We do exam preparation. If you have any question before we start with today's session, feel free to ask. Otherwise, I'm going to post the register in the chat. Please complete the attendance register. In the previous session, they were asked if they can put also their student number in the chat. I'm not sure how possible that is, but you can also share your student numbers in the chat, but they have posted their register on the chat as well. So you should be able to complete that. This week's session, we're going to deal with data handling, which is part of the statistics how to summarize the data into statistics calculations or into charts and all that. That's what we're going to be doing today. If you have any questions, feel free to ask before we start. Last time, I think we shared the three documents. I'm not sure if I was able to share them on the chat or where were they. I was not able to share them, and I don't think I'm able to share any attachment from... I can't share an attachment on the chat. Let me see if I can try to upload one of the question papers. The last time I couldn't... I still can't... There is no place for me to attach any document. Otherwise, we can try the other way around, which is where we are at. Let's see. There is no place for that. I can find an alternative to... I'm not sure if you are able to see the file section of our meeting. I've shared two documents there, but it might be that you are not able to access it. Just let me know if you are able to see them. Hi, Elizabeth. I can see the two files there on the chat. Thank you very much. They are there. Okay. Let me touch the last one so that then you have all of them. Actually, I'll put four. They are two more. I'll just put two more. So then you will have... Let me also open the last one. And then I will start sharing my screen. Just give me a second to share. So, yeah. As I said, welcome to today's session, where we're going to discuss data handling. So, in terms of your module, I'm not quite sure. But in terms of your module, you do a little bit of data handling, where you need to calculate the mean, the median, the mode, the coefficient of the quartiles and the quartile deviations. And sometimes they do ask you to describe what a frequency distribution is like, right? So, those are the type of questions that we're going to be looking at when we look at data handling. I just need to get there to those questions. So, these are some of those questions, right? Okay. So, in terms of data handling, so we do have a measure of locality, which are the measures that describe where your data is at. And those measures, we've got three of them. We have the mean, which is just the sum of all the values divided by how many there are. It gives you the average of the values. So, you add all the values together and you divide by how many there are. And then you have what we call the mode, which is the most frequent value, which means is the value that appears more than the other values, multiple times than the other values, not the highest value, but the one that repeats itself more than the rest of the other values. Then you have the median. With the median, before you can find the median, which the median is your middle value, your middle value, before you can find your middle value, it's always advisable to find the position first and by using n plus 1 divide by 2. So, it means you take the total number of values that you have, you add the 1 to that and you divide by 2. Once you have the position, then you can count towards where that position is and then that is your middle value. But before you also can do all this, you need to order your data. So, your data needs to be sorted from smallest to highest. So, it must be ordered in ascending order, meaning from the lowest number to the highest number, before you can find the position. And also, the other thing to take note of, if the position value that you get is the whole number, then the number that you find in that location, it will be your median, right? If it's a fractional or which is 0.5, if your answer is 0.5, therefore it means it is located between two values. It will be between two values. And when it's between two values, then you're going to take the average of the two values that the median position falls in. So, if we have one, two, three, four, five, six, and if the middle value sits between three and four, you're just going to say three plus four and you divide that by two and your middle value will be 3.5. That's how you will find the medians and the mole and the mean. So, you need to remember all that. So, let's answer the questions relating to that. Let's do more exercises. So, I'm just refreshing your memory because you would have gone through these things on your own before. So, questions like this. So, with this statement, we're going to answer six, seven, eight, and nine. The first one says, or the statement says, suppose a company has 10 employees, so therefore it means there are 10 of them. Our N is 10 when I was always referring to N. N is 10. 10 employees, one ending 160,000. The other one ending 120,000. Two of them ending 60,000 and one of them ending 40,000 and five of them ending 32,000. What is the mean of the company? So, before you can also even calculate the mean because remember the mean is the sum of all of them, right? So, you need to know that you have 532,000. So, I'm just going to write the 32, 32, 32, 32. Those are 32,000. I'm just keeping the thousands away because I'm going to run out of space. So, 32,005 of them. And it says, the next one is 40,000. There is only one 40,000 and there is two 60,000. So, you need to also make sure that you have all two of them, the 60,000. And it says 120, 120 and 160, 160. So, if I count all of them, they should be 10. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So, I've got 10. In order for us to calculate the mean, we know that the mean is calculated by the sum of all the values divided by how many there are. Therefore, you need to say 32, plus 32, plus 32, plus 32. How many 32s? Four. Plus 32, plus 40, plus 60, plus 120, plus 160. Let's see if I have one of them. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Divide by 10. And that should give you that. So, the answer that we will get there, we will have to multiply it by a thousand because we removed the thousand. Don't forget to do that. So, let's all calculate so that we can find out which one is our mean of this salaries.  Divide by 10. Multiply that by a thousand. And the answer is what do you get? So, I get 60 multiplied by a thousand. Give me 60,000. Do you also get 60,000? Yes. You? And that is? Yes. Yeah, that will be the mean. So, to calculate the median, we already have ordered our data from lowest value to highest value, which is easy. So, now let's calculate the position. So, n plus 1, divide by 2. n is 10 plus 1, divide by 2. What is n? It's 10 plus 1, divide by 2, which gives us 5.5. So, already I can see that it will fall between two values. Then you go and calculate our account until you get to 5. 1, 2, 3, 4, 5. 0.5 will be between 32 and 40. So, our median will be 32, plus 40, divide by 2. That will be our median salary. Remember, it will be multiplied by 1,000 at the end, right? Because I've taken away the 1,000. 32 plus 40 equals 72, divide by 2. It's 36 multiplied by 1,000. 32 plus 40 equals divide by 2 equals 36, multiplied by 1,000 equals to 36,000. So, it's 36 multiplied by 1,000 will be 36,000. Happy? Are we good? Okay. And the next question, they are asking, what is the mode? Remember, the mode is the number that appears more than the other numbers. So, let's go see which number appears more than the rest of the other number. The number that appears more than the other numbers is 32,000 because it appears five times, right? So, the answer is 32,000. And that is your mode. That's one thing. The next one that I didn't explain is measures of dispersion or measures of variation. Those are the measures that tells you how far apart your data points are from the mean. So, let's go back to our data. This is our mean. We found that our mean is 60. So, in order to know how far apart the data is from this 60, we calculate what we call a standard deviation, which tells us how dispersed the data is. It will tell us whether it's one standard deviation, two standard deviation, three standard deviation, and so on. So, how do we calculate the standard deviation? We use the formula, the square root of your variance, which is the sum of your X observation minus the mean squared divided by N minus one. So, what does that mean? It means we need to go back to each and every individual value that we have. Multiply that. The answer to that, multiply it by itself or squared. So, we take every individual minus the mean squared divided by how many they are. So, let's go and do it on this space here. 32, 40, 60, 120. I'm going to write down here. 32, 32, 32, 32, 32, 1234. 40, 60, 60, 120, and 1,<PHONE_NUMBER>0. So, we know that our standard deviation is the square root of the sum of your X observation minus the mean squared divided by N minus one. So, do it this way. Remember, in this instance, we need to also find what the mean was. So, the mean was 60. I'm going to use 60 instead of 60,000 so that then the numbers can fit on here.  So we need to go and solve the equation. So let's do that. I'm thinking about, you know, my calculator just takes itself. Okay, so 32, 32 minus 60 equals minus 28 squared. It is 784. I'm just gonna write it five times. 784 plus 784 plus 784 plus 784 plus 784, one, two, three, four, five.  Divide everything by 9. So now we need to go and add all of them. 784 plus 784 plus 784 plus until all of them, all five of them, it's equal to 3920 plus 400 plus 3,600 plus 10,000 equals 1,000 or 17,920 over 9. I'm just, I'm also going to show you shortcuts that you don't have to, unless those who don't know how to use your calculator, you can do it manually like I'm doing it and you can see how time consuming this is. So we need to take this, divide by 9 and we get the square root of 1, 9, 9, 1, 0.1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1. There are very lot of numbers of 1, repeated numbers of 1 and we need to take the square root of that answer and the square root is equals to 44.62. So let's go see if we get the same answer. Sorry, my bad. We remember also 46, we need to multiply this by 8,000. So the answer is 44.621, 8, and multiply that by a thousand. Always remember that, right? Multiply that, multiply the answer with a thousand and the answer is equals to 44,621.28. So let's see, 44.621. The answer is option number 2 and that's how you will answer the question, right? So let's now, actually let me stop sharing and go share my entire screen so that I can demo on the calculator as well the things that we just did. I've got two types of calculators. Let me know if you are able to see my screen.  Just give me a second because my laptop is dying. I must go get the laptop charger. But if you didn't have any calculator next to you, especially a cashier calculator or a sharp calculator, whether it's a scientific a financial math calculator, it's still going to work. Not any, but I don't have the same calculator as yours. So we should be... Okay, so I'll start first with the cashier calculator. I don't know how many of you are using a cashier. So we'll use the same information. Whether you're using a cashier or a sharp calculator, the basic thing that you already remember?  Is it on the screen? Because I can't see anything. Are you not seeing my screen right now? I only see the people that are joining the class. And now? Now I can see it, yes. Are you seeing my screen? What's most important is are you seeing my screen? Yes, sorry, yes. Now I can see it. You see my calculator, right? Yes, yes, I can. So the most important thing is always to read your question, decipher your question, make sure you understand what you are given, write it out, the things that are given to you so that you are able to highlight the facts within the question, right? So before you start using your calculator, like we did here, by expanding the statement into the actual values, even though I didn't write them in thousands, but at least I've got all of them here and I remember and I know that they are in thousand, that the answer every time I must always refer to them in thousands. So now let us answer the same question. The only things that we are able to answer using our calculator is the mean and the standard deviation. The more than the median, you still have to go and find them manually or whatever, whichever way you find them, we find them, like finding the position and then calculating or counting where it is and then allocating it and then calculating the average if it falls within the two values, the mode looking at the values and identifying the one that appears the most. The only thing that we're going to use for the calculation, the mean on your case, your or your shop, whichever one.  You should be able to identify those values. On button number four, there is your x bar, which is the mean. On button number five, there is the sx, which will be your standard deviation that you are calculating. And in order for us to reach those green button, we're going to use the alpha button. But that is that. So those who are having a scientific shop calculator or those who are having a financial meds calculator, don't panic, don't worry, they will use the same method. We will follow the same process with the same calculator that I will show you. Those who are using a case, that is the fraction button. It's going to be easy as well. Some of the case, they've got two buttons separated. They've got the ss dash sum and the s dash var on button number one and button number two. If you have that calculator, the method also is always almost the same, exactly the same. It's just that you are going to press different functions. So but if you have a calculator that looks like this, that has a fraction button, and you have on button number one, the stat button or the stat written in orange, we're going to reach that by using the shift button. But before you start with any of the calculator, you need to put it on stat mode. So you're going to press mode, and you're going to press the number that corresponds with s, t, a, t, which is stat, which is button number two. And you will get this menu. We're only going to be looking at number one, which is one minus var, press button number one, and you will get a table. Don't panic if your table, when you press it, it has x and it has a frequency on next to it. Do not worry about that. We're only going to look at the x value. In order for us to capture these values, here on your calculator, you can capture them the way you see them, like 32,000, 34,000, 40,000, and so on. You can use that or you can still continue and use the same method that I did, keeping them short. But always remember, once you get to the answer to multiply it by a thousand. So let's keep the whole numbers. It might take us forever because there are so many.  Equal, 32,000 equal, 32,000 equal, 32,000 equal. If you check your numbers and you see that you've captured something wrong, there is no problem. You can just go and say 32,000 equal. It will replace that number that you've captured wrong and then you just continue. And then we go to 40,000 equals and 60,000 60,000 equals 60,000 equal and 120,000 equal and 160,000 equal. And we have all 10 values captured. That is the most important thing. You need to make sure that you have 10 of them. If you have 11, where you see on row 11, it's black, it means there is nothing for that. It means if you have a value there, it means you've done something wrong. You just need to go and check your values. Once you have captured your values, then you can go press and go ahead and press the AC button. Your data is stored in the memory of your calculator. So you haven't lost anything. And on your calculator, it will show that you are in a state mode. And now we are ready to do state calculations. In order for us to do state calculation, we need to reach out to this function state by first pressing the shift button and then pressing button number one to call up the menu for the state function. You will see that you have so many other values or labels on here. If you press two, it will take you back to the data that we have stored. And you can go out again and press shift. You will always have to press shift instead to go back to the menu. When you look at number three, it says sum. The sum are these values here. The summations is this. So if I press three, you will see that it will give me sum of x squared. It means if I'm taking this value and squaring them and adding them all up, it will give me sum of x squared. If I'm looking for the sum on the mean function, the sum, which means adding all these values at the top, I can just press button number two and press equal and it will give me 600,000. And if I divide that by 10, because there were 10 of them, and I will get the answer of 60,000, which is what we got, right? That was the answer that we got. But that is not what I want to show you. So you can still find the summation, which is the sum via this method. What I want to show you is how we calculate the mean x bar. To calculate the mean, you need to go to this button called bar. So if I press, or before I go to the bar, you can also get the minimum and the maximum value by pressing button number five. If I press five, then they tell me what is the minimum value of this data set and what is the maximum value of this data set. That is if you don't know the minimum value and the maximum value. It can give you that. We're not looking for that. We're looking for the bar. So we're going to press button number four. And on button number four, we know n, because we know that it's 10. If you want, if you're not sure you want to validate, you can press one, and it will give you n is equals to 10. Number two, it will calculate the mean. We know that the mean is 60,000. So let's go and validate that. So we press two, and you press equal, and that is your mean. So there. In the exam, it will be a three step process, because in the exam, you are not going to be taking time explaining things to anybody. You are just going to just capture the data, press shift, set, and then four, and then the value of the mean, which is two, equal, and then that's what you will get as an answer. It will literally take you two minutes in the exam. So let's go and find the standard deviation. So go press four, and the standard deviation is on button number four. This one is for the population, but we deal with the sample. So we use four for Sx for the standard deviation. So we're going to press button number four, and we're going to press equal, and there is our answer. If I scroll to the standard deviation, we did find that the standard deviation is 44,621.86, and voila. So instead of calculating this whole big formula, you can just rely on using your calculator. It will give you the same answer. So let's do the same with our financial meds calculators. So those who have a financial meds calculator or the shop calculator, you can follow the same step.  I think it's an ENT. That's the button that we're going to be using to capture the data. So like any other calculator, you need to first put it into mode function. So you're going to press mode, and on your calculator, you're going to press the number that corresponds with step, which is button number one, and you're going to use SD. It will reflect SD on your sign, and then you press zero, and it will say you are in state zero. Your calculator is ready, and also at the top, it will have a state function. Now you are ready to get the data. So on your calculator, you will not get a table. So in order for you to keep track of what you are doing, you need to be very, very careful because every time you enter the data, it will tell you data set one, data set two, data set three, until data set ten. If you make any mistake, you have to play your calculator and start again, and I'm going to show you just now. So you need to be very careful when you enter your data. So how do we enter the data here? Because our calculator is ready. We go and say 32,000.  On the case show, it's an M plus. So I'll press M plus, or you will press E and T, and it will say data set one. So I need to continue, 32,000, and M plus. So you will always say enter 32,000. I must not forget M plus. So this is data set three, 32,000. Data set four, 32,000. Data set five. So now I know that I've captured all five of them. The next one was 40 and 260. 40,000. M plus. And then 60,000. M plus. 60,000. M plus. And 120,000. M plus. And 120. 160. M plus. And it says data set 10. Now I know that I've captured all of them.  If you make any mistakes, you enter the wrong number. You just press second function and the mode button, it will clear your calculator, and then you can start capturing the data again. Going back to the table, it's easy to clear your calculator data and capture them again anyway. But I will show you also later on how do you clear your data, because when you need to capture new information, you need to clear your calculator.  It's just two buttons to press. So the first one is to find the mean. We press the green button, which is the alpha button, and then you press button number four, which gives you the mean and always press the equal sign. As you can see, the entire is right there. 60,000. Doing the standard deviation as well is button number five. Alpha, button number five equals, and there is your standard deviation. 44621.87. And that's how you will answer questions relating to the mean, the median, the mode, the standard deviation, and so on. Easy, right? So let's say we are done with calculating all this, and we want to clear our calculator, clearing your calculator easy. Second function, CA will clear your calculator, and you will see if I apply the values, it will give me an error because there are no values on my calculator. For the case, you're clearing your calculator. No easy way of clearing your calculator. You just need to go and say mode, go back to your data. Sorry. You can clear it by pressing the shift, actually. Let's go back there. You can say shift, clear your calculator, and then say clear from memory. And yes, it's for cancel for clearing, which is equal and AC for any key button. And if I want to double check that I've cleared the calculator, let's see, because I needed to clear the memory. So if I go back to data, which is number two, it should have cleared the calculator, it didn't clear. So let's see if I come here and I say clear all three. What happens? So sometimes with your case, you just need to go back to the normal, but it will not work anymore because it went back to met. And you just need to go back to mode two. And you might find that the data is still there. Oh, yes. So that's all. If you want to use your calculator to clear. And that's it. In terms of the data handling, there are no more other questions on data handling on this question paper. So let's go and look at another question paper. There is another question. I'm going to give you time because you need time because these are huge numbers. There are about 25 numbers on here. You need to order this number from lowest to highest. Actually, no, you don't because we're dealing with no, let's not. Let's let's deal with the media, the mode, the way we see it, because they are not asking you to calculate the mean or the median and so on, especially the median. So look at this. Find out which value here is the mode of this data set. Therefore, it means which value from here appears more than the other values. I'm going to give you two minutes or five, three minutes. I'll be back. I need to get order. Hi, Yalla. Is he your man? What for me? Can you help me to show up? I only have a cashier. I don't really know how to work the shop. So I can't help you, unfortunately. Okay. Thank you. Do you have an answer? 140. It's 140. 140 appears more than the rest of the other numbers because 120 appears only once, 161. There is not even 161. 183. There's only one 183. So the answer is 14. Other things that they might ask you on is the way of sampling the data. How do we sample the data? So there are four different ways you can sample the data. We use what we call a simple random sampling, which means every individual in that population had equal opportunity of being included in the sample. So that is simple random sampling. So it's the same way as if I have names of people and I put them in the head. Every name that goes into that head has an equal chance of being picked from that head because it's a blind thing.  If you pick lazy, you could have picked Adam or Steve or Mary or John. Anyone who was in that group had an equal chance of being selected. So that is what we call a simple random sampling. Then you have a systematic random sampling. With systematic random sampling, it means you need to select the KM number that you want to start selecting from and after you need to start selecting that KM place number. So normally the systematic random sampling is done when you do surveys with the houses. So you can say I'm going to select the fifth house on the street. Every fifth house on that street will be included in my survey. Then you first choose the house and then you start counting.  That house is included. Then you start again one, two, three, four, five. That house is included like that. And that is what we call systematic random sampling. Then we also have what we call a stratified random sampling. So a stratified random sampling is when you put things into subgroups, into groups that are similar to one another. And from those groups, then you select your sample. You do your simple random sampling from those subgroups, which we call stratas. Yeah, we can call them subgroups or we call them stratas. And that is the other way of doing a random sampling. The other way of doing the random sampling is proportionality random sampling, where you select the sample based on the proportion of those people belonging to that sample. And the other one, which is the last one, it is what we call a cluster sampling. With clusters, you put them into different clusters. It's like the stratified random sampling. But yeah, you create groups with, do not even have to have the same commonality or same population groups or so. But it's just, you select, you create groups of people and then from those groups of people, you then go and do a random sample. And those are the types of random sampling or the types of sampling that you can get. So let's see which one of this statement refers to any of the types of random sampling. So the staff of a college consists of a professor, senior lecturer, lecturer, junior and administrative staff. A statistician wishes to determine the opinion of staff in the college on their salaries. He has a list of all staff members arranged alphabetically according to their names or their say names. He randomly selects a professor at the starting point of the list and then subsequently selects every fifth member of the staff on the list. The type of sampling will be, if you are listening to me, you should know the answer to that. Is it number one, number two, number three, number four, number five? Number three. It will be a systematic because of the cadence number that they would have selected. Anyway, it cannot be a quantitative sampling. I've never mentioned anything to that effect and the statement doesn't even count because there is an answer to the question, right? Okay, let's then move to question 25. The owner of a small company has 15 employees, three employees and 15,000 per month, seven employees and 10,000 per month and five employees and 7,000 per month. The owner's monthly salary is 25,000. The mean monthly salary of all people in the company is to two decimals. That is your exercise, not mine. May I please ask you, when you're done at the end, will you please show me how to work out with a sharp calculator because I didn't quite catch all of it when you were teaching so I apologize. We will do it now when we do this exercise because it also touches on that. Because you asked, let's do this together as well. But before we answer this question, we need to make sure that we have all the values. Right? So we know that there are three 15,000. So you should have 15,000, 15,000. There are seven 10,000. One, two, three, four, five, six, seven. And there are five 7,000. One, two, three, four, five. And the owner ends 25,000. I'm just going to put 25 here at the end. So there should be 15 employees plus the owner. They should be 16 because they say all people in the company also include the owner. Right? So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 18, 14, 15, 16, including the owner because they didn't say only the employees, they said all people in the company. So you also need to take that into consideration. So those who want to calculate manually, you can use x bar is equals to the mean, the sum of your observation, meaning adding all of them, dividing by how many there are. Those who are using calculators, then we can use the calculator. I will need the values because of these ones, there are so many of them. So let me just take it up a little bit and put the calculator right here. I'll have to move the calculator around. But there are 315, 710s and 570s. So let's capture the data. First, you need to put your calculator to mode, state mode. So you go mode. So I hope you are following right? Mode. Let's do this. Just going to move this to the site and move my calculator to the site.  Okay, so we know that we have, I shouldn't put it too big. Okay, so state mode, mode 1 and 0 for SD. I hope you are following. So we need to capture the data. When I start clicking on the calculator, it then changes. There's nothing I can do. I'll start with the values at the site, which are 7, 7. There are 570s. So you say, because it's 7,000. So you also remember it's thousands. So 7,000. And you need to press the ENT or the N plus. And 7,000. N plus 7,000. I've got more values. I can use the delete. 7,000. N plus 4. I need one more 7,000. Now, when it comes to the 10,000, it's going to be a little bit difficult because there are so many of them. I'll capture the 15,000. First, there are three of them, which makes it easy for me to count. 15,000, N plus, 15,000, N plus, and 15,000, N plus. And I can also capture the last one, which is 25,000. 25,000, N plus. Now, because I know that there are, how many are there, so that I know how many to keep in this? 15. So when I get to 14, I will know that I need to just capture one last 10,000. So now, I must just capture all the 10,000.  And they said 15, so I must go to 16, 10,000, N plus, and the last one, 10,000, N plus, because now there will be 16 people in the company because there are 15 employees plus the owner, which makes it 16. So I've got all the values captured. All I just need to do is calculate the mean. So let's calculate the mean. So the mean should be any of these values that we see here.  And the mean is option number four. 10,937.5, which is option number four. And you can do the same with your SHAP calculator. Let's see on the SHAP. I'm just going to put it on the side. And on SHAP, 25,000, and equal, 15,000, not 150,000, equal, 15,000, equals, 15,000, equals, and 10,000. So with the case, it's easy to see how many I have captured because in this one, I can always go and count how many they are. And if I captured more than I'm supposed to, it should tell me. One, two, three, four, five, six, seven. So I need to have seven of them. So 15, one, two, three, four, five, six, and one more. Seven. And then five, 7,000, 7,000, 7,000, 7,000, 7,000, and 7,000. And the last 7,000. And I can double check because one, two, three, four, five, seven, one, two, three, four, five, six, seven, thousand, three, 15, and 25. So I've captured all of them. And then I go to AC. And then we go SHIFT, one, and four, and two, and equal. And the answer is the same. 10,000, 9,000, 37, and 50 cents. Good, good, good, goodest. Good, good, good, good, good, goodest. So those are the questions relating to data, I think. There are no more other questions. Let's go to the next paper and see if we can find more questions. The other things that relate to data handling are what we call the counting rules. Because also with counting rules, you find a number of ways you do certain things. And in terms of data handling, these are part of what we call probabilities. But because you guys don't deal with probabilities, but there are counting rules that you also need to be aware of in your module. Counting rules are a way of telling us how many ways we can do certain things. So there is the multiplication way, which is your n times m way, which we call it multiplication way, where you have two restaurants and three parks and you want to go and visit any of them. But you don't know how many different ways you can go and visit those. So you have two options here that in your area, there are three restaurants and two parks and you want to go and enjoy any of those. So in order for you to know how many number of ways you can go to those two parks and those three restaurants, then we apply what we call the multiplication rule, which tells you the n times m ways. So if I have two parks multiplied by three restaurants, so therefore there are six ways that I can go and visit any of them. Right, that is the multiplication. Then we have a factorial way, which tells me if I have a race and there are seven positions in which you can come into this race and win a prize, then I can use the factorial way of getting to any of those positions, because if I have seven positions, then I can either be in seven position or six position or fifth position, did I forget to put the seven factorial or six position or fifth position or fourth position or three position or second position or position number one. That is the n factorial. On your calculator, you do have that. So let's, if when we continue with this, I can also show you a way to find all this information. So on the case your calculator, the factorial is in orange, so you're going to press the shift and then press the button that corresponds to that. So here we have seven factorial, which is seven shift and that will give you your seventh factorial and then press the equal sign. It's five thousand and four. Right, those are the number of ways, five thousand and 40. Or if you don't want to use your calculator in this manner, you can just go and say seven times six times five times four times three times two times one, even though it's one, it's the same. It will also give you the same answer. Those who are using the sharp calculator, you also have the same. So we need to go back to normal calculator on this one as well.  The same on the case, if you want to take your calculator back to normal, you just say mode and then you press one for comp and it will take you back to normal calculator. So in order for us to calculate the factorial on this one, so you say seven, and your n factorial is also written in orange is on button number four, so you're going to press this second function and press the button number four for n factorial and you can see there and press equal and that will give you your factorial. And that is factorial. The next one, it is what we call permutation and combination. So permutation and combination, before I explain what those two are, on your calculators, they, on the case, these are permutation and pr and this is combination where it says n, c, r. So permutation, combination, there is a slight difference between the two. In permutation, order matters the most. So how things are done matters. The order in which things are done matters in permutation. In combination, order is no matter. Preference has no matter. So where is my pen? Just give me a second. Oh, yeah, something. So n, c, r is your permutation and c, r is your combination here. Order or rank of how things are done, very important. If they specify, if you need to select a three committee member, they tell you that you need to select a chairperson, a president, a chairperson, a secretary and a treasurer. They told you the order in which you need to do the election. Combination, order, no issues. If they tell you you need to select a three-member committee, regardless of who sits in that committee, three-member committee, you can see that is the same question. The other one, they gave you the preference in terms of who you need to select. The other one, no preference in terms of who needs to be selected. So you need to be able to define or clearly read the question and identify which one is permutation, talks to permutation and which one talks to combination. So those are the two. So three ways. So if I have a ten committee member, so my n is ten and I need to select three positions.  The same way on this side, n is ten, x is three. So x is the same as r. r and x are the same thing. So how do we get that? So on your calculators, so we can start with the casual one. Let's calculate permutation. So you first need to put in the ten, the bigger number first. Anyway, the bigger number first. And because it's written in orange, we press second function and then we press the multiplication.  What did it give us? Seven 20. Now let's do the combination. Combination is the same. Bigger number first, second function, division and three and equal and that is one 20, which is ten. Three gives 120. As you can see, they both asking you the same thing of how many number of ways you can do this. And you will get the same different or you will get the different ways that you can do. If order was given, then it will give you 720 ways and if no order was given, it gives you 120 ways. So those who are using the casual, also the same on casual, also on the shop, sorry, those who are using a shop is written in orange. So the NCR is on button number five, NPR is on button number six. So let's do the NPR first, ten, second function, six, three, equal. You can see that it's 720 and NCR, it will also give us the same ten, second function, five, three, equal and it's 120. So okay, now let's answer this question of ours. So on this one, it might be a little bit trickier than the things I just explained. An investor has decided to purchase the shares in the stock market of three companies. One engaged in aerospace activities, one involved in energy development and one involved in electronics. After some research, the account executive of the brokerage firm recommends that the investor consider stock from five aerospace companies, three energy companies and four electronic companies. In how many number of ways can the investor select the group of three companies from their account executive? How many number of ways can they select? Because here we're talking about from the account manager, five, three and four. So the number of ways can only be five times three times four because there are four ways. How many ways can you select? You can select five times three times four times three. What is the answer? Five times three times four equals 60 ways. Those are the number of ways that you can select three companies. That is what we call the multiplication rule. Suppose the investor decides to purchase stocks in two IRO companies and two energy development companies and two electronics companies. In how many number of ways can the investor select the group of six companies? Now remember, there's a difference between the first statement where they wanted to know how many groups of three companies because that was a general statement because they are five, three and four companies to select from. Now he wants to select from this group of six companies. There are two, two, two, six, and he needs how many number of ways can the investor select the two groups of six companies from the investment from the recommended list of five, three and four. Now you need to take into consideration the following. How many number of ways can they select two companies from those five IRO space? Two companies from the energy, two companies from the IRO space, and that is is that a combination or is that a permutation? Because they telling you how they want the selections to be done, two companies from this, two companies from that, two companies from that. If they didn't mention that, two energy companies from development, two energy, they could have said two companies from any of the two companies from each of this and not tell you how many from each of these companies, then you will use what we call a combination because they are specific in terms of two companies from they telling you the order in which they want those companies to be or their preference or the rank in terms of those. So we're going to use what we call a combination or permutation, sorry permutation. So we need to calculate five, permutation is P, five P, two plus there are three and there are four. Let's see how many ways we can. So you say five shift times two equals 20. Probably even you don't need the plus it's a multiplication because it might be a multiplication. Let me first double check three shift two and six and the last one is four is 12 and or is it combination? But they told us how many they are because they need to double check if it's combination. It will be five, combination two and is two and three, combination two, three and four, combination two, six, six times two, it's combination. Okay. It's going to be combination not permutation and we're going to multiply. So it is combination. Then we can assume that because in the question in here they didn't specify. In the question it says how many number of ways can this investor select the group of six companies because they didn't say select energy companies. So they didn't give out. So we should have read the statement in that way not the other two parts of the question. So the important starts from there because they just say six companies. So we need to say it is five, combination two times three, combination two times four, combination two and then we can then calculate the values five, combination two equals 10 multiplied by three, combination two equals three and combination two equals six. You multiply that 10 times three times six equals 180 ways which is option two because if it was permutation it would have given us 1,140. So that's how you will answer some of the questions. So that is in terms of counting rules. Okay. So let's go and find the data handling questions. I don't think in your module this time you are doing the lapis and there is your question but that is so huge but they only want you to calculate the loaf of bread for Thursday, the mean of the loaf of bread for Thursday. That is your exercise and I'm going to give you five minutes to do that. So you only use the Thursday where the space it's a decimal. Are you winning? Are we winning? Remember you don't have to wait for me to give you the answers. You need to do the exercise yourself so that you can see where you're going wrong. I can go to the options to see if the one that you have is going to have. Okay. Just give me a second to complete my list 826. So let's see if you got the right values so we've got the same values. So the mean I've captured the data so already can just go and find the mean which it's quick, right? State of calculating things manually 803.44 which is option two. Also on the shop calculator I also did enter the data so it's just going to check 803.44. So both are right. So the next question is find the mean for Tuesday, the median. So that is the median, median for Tuesday. So sorry. So we need to go to Tuesday now. Remember you need to order your data from lowest to highest. I'm going to save you time as well. I'm going to calculate for you the position. So the position while you ascertain your data it's n plus 1 divided by 2. There are 10 of them so it's 10 plus 1 divided by 2 which then gives us 11 divided by 2 which is 50 or 5.5. We did do something similar to that. So it so you just need to order the data from lowest to highest. Let's see I can very difficult to do that with all this. So 784 is the lowest followed by no is not. That is the lowest. That's number one, that's number two, that's three, three, that's four, five, six, seven, eight, and ten. So we just need 5.5. It will be between five and six. So five and six are those three numbers so you can just add both of them and divide by how many they are by two. Not how many they are just divide them by two. So it will be seven nine five point plus seven nine six point three equals divide by two. Do you get the same answer? So let's see. Oh, maybe you are still busy writing the numbers down. Let me give you type. Maybe I also didn't look at the values correctly. My number might be wrong. So double check that as well. I'll just those who it should be option number four in that instance. Are we good? Yes. So the answer would be option number four, which is 796.10, which is 0.10. So the next one is calculate the standard deviation for the masses of low for Sunday because you are using your calculator. They give you the mean of Sunday to be seven nine because they assume that you need to go and calculate this manual. In the exam, you can use your calculator. You can use shortcuts in your calculator. You're writing a multiple choice question. So save as much time as possible by relying and using your calculator. So it means you need to practice and practice and practice on how to do these things on your calculator. So if we go to standard deviation for standard deviation for Sunday, so we need to go there. Sunday is this column. You need to go and capture the data. So I'm going to also carry on and capture the data. So I need to clear the calculator from any stored values and then start capturing again. Seven nine nine because it should give me data set one if I do this. So I'm capturing my data again. I don't have to put point zero because it's not going to even capture it on the calculator. It doesn't make any difference. I have all the data captured. I'll wait for you. I'll also go ahead and capture on the casual calculator. So because on the casual, it's very difficult. I'm just going to go back to the original. I'll wait for you. I can also go show those who don't know the answers. Let me go peep on the answers. They are the answers. 10.97, 8.61, 12.48. Those who want to calculate this manually remember the formula is s equals to the square root of the sum of your x minus the mean. So they gave you the mean. You write by n minus 1. I got 8.61. 8.61. So let's see if we also get 8.61. One, shift, step one, four, four, and equal, 8.61. If I get 8.61 on the other one as well. On and off. It's a written in green, alpha, five, 8.61. No more. The last question paper that we have probably will take us to the end of the session as well. I just want to go back to this paper just to double check if we didn't miss any, including also the counting rules because I don't think we also concentrated on that. So let's look at this one in terms of the counting rules. Six men, eight women, have volunteered to serve on the committee. How many different committees can be formed with three men and three women? Is that a combination or a permutation? So we just need to read the question.  No preference, right? Same as what we did previously. So you will, because there is no preference here, they're not saying whether they want the secretary or treasury or what not or how the committee should be formed. They just need to know how many different committees can be formed. So it means we need to use combination and multiplication rule because there are two categories here. So we're going to do a combination of NCR times NCR for men. There are six of them and they only need three men in the committee. For women, there are eight of them and they only need three women in the committee. So what is six combination three? It's 20. That will be 20 and eight combination three, 20 times 56. And that's how you will answer the questions. Question four, Adila is offering any four of six special options at the same price. On specially equipped car, how many different options of specially equipped cars do you have? Or will you have? Is this a combination or a permutation? Because they are not telling you what they're not giving you an order. So also this will be a combination question. So you can just use NCR. What is the answer? If you use NCR, any six and R will be four. And the answer is 15. Let's see if they don't have any sampling questions on this one. Other questions we can deal with them at the latest stage when we have enough time. Oh, yeah, we'll see how it goes because our session ends at two o'clock. Our session ends at what time? At two. So we almost, and they didn't even check the time. We have eight minutes. Okay. Let's look at this other one. Take some paper and see if there are any questions that we left out that deals with the topic of today. Nothing, nada. So the only question paper we didn't go through, it's 2016. So you are more than welcome to go through it and ask any question if you need to be because we only have eight minutes. Oh, I didn't pay attention to that. Let's see. Do we have any other question? Financial meds? Financial meds? Oh, they, so on May June, the questions are right at the bottom. So when they ask a question like the average, you must know that that is the mean. They're asking you to find the mean of this question by adding one of them and dividing by how many there are. Or you can use your calculator, just capture all the values and then calculate the mean. And also, they're asking you to find the mode. Within this six minutes, you should be able to answer all those two questions. If I can answer them, you can answer them. So let's go and answer the two questions. The 11, one, two, three, four, five, six, seven, eight, nine, 10, 11. They are 11. Charles, so are you winning? Yes. The other thing is when you capture the data, you need to pay attention to the need, to the numbers that you are capturing as well, because a once let mistake might give you the wrong answer. And you might find that that wrong answer is one of the options. And you might get too excited that you have the answer. You don't have. So what is the mean of the data set? The mean is alpha, mean, which is x bar equals. The mean is option number one. That is the mean. Let's see with the case you'll calculate that. What is the mean? The mean is shift state one, four, and two, and equal. And the mean is option one. That is the mean. The next thing is, what is the mode of this question? Which number appears more than the other numbers? So if we would have ordered this data, it would have been easier to see which one is appearing more than the others. But in this instance, I can see 1,580 or 80. What other number? It's only that number, right? Yeah, that's our 1111. So the answer would be option four. So, but do you remember it's 29 and 30 also? So with 29 easy, once you have captured your data, it makes it easy to find the rest four. And our standard deviation is four equals. And our answer is option number two. 770.096. And 96 is the same as 10. So it's option number two. Let's see on this one. Alpha five equals. And that should be the same option number two. Okay. Then the last one, which we didn't cover. Guys, we haven't covered something like quarter. And we left with one minute. I will show, ask you to just give me five minutes of your time so that we have covered almost everything because I don't know whether in the exam you will find questions like this or not. Since I'm not the lecturer and I don't cite your exam paper. So I just want to make sure that you have all the information and you know how to answer them. So, quarters. So, quarters are a way of dividing your data into four parts of 25% each. So it means when you are at here, you say you are 25% below, which is quarter one. When you are at this point, we say you are at quarter two. You are 50% below or above. And when you are at this point, we say you are 75% up. So you are quarter three. And at the end, all these values at the end, we call them the maximum and the minimum. And all these numbers, they make up a five number summary. In a way, what I've just done for you there is to tell you that in terms of a quarter, you will have a box whisker plot, which has your maximum, your minimum values within the middle. There is quarter two and quarter three. Sorry, this one is quarter one and quarter three, which is almost everything that I just explained there, right? So that creates what we call a five number summary. If I take the difference between quarter one value and quarter three value, there I create what we call an interquartile range, which gives me two three value minus two one value. The minute I take my interquartile range and I divide that and we get quarter deviation. When I take my interquartile range and I divide that by two, then I get the quarter deviation, which then gives me my QD, which is the value that we want to calculate today. So then how do we then find quarter one and quarter two based on the data that we have? Let's go back to our data, which is this data here. We need to order this data from lowest to highest in order for us to be able to identify those values. So for the quarters, sorry, that's the other thing. The data needs to be sorted. Let the same way as we deal with the median, the data needs to be sorted. So in order for us to calculate quarter three, we use three times n plus one divided by four to calculate quarter one. We use n plus one divide by four, or you can use the percentile, but I prefer to use this position values. So let's order the data. It's one oh eight zero followed by one five, one five eight four, one five four zero, one five eight zero, one five eight zero. So I can just cancel out the values that I've gone through so that then I don't get confused. And we have one thousand eight, one thousand eight hundred. Then the next one is two thousand two thousand and two thousand five hundred and then two thousand nine hundred thirty, three thousand, three thousand one hundred and twenty one, and twenty, sorry, and three thousand two hundred and eighty. One, two, three, four, five, six, seven, eight, nine, ten, eleven. So we have all of them. So we know that they are eleven. So it will be three times eleven plus one divided by four, which is I'm going to use the fraction, or I need to go back to my meds, comp one, so I can use the fraction, but three times eleven plus one goes bracket, go down, divide by four equals nine. That is the ninth position. Don't get it wrong to say this is the quartal value is the position. We find in the position first, that is located on position number nine. And here we have eleven plus three or plus one divided by four, which is twelve divided by four, which is equals to three, right? One, two, three, that is our QD will be one, one, two, three, sorry, I need to get first the position nine, one, two, three, four, five, six, seven, eight, nine. It's three thousand, three thousand minus position three, that is the third position, right? One, two, three, it's one thousand five, one thousand five hundred and eighty. We need to divide this by two initially. I'm going to explain why in the previous days your lecturers didn't divide the quartal deviation by two in the options. They just give it as the ital-quartal range value, not the deviation. So it's three thousand minus one thousand five hundred and eighty, which is one thousand one thousand four hundred and twenty. I just want to double check because sometimes they just use that value as they see it there, but you need to divide that value by two and the answer is seven hundred and ten. So always remember to do that. Divide this value by two and that will give you seven hundred and ten. If you don't divide it by two, you will still find the answer on there, but it will not be the correct answer, right? I think it must. So it will not be the correct answer, but the answer is option three and that is interquartal range. I hope you understand that in that five minutes that I first in onto you guys, but yeah, if you find more exercises, especially the recent question papers because I don't have like 20, 18, 19, 20, 21, those four years, if you have them and you want to share them with us so that we can use them, share them with us so that then we can use them in the activities, but those are the four past exam papers that I have and I hope they give you all the information you require in order for you to sit and write your exam on the first. So I just want to also share our next topic will be differentiation, which will be next week, Saturday, see you same time, same place. If there are any questions or comments, speak now or forever, hold your peace. Uh, the questions, comments. Thank you, Elizabeth and nothing from our side. Good. Remember always practice, practice, practice, play with your calculator in order for you to know what is happening and how to use your calculator effectively when you're writing your exams. Like I said, it will save you a whole lot of time if you use your calculator to answer any of the data handling questions, especially the mean the standard deviation. As you can see that the calculations are very long if you're doing it manually, but when you have a scientific calculator, it becomes a two minutes job. If there are no questions, comments, then thank you for coming. Enjoy the rest of your Saturday and the weekend. I will see you next week. Bye. Thank you. Bye. Thank you. Bye.


I'm here at ITU Telecom World 2012 in Dubai and I'm very pleased to be joined by Lina Ravindra Green who is CEO and Chief Consultant of Get It. Lina, thank you very much for being with us today. Thank you very much. Debates at ITU Telecom World 2012 focus on the current radical transformation of the ICT sector. What for you are the main challenges from this transformation? Well, first of all, I've been actually very fortunate to have taken scene telecom through many years. My first telecom was actually in 1987 and over the years have seen different technologies being the thing that was going to transform the home telecom industry from ATM to ISDN to the Internet. And as you know, everyone's been talking about convergence for a long time. But in some ways, what the radical transformation we're finding today is, right, that convergence we've been talking about for a long time about the telecom industry data and media is actually happening. There's a whole mesh of different players in this industry. Who's going to be making money? Where's the money flowing to? Is one of the key challenges that telecom operators are facing. The other thing is that we've had so many years of telecom deregulation and privatization that in many markets we just have so many players. So some of the key challenges people are talking about, is there enough spectrum to go around? Is there too much competition that margins are very low? How are we going to survive? Who's going to pay for the millions of people who are downloading YouTube videos, for example? Someone threw out the statistics about how in 2013 one day of traffic is what we had in one year of traffic in 2001. So the network demands are really, really huge. And this is what we are facing today, is this transformation that is finally kind of happening. And that challenge actually is very exciting because I've been involved in seeing this talk about this convergence and we've seen this convergence finally happening today. Okay, we've talked about the challenges. What about the opportunities are arising from this? So just like the challenges brought together many different players, that's actually where the key opportunities are going to be. As telecom operators, if they break away from thinking of themselves just as the pipes, the provider of the pipes and the access to the end user, but rather build into this transformation of this industry where basically voice is noise almost. It's all about data. It's all about, and also the beauty of the end-to-end paradigm used to be traditionally called the Internet. I found it interesting, one of the speakers at the opening ceremony talked about the Internet world and the telecom companies and how they are not working together. But largely what we're talking about is IP-based networks. Really, even media companies, a mobile company, all are IP-based networks. You've enabled this end-to-end paradigm and who produces content on YouTube? It's not YouTube. It's not the provider anymore. It's completely changed this whole paradigm of where the money is, where the content creator is. That's where the opportunities are coming from. What I see right now is telecom operators, three basic things they deal with. How do I keep capex stagnant? Of course, as you're building broadband, that's going to increase rapidly. That's an amazing opportunity here because now we have an area of huge innovations in, say, open-source routing technologies, green telecoms technologies that are really going to make your capex look very different. OPEX, how can I reduce my OPEX? Again, there are a lot of innovative ways people are looking at green ICT being energy efficiency and other ways in which energy OPEX. But revenue is where the excitement is. How can I create stickiness and loyalty in my customers? And how can I create new services? This area is very interesting as well because the opportunities may not necessarily be from me creating these services, but creating a platform of innovation that enables other entrepreneurs and young people and the power of the potential that unleashes that, that creates and adds value to my network. So it's a completely different paradigm of looking at where value is. I'd also like to ask you, what are your expectations of ITU Telecom World 2012? And what key message would you like to deliver here to our uniquely influential audience? Well, what I expect of any telecom that I come to, I think telecom is a very unique place in which the ITU has the ability to bring forward heads of governments and heads of telecom operators and vendors and create this conversation and dialogue in which people can exchange, how can we unleash that potential in the future? My passion has always been, I've been in telecoms for about 25 years and my passion has always been the bridging the digital divide. And as you can see, we're moving to broadband. That divide is actually redefined itself. We talked about bringing a telephone to walking distance of somebody. And now we're talking about bringing broadband internet to everybody. So the conversation has changed, but that passion and that mission hasn't changed. And that for me is where I'm involved in the two areas that I'd like to see governments and people be involved in. My last name being green, I'm very passionate about green ICT. I would like to see governments, for example, not not necessarily heavy handed regulation, but governments as procurers and governments off ICT actually lead the way in enabling green ICT to have that volume to make it, you know, basically sustainable ICT that we all are using. Using ICT as a leverage to help combat climate change and bring a more sustainable economy. So that's one area I'd like to see. The other area that I'm involved here at ITU telecom, which I'm very passionate about is the young innovators competition. I've been invited to be a panel on the judges, and I judge the second round as well, and I'm mentoring some of them. And that's actually one of the beauties and beauty spots I would say of ITU telecom 12. You have this experience of how innovators from different countries from Brazil, Kenya. That is what I want us to recognize. Because in telecom 99 Nelson Mandela stood up and he said, we need to have a South-South cooperation. And guess what? Necessity is the mother of invention. And especially in areas of bridging the digital divide, you can see amazing innovations coming out of scarcity of energy, scarcity of bandwidth, and the applications that people are coming up. That's what we need to help fund. That's how we need to innovate and allow them to go. And perhaps light handed regulation because at times, for example, in the last mile, they may be great partners for the telco. The telco may not have made business sense to go there. Then let the young innovators do it and partner with them. And together, we can bridge the digital divide in that way. That's what I would like to see. I certainly hope so too. Lionel Ruvindra Green, thank you very much for being with us today. Thank you very much.


On this channel, I've made videos explaining some intuition behind logistic regression, digging into the map. Now we covered topics like the sigmoid function, linear decision boundaries, and so much more. But I want to get a more visual take on this. And when we say we are training a logistic regression model, what exactly is going on? That's what we're going to look into today, so sit tight. Logistic regression, as you may know, is typically used for classification in machine learning. Did this passenger survive on the Titanic, or not? Will this person default on their credit card, or not? In each of these cases, we have a binary outcome. And typically in machine learning, we model such outputs as a probability. So instead of having our model predict 1, the person defaults, or 0, the person doesn't default, we instead have it spit out, something like, this person has a 60% chance of defaulting. Because of this, such classification models have a fixed range of outputs to spit out, between 0% and 100%. We basically need a function that takes in any real number, from negative numbers to positive numbers, and squishes it into a probability. Logistic regression specifically uses a function called the sigmoid to do this squishification. Fair warning, the implications of using sigmoid can get fairly technical, and I'd be happy to make a separate video for it. However, for this video, let's keep it simple and stick with the intuition that we use sigmoid because we want to squish any value into a probability ranging from 0 to 1. So the sigmoid function, what does that look like? Let's start by first drawing a number line with two coordinate axes. The x-axis will represent independent data points, and y-axis will represent the values of our sigmoid function. A simple sigmoid function would look something like this s-curve. And no matter what value x may take, the sigmoid always has a value between 0 and 1. This fact remains true for any such sigmoid curve. The y-coordinate always ranges from 0 to 1. So this verifies that we can use a sigmoid function to model probability. For example, this point could represent the conditional probability that y belongs to a positive class given a value of x. I'd be remiss if I didn't give a shout out to Grant Sanderson, aka 3Blue1Brown. The animations in this video were created using his math animation engine, Manim, and I'll leave a link to it down below, so check it out. We now have a function that spits out a probability. Great. But in the end, we want a binary output. That is a yes, a person will default, or no, the person won't default. Now, how do we do this from probability values? We assign this using a threshold. Let's take it as a standard 50% here. That is, if the sigmoid function for a given value of x is greater than 0.5, it should classify the point as 1, and if it is less than 0.5, it should classify the point as 0. But note, how you assign the threshold depends on the application. In cases where we want less false positives, a higher threshold is good. In other cases where we want a higher recall, a slightly lower threshold is good. It's up to the developer to choose this threshold. Now question, how do decision boundaries fit in? For those not familiar with a decision boundary, it is basically a boundary that the model uses to make decisions. Simple enough, every point that falls on one side of the boundary is categorized as 1, and points that lie on the other side of the boundary are categorized as 0. I'm speaking for the binary classification case, but this definition can be expanded to multiple classes in multiple dimensions. What you see here is a two-dimensional plane with axis variables x1 and x2. In a two-dimensional plane, we have a two-dimensional decision boundary in the form of a line or a polyline. If we look at the one-dimensional case with a single axis x, we have a point decision boundary. Every point on one side is classified as 0, and every point on the other side of this boundary is classified as 1. Now for multi-class classification, we could use multiple points as the decision boundary. Now that we've visualized decision boundaries, how exactly are they created? Restated in another way, given a set of these data points, how does logistic regression know where to put the decision boundaries that work best? Well, let's start with the one-dimensional case. We mentioned how a function called the sigmoid function is used for determining probability. In this one-dimensional case, the function looks like this. We've been here and done that. It squishes the linear input into a range between 0 and 1. In the context of logistic regression training, we could also write this equation. Just replacing the x with b plus wx. b is an intercept bias term. w is the weight of the independent variable x on some response variable y. If we give a threshold of 50%, then this final value is basically 0.5. Let's simplify this to get an equation. Take logs on both sides, and then we get this final form b plus wx is equal to 0. Or this corresponds to a point x is equal to negative b by w. But what does this represent? This is the point decision boundary in the one-dimensional case for binary classification. It is this point that splits your one-dimensional data points into two parts. If you don't quite understand, pause and think about how this is the case. When the value of x is at the value negative b by w, there is a 50% chance that y is equal to 1 for binary classification problems. Also, if x were greater than this value, the probability of y being 1 would be greater than 50%. And if x is less than negative b over w, then the probability of y being 1 would be less than 50%. I hope you understand this. Great, so now we know the equation of a point decision boundary. It's x is equal to negative b over w. But we don't have w or b. But we can find this out through the training phase of logistic regression. What does it mean to train a logistic regression classifier? It means finding w and b that maximizes the probability of seeing the training data. I made a gentle explainer video on logistic regression and maximum likelihood estimation before, so I won't get into the math details right here. My objective is to visualize logistic regression and training. During training, we can use a step by step technique of changing w and b until they reach their optimal values. This is done with optimization techniques such as gradient descent. It is a technique that involves changing the values of w and b ever so slightly to maximize the probability of seeing the training data. I'll get into the details mathematically in my next video, but know that the update equations for the weights and the bias look something like this. We start by initializing w and b to some random values, or just one in this case. For some m iterations, we change the parameters ever so slightly that w and b will convert to their optimal values w sub m and b sub m. Here alpha is the learning rate, aka how fast do you want to learn? n is the number of training samples. xi and yi are the ith training sample. x is the features and y is the binary label. And sigma is the sigmoid function. Don't worry if you don't understand this completely, but let's visualize what this math is doing. Maybe you'll get a clear picture. Let's consider the one-dimensional logistic regression case for binary classification. We have some training sample points. They are labeled as zero or one by color here. Let us just take the initial value of w and b to be one each, like we stated before. I'll set the learning rate to be a typical value, like 0.05, 0.01, whatever it is depending on your data. From this, we can determine the initial decision boundary. x is equal to negative b over w, which in this case is negative 1. And now we're going to start the training phase. We apply the update rules for w and b, and because of this update, the decision boundary changes. Continue to the second iteration, and now the third iteration. And if we keep applying this for some time, we eventually see w and b converge. So basically the decision boundary stops changing by much eventually. I'll play this entire training process again, but this time let's also visualize how the sigmoid function changes at every iteration. For every point x, the height of the curve above that point represents the probability of that point belonging to the positive class that is y is equal to 1. Pretty slick, right? Now during the testing phase, when we are past some data point and our models ask to label it, we first determine the probability that this point belongs to the positive class, and then assign its class depending on the probability. So what we did until now was visualize two major concepts, the sigmoid function and the decision boundary. But both of these are just for the one-dimensional logistic regression case. Let us visualize how these change for two dimensions when we have two features. We have the original sigmoid function here. For the one-dimensional case, the sigmoid function squishes the linear input of one variable x into the range 0 to 1. So we have this curve. But now we have two features, x1 and x2. The sigmoid should now squish a linear function of features x1 and x2 into the range 0 to 1. So we have a surface with this equation. B is again the bias term. W1 is the weight of the independent variable x1 on some response variable y, and W2 is the weight of x2 on the same binary response variable y.  Let's simplify this to get an equation in the same way we did for the one-dimensional case. Take logarithms on both sides, and we get the final form. B plus W1 x1 plus W2 x2 is equal to 0. Now this equation corresponds to the equation of a line with x1 and x2 as the two axes. So this is also the line decision boundary in the two-dimensional case for binary classification. It is this line that splits your two-dimensional data points into two parts. Once again, if you don't quite understand this, pause and think about how this is the case. For a data point x1, x2, when the value of the equation is 0, there is a 50% chance that y is equal to 1 for the binary classification problem. Also, if this equation is greater than 0, the probability of y being 1 would be greater than 50%. If the equation was less than 0, then the probability of y equal to 1 would be less than 50%. To find this line, we need to know the coefficients B1, W1, and W2, and this is done during the training phase of logistic regression using the same gradient descent algorithm. So here was the algorithm for the one-dimensional case, and now here is the update for the two-dimensional case. The only difference that I made is now making W and x bold. This is just simple math notation to show that these are vectors. In the two-dimensional case, they are vectors of two dimensions. For a clearer picture, like we did for our one-dimensional case, let's visualize how this two-dimensional line decision boundary is created. Back to our black screen. Consider a 2D plane with axis x1 and x2. We have some training sample points that are scattered in some way in this plane. Remember, for each point we have a label indicated by colors. Let's take the initial values of the weights and bias to be 1 each. I'll set the learning rate to be some fixed value. From this, we can determine the initial decision boundary, B plus x1 plus x2 is equal to 0. To start the training phase, we apply the update rules for the weights and bias to get that slight shift in values. Now the second iteration of gradient descent. Now the third. And if we apply this for some time, we eventually see the weights and bias converge. And look what we have here. We have a decision boundary in the form of a line. And guess what? This line has the equation of the form B plus w1 x1 plus w2 x2 is equal to 0. Now, where does the sigmoid fit in? I'll play this entire training process again. And this time, let's also visualize how the 2D sigmoid function changes at every iteration. For every point in the x1, x2 plane, the height of the sigmoid function would represent the probability of being in the positive class that is y is equal to 1. So during the test time, when you are given a point x1, x2, you first determine the probability value. Then assign the predicted class based on that probability value of it being either the positive class or the negative class. Great, we visualize the sigmoid curve and the creation of a decision boundary with logistic regression for the 1 and 2 dimensional cases. But this begs the question, how do we go beyond this? How can we think of multiple dimensions greater than 2? Now, I can't create visuals for this myself because I can't effectively demonstrate 4 dimensional plotting on screen. But we can easily expand our intuition on this. For the 1 dimensional case, we have this. And for the 2 dimensional case, we have this expanded equation with w1 x1 and w2 x2. But we can also rewrite this in vector notation treating w1 and w2 as a w vector. And also x1 and x2 as an x vector. Now for some arbitrary m dimensions, we can write the same equation form. But x and w, instead of being vectors of 2 dimensions, they are now vectors of m dimensions. And that's the only difference here. We can use the same argument for every process. Gradient descent, for 1 dimension, we have the updated equations that look like this. In the 2 dimensional case, we introduce w and x as being bold as they are now vectors of 2 dimensions. And for the higher dimensional gradient descent, all the equations are the same. All we do is replace the 2 with the order of dimensions. And that's it. I hope this video helped you visualize logistic regression training in 1 and 2 dimensions and gave you an intuition on how you can think of higher order dimensions. Note, there is a lot more to logistic regression and I'm going to be deriving all these equations from scratch in my next video. So that's going to be fun. Some learning resources are down below, but stay subscribed and I will see you in the next one. Buh-bye!


Hi guys, what is up? Welcome to my channel. In today's video, I am going to be doing a shopper drop. If you don't know what shopper drop is, it's basically when I go through Trend Mood's Instagram and I show you all of the new makeup releases and telling you my thoughts on them, whether or not I will be purchasing them and just this, that, and the other. So if you want to hear my thoughts on the new holiday makeup releases, then just keep watching. It's been so long since I've done one of these. I am so behind on these makeup releases. I think just with me being so busy, I haven't even been really able to fully keep up on everything. So today we're going to have a thorough look into Trend Mood without this video being 40 minutes long. We'll see. So if you are not following Trend Mood 1 on Instagram, I would certainly suggest that you do. Trend Mood has everything you need to know about what's up in the makeup world as far as releases go. Let's see how far I want to scroll back. I wish I would have done a few in between, but I just haven't had time because there's been a ridiculous amount of new makeup out. So one of the things that I first wanted to talk about was the color pop pretty fresh line. And in that line, you're getting a tinted moisturizer. There's also a setting mist, a beauty sponge, or you can purchase the whole kit and it's only $25. I definitely plan on getting this. I know a lot of people have purchased and reviewed this already. I'm kind of late on it, but Kathleen Light seems to love it. I'm really into light coverage makeup right now, as you can see, right? And yeah, it's only $14. I think I'll just get the whole set, but I don't know when I'm going to order it, but next time I place a color pop order. I do plan on getting it. I haven't had the best luck with color pop base products. I just, they aren't very good for me personally. So maybe this will change my mind. We'll see. Oh my gosh. So Too Faced has, and this is available now. I'm all the way back in September 27th at the moment, but this is the Too Faced pumpkin spice and everything nice makeup set. So you have a palette, a better than sex mascara, and what looks to be a lip gloss. Absolute pass. Now I do pick up a few things from the Too Faced holiday collection every year. Not everything is horrible and just interesting to me. Honestly, everything is cute every year. I really like the stuff, but the quality is just almost never worth it. Normally they'll release an eyeshadow palette in tin. That's usually good quality. That this year is the gingerbread extra spicy palette, which I thought was pretty nice. In my last video I used it and I created a sudden look. But anyways, this looks like it's horrible. Just given the pattern of Too Faced products in this packaging tend to be made in China and the quality is not as good when they make items from there. So it's just another repeat of all the other Too Faced palette colors that exist. Cute palette though. Next we have this Lunar Beauty the Moon Spell collection. Now I've personally never purchased anything from Manny's line. The palettes do look really pretty though. I have seen them in store. I haven't swatched them ever, but this palette, I'm digging it. It looks super nice. So this launched August 4th. That was so long ago. Okay. No, this looks so stunning though. I love purples. Okay. I probably won't purchase it right now because I am swimming and new products that I need to try and review. But I can see myself like once this comes to Morphe or next time I go to Morphe, picking this up because it is super pretty or if it comes to Sephora, whatever. I'll need to see it in person before I decide, but looks stunning. So now we have the Anastasia Beverly Hills and the Carly Bible collaboration. I feel so horrible for Jackie Aina and Carly Bible. Jackie Aina in particular, I think, got the shortest end of the stick on this one because this is ABH's like fifth palette in two months, which is absolutely ridiculous in my opinion. It's just they're doing it to make money and blah, blah, blah. And don't get me wrong. The Norvina palettes that have come in between them are super good. Purchased all three of them. Like I'm complaining about the brand right now, but the palettes are really good quality and I did buy all of them. So five palettes. It's very overwhelming to consumers. Jackie Aina didn't get her moment to shine with her palettes. She had about like two weeks before the next palette released and it was a big deal. And I know these palettes were supposed to have been leaked, so they were released early, but regardless, why are you making so many palettes and then have the Carly Bible palette on top? It's just not fair. Jackie Aina didn't have her time and Carly Bible is not. I don't know. I think it's kind of gross of the company to do that. And it just, it's like, I don't know. It just leaves a bad taste in my mouth. But that being said, I did purchase this palette. It looks absolutely stunning. I really want to support Carly anyways because she seems to be really sweet, really grounded and the colors are really pretty. I have her BH cosmetics palette still. That is so nice when she did a collaboration with them. So so excited to see kind of another light and airy palette. This is on the way right now and I'm really excited about it. I don't know about the packaging. It looks a little tacky to me, but I'm not a BH, but I still bought everything. So whatever. Okay, let's move on. So Charlotte Tilbury released like a huge holiday amount of stuff. So I already purchased the big long palette and reviewed that a few weeks ago, but a bunch of new stuff has come out. Let's see. There's a whole $200 advent calendar. Normally that would excite me, but I have so much from Charlotte Tilbury that I feel like that would just be a bunch of repeats. You have this hot lips luxury wardrobe where it's a bunch of hot lips, lipsticks, which that's I already opened that entire line, but that's a really good gift to give somebody. It's an amazing range of colors. So if you had your eye on it, I would definitely recommend you invest in it because it was one of my favorite lipstick lines to release this year. You just have so much the instant look in a palette. I've never purchased any of those from Charlotte Tilbury. That just looks so boring to me if I'm being honest. I saw Kinky Sweats use it and it looked great. So it did get me a little bit more interested in it, but I'm really just like not interested in it. I don't know. But honestly, just taking a look at everything, that little palette of pops I wasn't interested in. I saw it in stores. Brush said I'm not interested in. I mean, I'm not purchasing anything from the new releases from them. I already purchased so much from her, so she'll live. Alongside ABH is Carly By-Bow Palette. There were a few other things that they're releasing for the holidays. So you have a glitter set and I did stop and look at it for a little bit, but I don't need glitter. I don't use glitters as much anymore in my looks. A lot of brands seem to have catered to that by adding glitter shades in their palettes and I don't know, it's just a bit of a hassle. So I haven't been using glitter lately. It hasn't been my favorite thing, which is something I thought I would never say, but passing on that, but I did eyeball it. Like it's the kind of thing where if somebody bought it for me, but I'm not about to spend my own money on that. We have a liquid lipstick set, which I will not be purchasing. It looks like there's a gloss in there as well. Yeah, liquid lipsticks and one gloss. I'm so done with liquid lipsticks. I own so many. I don't need any more. So that's just not a set that interests me. And then you have the highlighting trio, which I did look at maybe if it ever goes on sale, I will purchase it because I do like loose highlighters and I want to try the So Hollywood one in particular and you might as well get the trio. So if that ever goes on sale, I will definitely be picking that up. Oh yes. So a lot of you guys have been messaging me about the Huda Nudes palettes and I purchased the ones that came out this summer, all three and doing like three palettes in one video is always such a big production because I like to do whole looks. I just don't do the eye and I just have so much new makeup for the holidays that have to skip it on that one for now because they look absolutely stunning. Right now, I'm very overwhelmed with all the amounts of products I have to work with. So I'm not purchasing those right now. I'm going to wait and see if there's a sale. Then I will pick it up because they're gorgeous and I at least want the light and the medium. Probably not the dark. It's just a lot of rich tones. I don't know. I might get all three if I see them in store. So I eventually plan on purchasing at least a couple of those but right now those are not at the top of my priority list. Some of you guys know this but as far as the Pat McGrath Labs Holiday Collection goes she revealed a new eyeshadow quad and a couple of highlight creams. So I did order that Rose Astral Quad. It's supposed to come in on Monday and then there's two of those highlight creams. One is blue and one is gold. I did order the gold one so that I could play with the formula because I'm not going to use the blue one. That's just not me but I'm very excited to use it. I think it'll probably be something I'll end up using on my eyes, not necessarily my face. But you guys know I got to give it a try because I love Pat McGraths. So those are on the way. This interested me. This Farsali Holiday Mini Set of like their oils and skincare stuff. I've never wanted to shell the money for an oil that I thought was kind of gimmicky if I'm being honest. But now it looks like there's a few different serums and oils for $25 and I almost purchased it the other day. I didn't. I'm holding tight to not do it. But that's something I want. I think I'm going to ask that for that for Christmas. That's a good set to get. So Dominique Cosmetics revealed this six-pan matte eyeshadow palette which she's cute. She has like lilacs, pinks, kind of more cool toned colors. We have like a warm yellow brown. I'm not going to be picking it up. It's the last thing that I need but it's really cute. But I'm not picking it up. Colourpop revealed this like super grey smoke show collection and palette. It's kind of pretty. I kind of want it because it's really unique. I won't wear it. I rarely wear colors like that but I don't know. Maybe I will. This is something that I'll probably add into my basket when I order the tinted moisturizer to be honest. So I'm not in a hurry to pick that one up but I can see myself just kind of tacking it on to my order because it is really cute. And I'm a sucker for those monochromatic palettes from Colourpop. Jouer had a couple things coming for their holiday collection. So it looks like this is called the Rose Cut Gems. I'm so disappointed in Jouer. Is it just me or does this look cheap and not Jouer quality? I mean obviously I cannot speak on the quality but it just looks cheap to me. Like their Rose Bouquet blush palette that came out earlier this year was so perfect. It was beautiful and dainty and it looked feminine. I mean it didn't look expensive but it didn't look like this. Like this looks cheap to me and I won't be getting that lipstick or whatever. No I'm not interested in that and I had a little bit of higher expectations for Jouer.  This Fenty, what's it called? What's it, what are you called? Glossy Posse set. It has five mini shades of the gloss balm formula which is amazing. There are fireworks going on. I hope you don't hear them. Yeah, there's five shades of lip gloss and only one of them is a repeat I believe. So I already purchased this. I was so excited to have different colors of that lip gloss and then also there was this set with a little mini highlighter and another lip gloss in Sand Castle. I purchased that as well. I don't know what got into me. I just really wanted them. I don't get as excited about Fenty releases as a bunch of other new people. They got good glosses. I love throwing them in my purse and their highlighters are good too. So I purchased that. Also, Charlotte Tilbury has a new highlighter out. Magical Star highlighter. It looks super pretty and I'm sure it's gorgeous on the skin but I don't need to be spending $45 on highlighter. Not that I haven't done it before, I know but you've seen how much money I've spent. I got two. This MAC lipstick set. Their lipstick set's always catch my eyes. I'm a sucker for sets but this set looks ugly. It just does. The colors in here are ugly. Don't feel offended if you purchased it but those just aren't colors that I would wear. Is this something else by ABH? I have not seen this yet. This is a holiday glitter kit. First of all, it's a really cute idea.  I'm sure some people could actually use that but I'm mad at ABH so I'm just gonna move on. I have done videos in the past on the Jeffree Star mystery boxes because it was a mystery I wanted to know. I passed this time. They're Halloween-y and I don't need Halloween colors. I won't use them. So I just saved my money with this one. I didn't do it. My cousin did it though so I'm excited to see what she gets. And then Morphe came out with a mini version of the James Charles palette. Genius. I think that was a really smart marketing strategy on their part because now I think more people will buy it. I wish I could return my big one and exchange it for the little one because I still haven't used my big one. I bought it. It's in my collection. I'm keeping it forever but I like the idea of a mini one because that palette is so big. It's huge. It takes up so much space in my makeup collection. So I want the small one. I'm so sorry for these fireworks. There's a pumpkin patch by my house and they do fireworks every night in the evenings. We're gonna push through. Wait a second. Lorac X Raystrol Zote. These palettes look adorable. $15 each. So they're probably not very good quality but they look so pretty. I feel like I used to love Lorac. I used to purchase all their palettes every year but they kind of just stayed in their lane and never went with the trends and put themselves in a hole and it looks like they're trying to dig themselves out. This collection is cute. It looks a little cheap. I'm interested to know where it's made but that looks really cute. I might just like want to purchase that just to support the brand because I have a soft spot for Lorac because I used to love their shadow so much. I don't know though but it looks really cute. Just so you can know my thoughts on the whole Shane Dawson and Jeffree Star thing. I know a lot of you probably don't pay attention to it but I have been watching the series because I find it very interesting. There's a lot of the parts that are boring and I wish weren't in there. I'm not like a fan so much of Shane Dawson and really not Jeffree Star either. I just love his passion for makeup and how he does treat his makeup like art. The series has definitely been interesting to me in some parts. Just kind of seeing the development of a palette and what happens. Just getting a little bit more insight into that process. I might when this palette comes out I'll probably pick it up just because I took the time to watch the series. It's very well done. Some of the colors really do look nice and I really do feel like Jeffree puts a lot into his makeup brand. How he is as a person, whatever but yes, he treats his makeup like art. That's how I see makeup and yeah, I think I'll pick this up because it's just fun. All right. So that sums it up for today's shopper drop. Let me know your thoughts and opinions of the releases that I talked about. What are you picking up? What are you passing on? If you have not yet taken the time to subscribe to my channel. I do hope you take the time to do so and I will see you guys in my next video. Bye guys. Have a good one.