me five just for just me being five you can have thousands of possibilities that means for every scenario you'll be needing a table to handle them because the normal that has five and 1.5 will not obviously work for that or five and 1.2 two will not obviously work for five and 2.5. So you'll be needing different tables for every possible scenario which is not feasible and that's the reason why in statistics we don't deal with what we call a normal table. No, we deal with what we call a standard normal table.
All right. Now another word be used to identify a standard normal table is Z table.
So when you hear the word Z table, you are dealing with a standard normal distribution. Z table. All right. So now I told you guys already that a standard normal distriution is a normal distribution that has mean of zero and standard deviation of one. Good.
All right. If you look at something and you will be able to understand something going on here. The reason why we have a standard normal a standard normal distribution of course which will make us to now have a standard normal table is that when we have all these many different possibilities rather than we trying to solve all these many different possibilities all we have to do is something and this is the word that's The big word, the very important word when dealing with normal distribution. We standardize.
We standardize. So it's a word you'll be using very often while dealing with normal distribution. So we have several possibilities. When mu is five, when standard deviation is 1.6, when mu is five, standation is 1.7. When mu is 10, standation is 3.8. Different possibilities infinity. I can't you can't none of you here can tell me the number of possibilities that existed but rather than we focusing on those possib different possibilities all we have to do is whatever you have just make sure you come back to standardize. So all you have to do is when you have a question like this go on and standardize it go on and standardize it. The meaning of standardizing is for you to conform that normal distribution to a standard normal distribution.
So you have to go on now and confirm that's your normal deviation to a standard normal distribution. Meaning that you will now be able to make use of a standard normal table to solve any problem normal distribution. So you will not be needing to be getting several tables for different scenarios of normal division.
All you have to do now is to go on and just whatever normalization you are dealing with standardize it. Then you can use just that table I sent to you.
That table I sent to you is a standard normal table. That's what we call a Z table. So you these are just few possibilities. When we even start solving questions, you will see all the many possibilities. Your mean can be 100, your mean can be 50, your mean can be 80, your standard deviation can be 5.3. Different possibilities. So the thing you just need to do is to standardize that conform it to a standard normal distribution and where thereby you'll be able to use a Z table to answer it rather than you needing a special table for it. That's the word you need to uh get familiar with. Every time you have a normal distribution you always need to do something you need to standardize it. All right you need to standardize it. So the next question which is a very big question is okay all right you have said all these many things now we understand what you're saying can you now talk talk about how do you standardize a normal distribution how do you standardize a normal distribution very simple very simple and straightforward very very simple it's as simple as ABC it comes with no stress at all it comes with no stress at all the only thing you need to understand the reason why you use it and how to use the table which is all very simple.
All right. So by standardizing I'm sure you understand what I'm trying to always do now when I'm trying to standardize anything.
That means I'm trying to make that thing come to having a mean of zero sorry come to conform to what a zed uh distribution entails.
Now let me teach you something that happened.
Let me tell you something that happened.
You need to pay attention to what I want to say next so that you don't feel any form of magic was done. You can see right here that I was very uh I did not act like I wanted to write X here. I went on to write zed. I wrote it clearly. Meaning that something happened here. Nor we said this was a random variable X before. But now once we see that the random variable X has a mean of zero and a standard deviation of one, we turn that X to Z. We said that's a standard normal distribution. What happened that made X turn to Z. This is what happened guys when we had X before.
When we had X before, if we subtracted mu from X, what happens there is we have a mean of zero.
And if we now go on to divide it by sigma, we will be having a standard deviation of one. Many of you will be trying to say, can you prove it? What's going on?
You don't need to bother about that. I'm just telling you what happens that turns x to zed is that this x turning to zed at this point zed all right let me just say this way I'm going to take this off this is zed just let me say that way for you to get zed that mean for you to get zed which I told you had a mean of zero and a standard deviation of one x you subtracted mu from x when you subtracted mu from X that's the mean from X you got a mean of zero when you divide by sigma you get a standard deviation of one all right that tells you already that for you to standardize you don't need to start sweating over anything whatever was your x value whatever was your x value whatever it was if you want to turn that uh random variable if you want to turn random variable to a Z uh uh variable, you just have to subtract the mean, divide by the sigma, and that's all. So the process of standardizing is what I just wrote right here.
This is how you standardize.
What how you standardize is that whatever your X variable is subtract mean divide by sigma you are done you stand that is it it's so simple nothing else nothing more all you have to do is when you have anything you will see an example now you know there's no way I leave this place without giving you examples to saw so that you get familiar with standardizing is the only major skill you have to learn under normal education apart from you being able to identify your mean and var and standard which is always very clearly said in the question. All you need to know how to do guys is to standardize and then learn how to use your table. It's more easy than easy can be. I can assure you and you I told you guys at the beginning you don't need to always shoot yourself in the leg when you are just learning something. Just get patient follow through. You don't you can't go wrong but you don't expect that the first time when you're seeing X turning to Zed that at that very moment of course if you are very sharp or you have an idea of what is going on. You may but at this point you it's still premature for you to be blaming yourself for not understanding something. So just try and be patient and then you will see what goes on very soon. Now I just gave you something very important. I'm emphasizing on it so that you understand how simple this is. If you want to standardize, go on, find the mean.
Get the mean. Of course, I'm sure you guys are smart enough to know that. One of the one of the uh trick that they can give you here is for them to give you a question without giving you the mean, expecting you to find the mean. Of course, that's a story for another day.
But whatever you can do, identify the mean, identify the standard deviation.
You say, "I want to standardize it."
They gave me let's say an instance now.
Let me see an instance now. When I was dealing with this random variable, this one that has mean of X and standard deviation of 1.5. I said okay good.
Let's say they told me my random variable has u or let me just say okay let me just say this way very I would prove to you guys how simple this thing is. If I tell you something is simple believe me it's simple. If you don't believe it's simple that will be your own decision. But it's quite very simple. I have a question that said my value is between uh let's say 15 or let me just say let me put 15 here and let me put 35 here. Look at the question right here. Find probability that X variable X is between 15 and 35.
Don't forget I told you guys that we're using we're using this particular uh normalization parameter that means stand of five and standard deviation of 1.5.
Let's say I of course I'm not trying to build up a word problem around this. I just want to give you the the cheap the the how very easy this is. You have a question like this. You have this parameter. You have x variable here. You don't have a table that can handle normal division with this X. You only have a standal table with you. How do you standardize this that makes this achievable with a normal a standard normal table which is what you have with you? Very simple. I want to find probability of X being between 15 and 25. I can't find any table that handles this. I said you want to standardize this. You need to turn it to X. And if you're turning this to X, you turn every you in mathematics. You don't do something here and don't do it to the other places. So this is what I'm going to do. I'm going to say P. I'm going to start with X. I want to standardize X.
Look at what I told you. For me to turn X to Z, I have to say X minus mu over sigma. I'm going to decide not to put anything here first. Meaning that let me clean this up.
I wanted to standardize this. Now I had X in the middle here. I want to change it to Z. I go on to subtract mu divide by sigma. meaning that once I do this I told you immediately that becomes zed now but in mathematics you don't say I do something to the middle I don't do to the other side so you would have to do it to these two side but in these cases now you don't have to be using minus mu or over or or just writing it in alphabet or something in in in this way you have to go on to fixing the values you have known so my mean was five so I would say 15 - 5 over what what's my sigma 1.5 I put that here on this side I have 35 - 5 of course mean is 5 over what over 1.5 now can anybody press the calculator quickly and tell me what this is this is 10 over 1.5 what's 10 over 1.5 anybody quickly can you give me the answer quickly uh 10 by 1.5 would be 6.667 6.67 right 6.67 67 probability. Okay.
And this other side we have 35 - 5 that's 30 / 1.5 that's 20.
Now look at what I'm saying right here.
A very typical example. Now this is something you need to know. I had a question. The question said I find probability of X being between 15 and 35.
Now you don't have normally I told you already this is if you had a normal diffusion that had this that you had a normal that had this this whole feature you just go to normal and use this of course don't yet start killing yourself on something you have not been taught I've not taught you how to use a normal division so you probably feel it's difficult it's not difficult at all it's so simple you just check it it's more like looking at the horizontal and looking at vertical you answer it's very simple very very simple But you don't have that table that can handle this because you don't have anything. You only have a table that can handle Z. That's the stand normal distribution. So what you have to do is go on to standardize this. And how do you standardize this? You just go on to do this exact thing. X - mu / sigma everywhere. Now for 15 for X45, you have to go and subtract the mean divide by sigma. In this case now when you subtract the divide divide by sigma you getting zed in this place. Now you do the same. Now look at what we have got.
Now we no longer having x between 15 and 35. We are saying zed between 6.67 and 20. Now I told you guys already just stop right here. Don't yet talk about how to get this. That's a that's a lesson for another day. How to get this is just by looking into your table which we'll be doing sometimes later.
When we get to that place, you will see how easy it is to bring this out on your table. But what I wanted you to learn today is to learn how to make this become this because you cannot solve using any table from here. You have to standardize. How do you standardize?
Subtract the mean, divide by sigma, do the same, and that's all. Very simple.
It's that simple. There's nothing else to this than what I have said. So, let's go on to some questions. I don't want us to just be let's use let's use some questions to and to to get yourself very familiar with this guys just uh if you have a pen please try and jot this down because I would not be writing this on the board of course I would probably do that when I want to solve this completely but for now I just want to teach you how to standardize But please write this down so you understand how these are the kind of that come out. You see it a lot a lot in your exams.
Suppose the height of adult American male is approximately normally distributed with a mean of suppose the height or do you want me to write it? probably I try to suppose the height suppose the height of adult American male is approximately normally distributed with a mean of 176.3 and a standard deviation and a standard deviation of 7.1 cm. They're both in cime by the way.
cm if you want to write centimeter cm.
All right. So this is the question now.
What is the probability a randomly selected adult American male adult let me write as adult American male am so we don't have to write that over and over again is taller than 180.0.
This is a very good question for us to start with.
Suppose the height of adult American male is approximately normally distributed with a mean of 176.3 and a standard deviation of 7.1. What's probability that randomly selected adult American male is taller than 180 cm? This is a very uh typical question you should be expecting. You give you a word problem and that's why I don't want to do anything less in teaching you. We cannot just be using just figures alone.
Let's use word problems to tackle this.
So calm your mind and look at what we have right here. Suppose the height of adult American male is one is approximately. Look at what they told you. It's normally distributed. So I told you that normal distribution is a very important decision in statistics because it mirrors day-to-day living.
It's a very it it has a lot to do with real life experiences. So they said the height of adult American male is approximately normally with a mean. Look at they told you that's what I told you.
They always tend to that's what makes this kind of question very simple. They will give you what for what they are saying the mean now they said is 176.3.
All right.
And they said the standard deviation, you can see it right here, standard deviation of 7.1.
You see how they are so direct with what they saying? They're not mining words at all. Sigma is what? 7.1.
Simple. They went straight to the point.
Now they've told you that means now the if we if we say X represent the random it's a random variable representing the height of course X represents the height of adult American male. So that means X follows because they told you it's approximately normally distributed. So expose a normal distribution with mean of 176.3 and sigma of 7.1.
Good.
That's what I was telling you. It's so direct. They gave you exactly what they are doing. They are not trying to uh um put anything technical or anything.
That's all they from here to this point.
That's this is the summary that X follows a normal deviation 176.3 as a mean some.1 and standard deviation. Of course, I define my random variable X as representing the height of adult American male. Simple. Now the next question said what is the probability a randomly selected adult American male is taller?
What is the probability? What is the probability? probability of a random randomly selected adult American male of course since I have said x represent the height of adult that means I can write x now instead of just writing it word for word and I explained to you at the beginning of explaining random variables and that's the main essence of us talking about random variable is that it helps us to be able to write things rather than writing height of adult we can just write it in a numerical way so what's the probability a randomly selected adult North American male of course of X is taller than is greater than taller greater than 180.0 zero.
Simple. That's the question. It's no even I think even a J a a a newborn uh secondary school student should be able to solve this like to interpret this probably not solve this but interpret this when they said property selector is taller. Taller means greater than obviously greater than 180 taller than if they say it's less than that would be less than very simple. So what's probability that a randomly selected adult American is taller than 180.0?
Simple. Now this is where I was explain this is where the use of standardization comes in for all the many things I have said.
Imagine you had this question and you don't know about standard normal distribution. What you'll be needing to do will be to go and be looking for a normalation that has a mean of 176.3 and standard deviation of 7.1 which is not feasible. You don't even have a table like that. That's the reason why we have to do what we call standardization.
So what we just have to do now is say okay come we're not going to be using a normalization. What we're going to do is to use a standal and the way to confirm my noration to a standardization is by standardizing. So what do I have to do here now? Rather than trying to find this using a normalization which I don't even have I have to go on now to standardize this by just doing this simply I want to standardize this. Now X will have to change to Z. And how do I change X to Z?
I change X to Z by saying X - mu as I told you already divided by sigma.
That's the way I'm going to change X to Z. I subtract the mean divided by the standard deviation. That's the way I'm going to standardize.
And then I have to do the same thing for both sides. So obviously I'll be having 180.0 now minus what was our mean? 176.3 divided by what's my standard devation?
7.1.
Simple.
Now obviously this will be changing to zed. Now then you can go on to do your calculation for that.
If you go on to do that 180us 176 you'll be having 0.521 or so.
Now getting to this point, you're almost done with all your work. I've told you guys this is so simple. It comes out a lot, but it's a bonus mark for you if you understand this. If you just understand everything I've said, it comes out a lot because they believe that it's a technical aspect of statistics. But it's not. It's a very simple aspect because you just need to go on interpret your question the way I did from here to this point was interpretation of question. Very simple to interpret. Then I go on to standardize. How do I standardize?
subtract the mean divide by standard deviation and that's all. No need to do anything more. Just subtract the mean and divide by standard deviation. I do that I get my answer. Now this part now is the part where you be needing a table which obviously I'm not touching and I'm going I'm not going uh I'm not going to be dealing with this in uh with uh in this class. I'll be doing this later on probably much later. But I want to emphasize more on uh having being able to standardize. So this is it. If you have if you have a table now, you can use this. You can find this with a table straight up and directly. All right.
All right. Uh now we have to we have to go on to answer the next question which is uh for you to find the probability. Now look at this was the first question. Let me now I'm going to give this out to you guys to solve. I want you to find a way to get this uh this is the next this is the question I'm going to ask you to solve. Now all I need you to do now is to standardize this. All right.
So the question I want you to try and solve of course I don't need you to check a table here is what is the probability a randomly selected adult American male has a height It's between 170.0 and 180.0 cm obviously.
So this is the question I want you guys to try to answer and all you need to do is to give me uh the standard form of this. What's probability a randomly selected adult American male has a height between 170.0 and 180.0 randomly selected adult American male has a height between 170.0 and 180.0.
Can you give give me what the standard form is going to look like?
Do we have any answer right now?
Do we have any answer right here? What's going to be your form?
Now, don't forget this question is still related to the formal question where you have a mean of 176.3 and your standard deviation of 7.1.
All right, I have an answer right there.
And the person said Zed is between 0 sorry minus 0 87.
Now look at how you're going to answer that. Of course.
Uh you're told to find the probability a randomly selected adult American male has a height between 170. Of course, you want to first of all interpret that question. This question number B.
The probability that a random selected adult American male has a height between this.
That means your random variable X of course we're dealing with adult the height of adult American male we have 170 and we have 180.
Sorry.
So this is the question you're trying to answer.
The probability a randomly selected adult American male has a height between 170 and uh 180.
All right, this is the question. This is just interpreting the question, nothing more. Uh you have to find probability that X is between 170 and 180 just like they said. Now you cannot solve this uh because you don't have a table that can give you this answer. You have to find a way to standardize this. And I told you already how to standardize anything.
What you have to do is to go on to subtract the mean and divide by the standard deviation. So simple. Subtract the mean divide by the standard deviation. So uh of course you have to do to every side. So let me just uh write this here. That mean you'll be having 170 minus the mean was 176.3 the standard deviation was 7.1.
The mean was 176.3.
Standard deviation was 7.1. So I'm going on now to subtract the mean and divide by the standard deviation. That's all I have to do. All right. And if you go on to do this, you see that you get a probability of minus0.887.
Of course, this will turn to zed and this will turn to 0.521.
All right, that's just very simple. This is how you standardize anything.
So, uh the whole idea around this is to find a way to first standardize them at the end of all this. Now, I'll be explaining in a very much later class how to find this on a table. But the essence of this is to ensure that you guys are very sound with standardization. So let's try this question also to perfect your knowledge on that. Take this down. If a random variable has the normal division with if a random variable if a random variable has the normal distribution with mu equals to 82.0 0 and uh the sigma is equals to 4.8.
All right. So the question is for you to find the probability it will take on a value.
So the very first question is the value less than 89.2 B greater than 78.4 4 C between 83.2 and 88.0 and then the last one which is D between 73.6 6 and 90.4.
All right. So, I want you to find a way to do this.
Uh, you have a question that says if a random variable has the normal division with mu of 82.0 and the sigma which is standard deviation of 4.8 but the find probability it will take on a value less than the first question is this. All I need you to do for me in this question is to just bring out rearrange this and make it conform to a standard form. Of course, we will not be solving a question perfectly because we would not be talking about the use of a table in this class. All I want you to do is knowing how number one to interpret the question and knowing how to standardize it. So, can you give me the solution to these different uh questions, the four of them very quickly?
So the very first thing you have to do is to bring out your data and you have your mean is given as 82.0 and your sigma is given as 4.8.
That was clear. Of course, a random variable has the normal distribution.
That means we say a random variable X follows a normal distribution with mean of 82.0 sigma of 4.8.
All right. So the very first question now is for us to find the probability to take on a value less than 89.2. So the first question is saying find the probability of X being less than 89.2.
This is the very first question you have to answer.
The second question said you should find it being greater probability of X being greater than 78.4.
All right. And C X is between 83.2 and 88.0.
and D X is between 90.4 and 73.6 All right. So for you to find uh the very first one, all you need to do is to standardize this. That will be probability of x - mu over sigma less than 89.2 minus your mean is 82.0 over sigma is 4.8.
So when you are done finding this what did you get as your final answer for a of course probability of zed because your x - mu divided by sigma will be resulting to zed Yes.
So that's the first answer. All right.
The second one the question was for you to find greater. So we'll be having of course our x - mu / sigma greater than 79 sorry 78.4 we having 78.4 minus 82.0 0 over 4.8.
This will give you probability of Z greater than when you go on to solve that.
All right.
Uh what did you get as the next one? The third one.
Of course, for the third one, you'll be having x - mu / sigma And uh you're having 88.83.2 that'll be 83.2 - 82 over 4.8 less than you you have 88 on the other side. That'll be 88.0 - 82 over 4.8.
the probability of this. So we go on to solve that.
What is the answer you got?
You go on to solve question number C.
What did you get as your left hand side?
83.2us 82 is what?
Okay, thank you. 0.25 less than zed now because this will be changing to zed less than what? 1.25.
All right, thank you very much.
A probability of this. And for the last one for D, this is what you have. Of course this will be probability of 73.6 minus your mean is 82 over your sigma is 4.8.
Of course x - mu / sigma is what we're doing. Then we have 90.4 minus 82 over 4.8.
This is what we call the process of standardization. I'm trying to get you guys very familiar with heat before we start solving question on heat. Once you are very good in standardization, all you need to know is how to that will be probability of minus0.75.
Yes, is that correct? Sorry. 1.75 less than Okay.
All right.
So this is just it. You can see the process of standardizing a normal division. Of course, I told you guys already the main reason why we go about standardizing our division is for us to be able to use a standard normal curve. So let me do a quick recap on everything we have done so far. I started this uh topic introducing you to the concept of normal distribution and I told you guys that normal distribution is a symmetric distribution which is symmetric about the mean.
All right, I said it is symmetric about the mean.
One of the very first thing you need to note is that of course we call when you say something symmetric it has what we call a bell shape like this.
You see something like this. Now of course when I say symmetric about the mean I told you that that means the dividing line that divides the this thing into two equal parts. Two equal parts is the mean.
Now that mean of course as you can identify will also be the median and also the mode. So one of the properties of metrication is that the mean equals to median which is equals to the mode. All right, note that very important.
Now I went on to of course which something we have done before which we say the I asked you about the total area under the curve. So area under this curve is one. Note that area under the curve is one. And uh I wrote the PDF of a normalization for you which I told you guys that you would seldomly use almost not never use under in this your Jupiter program. But um of course if you have to do more into normalization you would have to use them and I explain to you all the component of the PDF.
Now uh moving on I went on to identify uh some of the special features of the cough. Of course since you know that the whole area is one don't forget that the whole area is one. Meaning that from here the whole area will be one. But I told you guys one of the special feature of normalization is that if you have one standard deviation above and below something like this this area that is covered here of this is mu plus sigma this is mu minus sigma one elevation above and below the mean.
I told you that that area is 0.68 that means it's 68% of the whole area.
Of course if this is if this whole place is one is 0.68 that means the rest place that we have not shaded will be 1 minus 0.68 you can understand that because the whole area is one and uh this place we have shaded is 0.68 68 and then went further to talk about what if we have to discuss about two standard deviation above and below that means probably we move a bit further that be mu + 2 sigma and then we talking about mu - 2 sigma so we have to talk about this area now the area that is bounded by two standard deviation above and below the mean I said the area that is covered by this band is 0.95. That's 95% of the whole area. You don't you don't you you need to understand that I may not be drawing the whole diagram to scale. So just understand the whole concept. Of course, it may not be very perfect since I'm using an N sketch. Now um for that of three standard deviation mu + three standard deviation and mu minus three standard deviation of course the whole area bounded by this space of course three standard deviation above and below I told you that is 0.997 that's 99.7% of the whole area that's just a special feature you need to note in case you are being examined on that then I went on to now tell you about how you identify a normal division It has a mean and a standard deviation and we say X follows a normal distribution with mean and standard deviation. All right, that's just a way we write it. Now I told you that ideally if you didn't have let's say you had a function you drew a diagram and you had a function what we have always done when dealing with uh functions is that for us to find the area under the curve what we always do is to integrate that function.
So say we have a function let's assume a function 2x² produced this. Let's assume if we wanted to find the probability let's say we have one let's say this was five this was 10 15 20 25 just an instance 30 now if this was a function that produced this curve and if you're trying to find the probability of the random being between let's say 15 to 25 if we knew all these things what you can uh is that if you're trying to find this probability all right what you have to do is to find the area under the curve if you could that means the area bounded by 15 to 25 between 15 and 25 that means you want to find this area if you could using geometry but it's not very possible to do that and that's the reason why sometimes if you're dealing the continuous random variable of course you have to go on to integrate the function that means imagine 2x square was what brought about this whole c you'll be integrating 2x I said imagine I did not say it was you would be integrating 2x² with respect to 15 and 25 being the the respective limit when you integrate this function with respect to this limit you will be getting the area under the curve that's the way you find area under the curve but I told you guys that one other way you can find the area under the curve Apart from using integration is by checking a table. All right, you check a table. A table is a very beautiful way, simple, easy way of getting area under the curve. Now, how do you use a table to get area under the curve? Now, you have a normal division. Let's say for instance, that moves with a mean of 40, salvation of 2.5.
All right? What you want to do if you had different normal division table is to go on to use your and let's say I had a question like this and I wanted to find probability of maybe uh x been between the 15 and 25 as I mentioned earlier what I would have needed to do would have been to pick up a normal region that has this feature and if I pick up that normal region that has this feature I would just go and use it to find this straight up but the thing is as I told you there is No random there's no normal table that has this feature. If you had to be using depending on using a normal division table you would have to create millions of normal division table because there are different scenarios that can happen.
We can have 40 and 2.5. You can have 88 and 2.8. You can have 20 and 4.8. There are too many you it's uncountable. All the possibilities of this. That's the reason why we don't try to even use a normal division table which don't even exist. What we do is to deal with this and make it conform to what we call a standard normal distriution. And the process of making it conform is what we call standardization where we try to make it conform to the feature of a normal a standal. What do we do in that case? uh a standalution is a normal decision of course that has this feature. The mean is zero. A stand is one. Of course, we have a table that has this. So instead of us trying to look for many tables, we just find a way to bring this to this form. Now if I now wanted to find a probability of X being between 15 and 25. Now I can now do one thing. Now I will confirm this to a normal to a standard normal division and then use a standal table to find it. So how do I confirm it to a standal division? I go on to subtract the mean and divide by sigma. Once I do this automatically I'm changing my x my x is changing from x to zed. And what I just need to do is to go on to subtract of course the same whatever I had before.
If this was the thing, I would be saying 15 - 40 / 2.5. I do the same thing for this 25 - 40 over 2.5. This is what I'll be solving. Whatever that gave me, this is what I will now be using on my standard normal distribution. Is that simple? It's that simple. That's the summary of everything we have done today. Is that simple? So in our subsequent classes now, we'll be talking about how to get the final solution. how to get the final solution after you have been able to standardize and that will be by using a standard normal distribution. Guys, this is where we're going to stop for today. I believe I've been able to give you a great introduction to normal distribution and um I really wish you guys the best in your understanding.
Make sure you take good care of yourself and enjoy the night. All right, guys.
Bye-bye.
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