 Nothing. No comments, no question, no query. Are you guys unhappy? Hi, Lizzie. Hi. Anastasia, sorry I missed the first three sessions. I had some difficulty getting online. That's why I'm joining you guys as of from today. Just a quick question. The first three sessions, is that recordings available or not? The recordings are available here on MS Teams if you have joined the channel. You are able to also check the recordings on the top banner. I will show you at the end of the session where to find the recordings. I do upload them there as well. And also they are available on my UNICEF. I will also show you that at the end of the session to show everybody where to find those recordings as well. Thank you so much. Okay, so if there are no questions or no queries or comments other than the one that Hendrik posted online is happy. So I'm assuming that everybody is also happy. So we can continue with today's session. So let's first quickly recap on what we did on Wednesday. We looked at central tendency, the measures of central tendency, the measures of variation and the shape or distribution of your data. And we also looked at the practice and how we identified the five number summary and construct a box plot. Okay. So we said the mean is the most commonly used measure of central location. It is the average. It is calculated by the outlier. And if we calculate the mean of the sample, we use X bar because it is a sample statistic. We always use the normal letters like the normal known letters like X with a bar on top. That gives us the sample mean and is the sum of all observation divided by how many they are. We also calculated the mean for the population, which we use the population parameter mu for all the population parameters and the sign is always represented by the grid letters. So for the population mean we use the mu and it's also the sum of all values divided by how many they are. We also looked at the median and we said the median is your middle value. You have to sort your data, order your data from the lowest value to your highest value. And the middle value from the asserted values is your median. Sometimes when you have even numbers like the count when it's 10 values that they gave you or the data values, there are 10 of them from the sample of 10. Then it is an even value. So therefore it means your median position will be in the middle of two values. What you do is you take the average of those two values and divide them by two. So you add the two values and divide by two. That is taking the average of the two values. And to find the middle value, we first use the position and we know that that position formula is n plus one divided by two. And the median is not affected by extreme values or outliers. We also looked at the mode and we said the mode is the value that appears more than the other values. And we said the mode with the mode, they can be normal because the numbers like they are one, one, one, one, one, they appear the same way. Or we can have a bimodal where we have two modes, two numbers repeating more than the other numbers. Or we can have one mode where there is only one number that is repeating more than the other numbers. Or we can have a multi-mode or multi-modal data set where there are multiple values, more than two values, which are multiple values that have, or that are repeating more than the other values. And we also see the distribution of the data. Please make sure that you always mute yourselves. So we looked at the distribution of the data where we said if the mean is less than the median, then your data is left skew or it is negatively skewed because the tail of your data is to the left. And we said it is symmetric when the mean and the median are equal, then the data set is symmetric. We said it is right skewed or left skewed, sorry, right skewed or positively skewed when the median is less than the mean because the tail also is to the right. The measures of variation, we said they are the range, the variance standard deviation and the coefficient of variation. And we said the measures of variation tells us the spread or the dispassion of, or gives the dispassion of your data around the mean, or the variability of your data around the mean. And we said the range is your largest value minus your lowest value. And we also looked at the standard deviation and the variance. We said the variance is your sum square deviation of your values from the mean. And we looked at the formulas for the sample variance, which is the statistic. We use s squared and is the sum of your observation minus the mean squared divided by n minus one. And for the population variance, we use sigma squared, which is the sum of your observation minus the mean squared divided by n. And you should notice the difference between the two formula. One has n minus one and the other one just n. And then we also looked at the standard deviation and we said the standard deviation is the square root of your variance. And it shows you the variation of your data around the mean. And we said, because it is the square root of your variance, therefore the standard deviation for the sample statistic. It's represented by s. And for the population statistic, it will be represented by the sigma. The other thing we looked at was the coefficient of variation. Remember, we said we represent the coefficient of variation as a percentage. So your standard deviation as a percentage of your mean. And we use the coefficient of variation to compare the variability of two data sets. Then we looked at the quotas. We said the quotas divide your data into four groups. But you need to order your data like we did with the median. You need to sort your data from highest to lowest, from highest to lowest. And in that way you will be able to identify where 25% of the data falls with 25% of the data are less. We have all within Quartal 1. Half of the data that are less than or they are above Quartal 2. And 75% tile will represent by 75% of the data will be less than Quartal 3. And only 25% will be greater than Quartal 3. And we also looked at how we locate those positions or those Quartals. Not the position, but the values. How do we identify those Quartal values that will give us a guide in terms of where do we fit in the Quartals? And we said four, finding the Quartals values. To find the Quartal values, you need to first use the position to go find the position where that Quartal is. And to find the Quartal 1 position, we use n plus 1 divided by 4. And to find Quartal 2, we use n plus 1 divided by 2. And we said Quartal 2 is the same as the median, and that is why the position, you use the same formula as the median to find the position of the Quartal. And to find the third Quartal, we find it on the position 3 multiplied by n plus 1 divided by 4, where n is the value of your sample size that you would be given. And then we also said we need to be mindful of the rules when we're looking at the position of the Quartal by looking at the answer you get from the position. If the position of the Quartal is the whole number, like 3, 4, 5, 6, 7, then the value that is located at that position will be your Quartal. If it is a fractional, so your position answer is in a fractional, which is 0.5, then you're going to take the average of the two values and divide, or taking the average of the two values, meaning you're adding the two values that where the position is located, you average the two values, or by adding the two values and dividing them by two. And if it's a non-fractional half, which is 0.25, is the answer that you get or 0.75. So when it's 0.25, we're going to estimate to say it is closer to the lowest. And so if it was 2.25, therefore it means we're going to say the position is at position 2 when we go look for the value on the table. But the position will still be 2.25, but we're going to locate the Quartal value at position 2. When it's 2.75, we're going to locate the position closer to 3. So we're going to locate the position at 3, but your position of that Quartal will be at position 2.75. And then we use the estimate to get to 3. And that's how you will locate your Quartals. And we also looked at when we do the Quartals, there is what we call the Inter-Quartal range. And your Inter-Quartal range is your Quartal value 1, sorry, the difference between Quartal 1 and Quartal 2. So it's your range of your highest Quartal minus your lowest Quartal. So it's Quartal 3 minus Quartal 1, which will give you the spread or the range of your data set. And we also looked at the five numbers, which are the minimum value, which is the smallest value, Quartal 1, Quartal 2, Quartal 3 and Quartal and the maximum value. And we also said from when you have the value of your Quartal 3 and the value of Quartal 1, you are able to calculate Inter-Quartal range. Not the position, but the actual value. You can calculate the Inter-Quartal range and the five number summary makes up the box whisker or the box plot graph. Then lastly, what we looked at was to look at the distribution or all the shape of your data based on the Quartals. And we said it can be left skewed when your Quartal 1 minus the smallest value is bigger than your largest value minus your Quartal 3, then the data is left skewed. If your Quartal 1 minus the smallest value is the same as your largest value minus Quartal 3, then it is symmetric. If you write it's right skewed, if Quartal 1 minus the smallest value is less than your largest value minus Quartal 3. So you need to know all these things that we discussed now in order for us to do the activities just now. So let's start with the activities. So we're going to start the session today looking at everything that we did on Wednesday after we just did the recap. We're going to look at the activities and look at how we use our calculator, our scientific calculators to calculate the mean, the mean, the mean, the standard deviation and the variance. Okay, so and welcome to those who just joined us. Take out your calculators and we're going to calculate that data set. Consider a sample of 12 monthly sales of bicycles sold by a bicycle dealer and the data is 10, 6, 5 up until 6, 6. We're going to calculate the mean of this data set, the variance of this data set, the standard deviation of this data set, and we're going to calculate the coefficient of variation. So everybody has a different calculator I'm going to assume. Some of you have the shop business financial calculator EL738 or the latest one. I don't know the number for the latest one, but it is a shop calculator. Some of you has a cashier calculator that you are using. I'm going to show you with those both. If you have an HP calculator, unfortunately, I do not have the steps of an HP calculator. So can I get an indication of who has an HP calculator? Okay, so there is nobody with an HP calculator. So I'm going to assume that the majority of you have a shop calculator. Or they have a cashier calculator. So you either have a cashier or you have a shop calculator. It doesn't matter what model of a calculator is, but I am going to start with those with a financial calculator. So let's do that. Anyone with a shop business financial calculator unmute and let me know so that I don't have to. Okay, I can see Lungile says she has a cashier. How many people has a cashier just like Lungile or Hendricks number? It doesn't matter which financial calculator you have. Just let me know. I just want to know which calculators you have quickly. Can I get that so that then I can use the same calculator that you have for this session? Miss Lizzie, I'm using a cashier. Okay. I'm also using a cashier. Okay. She is using a shop. Using a shop calculator. Does it look like the one that I have in front there? Yes. So those who are using shop, okay, we will use a shop or a cashier. So I'm going to assume that those are the two calculators you have. So those with a shop calculator and the steps that I have on the handouts are related to that calculator. So easy to follow. You need to put your calculator to a mode. So I'm first going to show those who are using a shop calculator. So we're going to do with the shop, then we move into cashier. I'm also going to demonstrate it online for you. Don't worry about it. But I want you to understand the steps. With all the calculators, we will need to put our calculator to state mode. Different calculator, different modes will appear or different functions to get to the mode state mode. So for the shop calculator, when you press mode on your calculator, your mode should be somewhere on the CA button. It's M-O-D-E, the mode. Then there will be some words that come and say state or something and something and something. We need to press one for state. So you will press button one, like your normal button one. And it will write on your calculator SD var reg and all sorts. We're going to use the SD for descriptive statistics. SD represents descriptive stats, right will represent the regression mode and all that. We will do those later on. But for now you're going to press zero for SD and your calculator will say STAT zero in front on the screen. And then it means your calculator is ready to capture the data. If you see where I am on there with my mouse, let me get a pen to highlight. So there are a couple of things that you need to know about your calculator as well. Right here next to, on top of the close bracket, there is an M-plus. That is the button we're going to use to capture the data. So you're going to put 10 and then you press that M-plus. Not the plus sign, but that M-plus. Oh, we start with the 10. Yes, you start with the 10. So we're going to enter all of them. So we first start with 10 and then you press the M-plus. Like I said there and your data, your calculator will say data set one. Then you move to the next one, six and then press M-plus and your data set should say data set two and so forth and so forth. You continue until you get to six and you should say data set 12 when you are done. I'm going to stop right there because I want to demonstrate it online. Sorry. Don't panic. Must you continue entering the values? You can continue entering the value. If you make a mistake, if you made a mistake when capturing your data, you can clear your data by pressing second function, which is the orange button and the mode button. It will clear the data that you already stopped on your calculator and then you need to start again. 10 M-plus, six M-plus until you get to the end. So let me open a calculator. Sorry, you said today we must press a second. Shift, which is second function, also the second function and the mode button. Okay, thank you. Let's hope not this one. Also not this one. Okay, this is the one. Okay. I am going to use this site. I am going to use my screen so that we have the data and I must have my calculator. Okay. So if you look at this calculator, you will see that it looks almost exactly the same as your calculator that you have, even though it, because this is an emulator, it might not be the replica of what you're seeing in front of you. But there's my mode. There's my second function. 10 M-plus and I am going to capture my data. First, press the mode. So if I follow what I just told you to do, so if I scroll, scroll. So I said, press the mode button, then press one. So if I press mode, my calculator shows mode normal and set. Then I press one for set. And then I press zero because there is SD line and quad that I press zero for set zero. And in front, my calculator will show set zero. And then I need to capture the data. So capturing the data, I'm going to start with 10. So it's 10. Press the M-plus and your calculator should say set one or data set one. And then I continue six M-plus, data set two, five M-plus, data set three, 10. And you must capture them in an order that you see them M-plus, data set four, nine M-plus, data set five, seven M-plus, data set six, ten M-plus, nine M-plus, six M-plus, eight M-plus, six M-plus, and six M-plus. And then my data is stored on the calculator. I can press the on and off button. And once I'm done, then I'm ready to calculate any of these measures that I see here that I need to calculate. The mean, the variance, and the standard deviation. Now, how do we calculate that? On your calculator, you will notice that you've got on the buttons, on the number button, you have letters on top that are written in green or blue, whatever the color that is. In green, like on button number four, it is X bar, which will calculate the mean. Whether the mean of a population or the mean of the sample, remember the formula is the same. Your X bar will calculate the mean. On button number five, you will have SX. SX will calculate the standard deviation of a sample. So we will use SX to calculate the standard deviation. So for this one, we will use X bar. For the standard deviation of the sample, we will use button number five, which is XX. SX. Then on button number six, it has SX, which calculate your population, not coefficient of variation. So population, standard deviation, we will use button number six, which is SX. That will calculate your population standard deviation. If they ask you to calculate population standard deviation, you will press that button. So because they are written in green, then it means before we press four, we need to press alpha. It means for all these values for the mean, we first need to press the alpha. And then for, let's say we calculate in the mean, we will press alpha four. It will give us the value of the mean. So let's go and calculate that. Alpha four equals. As always remember to press the equal sign. And that answer is, you must write it, it is seven comma six seven. I'm going to leave it at two decimal, seven comma six seven. That is my mean. So those who are using the case, you must also take into consideration that I just spoke about. So that we don't have to go and validate the answer again and again. So let's calculate. I'm going to skip the variance. I haven't spoken about the variance. I'm going to skip it for now. The standard deviation for the sample. Let's say we calculate for the sample, the sample standard deviation. So for the sample standard deviation, we press alpha. You don't have to clear your calculator. You don't have to say on and off. You can say alpha and press five and then press equal. And it will tell you it is one comma. So your sample standard deviation is one comma eight seven. I'm going to leave it there. Yes. So if you don't get better, get the values as yours. It means your values are wrong. So if, for example, you can check how many you have. If it didn't say data set 12. So if I got alpha and I pressed N, I should get equal. I should get 12. If you don't get 12, it means there's something wrong with your data set. If I press alpha N. Alpha and zero. What do you get? What is your N? N zero. 29. There is your problem. Your N should be equals to 12. So it means you have double counted your data. You need to go and clear your data. Press second function and press mode. Second function mode and capture your data again. So that's what you will need to do. Okay. So let's calculate the population standard deviation. So calculating population standard deviation. We go alpha and we press six. And it will be equals to population standard deviation, which is sigma X gives us one comma seven nine, which is one comma eight zero. I'm just going to round it off. One comma eight zero. Okay. So how do we calculate the coefficient of variation? Remember the coefficient of variation is your, your coefficient of variation is your standard deviation. Standard deviation. Divide by the mean. Multiply by a hundred. Remember that. So we have our standard deviation, which is S X. So we're going to say alpha S X. Divide by alpha four equals. Multiply the answer by a hundred equals. And that gives us our coefficient of variation gives us 24.45. Now that is coefficient of variation. We set the coefficient of variation. This is your alpha, but in, but in five alpha five. Divide by alpha four equals. Multiply by a hundred. That's what we did to get to the coefficient of variation. Now let's calculate the variance. Since on your calculator, you do not have a variance as a button that you can calculate. But we know that. What do we know? We know that our variance, if we calculate in the variance of a sample, so let's say this is our samples X squared. The variance will be the square root. Oh, sorry, not the square root. The variance, we said the variance is the square root. Is the square root. Sorry, the standard deviation is the square root of the variance. Therefore, if we have this standard deviation and we want to calculate the variants, therefore we need to square our standard deviation. So we have our standard deviation. In order to calculate the variance, then we need to just square the variance. To calculate the variants, we just need to square the standard deviation. And how do we square the standard deviation? We square the standard deviation by pressing the X squared button. On your calculator, your X squared is just there. Next to the square root, there is an X squared. We first need to calculate the standard deviation. So standard deviation, alpha, five give us the standard deviation, and if I press X squared, as you can see there, it's S X squared, which will give me, if I press equal, it gives me my variance, which is my S squared of 3.52. 3.52. And that is how you calculate the variance. So any question, those who are using the shop, are you happy? No response. So let's use our case here. I'm going to open the case your calculator quickly, so that we can do the same thing with the case your calculator. With the case your calculator, move this for now, so that I can have space for the case yours. So with the case your calculator, your calculator does not have everything on there. So some of you, your calculator has the state conversion and state something. I hope your calculators look like this. If your calculator does not look like this, let me know. But they should be state on button number one. Some calculator has some, I think it's S sum and S var on button number one and button number two. And I hope your calculator, all of you has that fraction button, and it looks like this. The state is on button number one. So for your calculator, we also need to put our calculator to state mode. You need to press the mode button, and we have one for state. So you're going to press, oh, sorry, we have two for state. Depending, you must look at your calculator way or ST, sorry, because I'm sharing my whole screen. Things like that will always happen because I didn't close all my applications. Okay, so you must look at your calculator, what it says you press the number corresponding to a state is. So on this one, my state is on two, so I'm going to press button number two. Then I get this other main, which has one and one minus var. I'm going to use the one minus var, which will give me the descriptive statistics. Number two, a plus bx is your linear regression. When we do that, we're going to do number two. But for now, we're only going to use one. So you're going to press one and you get the table. Some calculators, you get a frequency table on the site. Ignore that, we're only going to work with an x. Capturing the data here, you just press the number, the value, let's say it's 10, and then you're going to press equal side. Then you press six equal side, press five equal side, press 10 equal side, and until you get to the end of the table, you will see when you get to the end of the table, you should have at number 12, you should be having six. So let's capture our data. 10, sorry, I must activate. 10 equal, six equal, five equal. I'm not going to say that again, 10 equal. I'm just going to capture the values as they are. And with the ratio, you are able to see when you capture the data that you did something wrong. Easy to solve. If I capture that one wrong, instead of 11, instead of 10, I put 11, you use your arrows, you go one up and you can just overwrite that value again and say equal, it will overwrite the value. I'm on 10, and I must say nine equals six equal, six equal, eight equal, six equal, six equal. And I can see that I am on 12 and I have six, so I've captured all my data set. Also, you can press the AC button and it goes away. The table goes away. The table is not in your memory, so it doesn't really actually go away anyway. It is on your calculator. It is stored in your memory of your calculator. So now, to calculate the mean, the variance standard deviation and the coefficient of variation, we're going to use the STA. So because it's written in orange, we're going to press the shift button. So you press shift and then you press button number one. And that will give you this menu. On this menu, one will give us the type of data we just captured. We're not interested in that. Two will give us the table. If I press two for now, don't press two. It will take us to the table. If, oh, sorry, I need to go out, then come back again, shift one. Number three, I forgot to show those who are using the cashier. If I can bring back the cashier on here, you have the sum x, the sum y, the sum y squared here, the n and the sum x and the sum x squared. Not for today, but you do have that. So on the cashier, those sum sums that I just showed, if you press three, you don't have to press three. So there you can see them, the sum x squared, the sum x and sum of x and sum of x, y. What we interested in is the var. So we need to press var, which is button number four. Then it gives us one for n, two for x bar. So we're going to press button number two. We're going to press two for the x bar. We're going to press for the sample statistic. We're going to press sample standard deviation statistics. We're going to press four. And for the population, we're going to press three. Oh, sorry. And for the population statistic, we're going to press button number four. So let's calculate the mean for now. So calculating the mean, calculating the mean, we just press two and you press equal sign. So you will just say two equal sign and you can see that we get the same value as we got with the shop calculator. And to calculate the standard deviation, we do the same for the cashier. You need to always press the AC button and go again, shift one, four, and then press four again for the standard deviation and press equal. And that will give you the standard deviation. As you can see, one comma eight, seven. And for the population standard deviation, AC, shift, stat, four, three, equal. And that gives us the population standard deviation. To calculate the coefficient of variation for you, it's going to be a little bit different. So you say shift, so you will follow the same steps. We want this sample standard deviation, which is four. You press divide by and you go shift again, stat, four, and you press two. And that will say sx divided by x bar. And you press equal and you press multiply by a hundred. So to calculate the coefficient of variation, you say, oh, sorry, you say shift one, four, you press one, you press four, and then you press four again and equal and then you press divide by and you repeat the steps. Shift one, four, two. Oh, actually you don't press equal. You say shift four, four. So shift one, four, four. Divide by shift one, four, two. Four, two equals. Then you say equals and then you multiply by a hundred. Then multiply by a hundred and not a thousand, a hundred. And that will give you 24.45. And that is the coefficient of variation. To calculate the variance, say you're going to use x-squared. There is our x-squared button, but first we need to calculate the sample standard deviation or the population standard deviation. So let's calculate the variance of the sample. We say shift, stat, four, four for the variance, for the standard deviation and then we press x-squared and then you press equal and that gives you the standard, the variance, the variance of the sample standard deviation. So you will say the steps you follow for the variance, shift one, four, four, x-squared button equals. That will give you the variance. And that's how you use your calculator to calculate the variance. Now, both of all the steps that we have just done, I want us to look at the data set and calculate the same things that we just did right now using a different data set. So let's go to the next one, make this bigger. Here we have another exercise that we need to do. So I'm going to use both of the calculators to calculate this. This will take us long to capture. You can ignore the first, the number at the top is just the count to show us that there are 50 values in this calculation. In this data set that we have. So you will need to capture all these values that you have here. We're not going to do it now in class because it will take us the whole hour to do that. I want you to use the same steps that you have to go and practice with your calculator to calculate all these values. We can discuss this on WhatsApp or on my Unisa. All the other days. Yes. Can I assume for the question that you just go share or number nine and nine and eight or? Pardon? To clear the data. Go shift number nine. So, yes. So to clear your data from this, you say shift, you will press nine. And it will say clear all, so which is three. And it says press equal for yes, because yes, we want to clear everything and press AC or reset all by pressing AC. And there we go. So if I press shift and I press stat, it should not work because I've cleared my data is no longer in state mode. So when they cash with the shop to clear this information, you press second function and you press CA. It clears your calculator. If I say alpha four equal, I shouldn't get any value because I've cleared my calculator. You always have to clear your calculator before you do the next calculations. You always have to clear your calculator before you do the next calculations. So you can do this at your own time. You can calculate the mean, the median, the mode of the data set that you have. The standard deviation, the sample variance, the population variance, the coefficient of variation. And we can discuss the answers. You can post your answers on WhatsApp during the course of the week. And during this weekend and the course of the week before Wednesday, because on Wednesday, we're not going to discuss this again. We continue with doing probabilities on Wednesday. So you'll have time between now and then to do this and discuss on WhatsApp or on My Unisa. Okay, so let's continue with the activities. Which one of the following descriptive standards is not a measure? It's not a measure of central tendency. It is the range because the range is a measure of variation. Which of the following statement is correct? Remember, you can also post on the chat. Okay, let's go through each statement. Which one of the following statement is correct? One, a high concentration of the data values about their central location indicates low reliability. Remember when we were doing the coefficient of variation? We had, let's go there, and we said that the variability means your data is closer to the mean or it's got how far your data's got around the mean. We said this one has, stock B is less variable than stock A because stock B has the lowest variability and stock A has the highest variability. So if this is the mean, so stock A will be somewhere here next to the mean and stock A will be there. So this will be B and this will be A. So stock A will be closer to the mean, stock B will be far away from the mean. So let's go back to the questions. Number one, a high concentration of your data about their central location indicates low reliability. And the other thing you need to take into consideration when you answer this question is that central location. What are the measures of central location? We know that there are the mean and the median. Reliability. Reliability are those measures that we get from. We calculate those. We can test the reliability of your data using the variance. We're looking for the statement that is correct. So does number one and number two correct? Are they correct? No. Number one and number two are incorrect. Number three, the measures that are commonly used to describe the data dispersions are the range, the variance, the standard deviation, the coefficient of variation. Dispersion of variation. So the measures that are commonly used to describe the data variation or dispersions are the range, the variance, standard deviation, and the coefficient of variation. Is that statement correct? As opposed to the range, it's supposed to be the mean. And then it would be correct if that's the case. Measures of central locations are? Mean, mode, and median. Okay. Measures of dispersions are? Range, variance, standard deviation, and coefficient of variation. Measures of variations are those ones. We're going back. So is statement three correct? Yes. Statement three is the answer that we are looking for. The range is the difference between the lowest and the highest. That is correct because the range is highest minus the lowest difference means highest minus the lowest. The symbol sigma is used to define the sample standard deviation. No. It's used for population standard deviation. This one is also correct. So I'm assuming they made a mistake some way here. Because number four as well, it is correct. Leslie? Yes. Maybe regarding number four, normally we normally say the highest value minus the lowest. Maybe they put in purpose, they started with lowest so that it can be correct. I'm not too sure. And it doesn't really matter because it says it's the difference between the lowest and the highest. Difference means subtracting. And I think this comes from one of the past exam paper. Hmm. Leaves more to wonder. The distribution of the following data. This is the data that you have. Is this data symmetric, skewed, binomial or poison? What do you need to do? Order the data from lowest to highest? The first thing you need to do is calculate the mean of this data. Because if I look at those three answers that I have, for now you can ignore the two because at this point we haven't touched those. So you can ignore those three. So it's either one or two or three. So you need to calculate the mean of this data set. You need to calculate the median of this data set. If the mean is less than the median, we say it is negatively skewed. If the median is less than the mean, we say it is positively skewed. It's symmetric. The data set will be equals to the median. And that will say it is symmetric. So first calculate the mean. Sum of all the values divided by how many there are. There are one, two, three, four, five. What is the mean? The sum of all of them divided by how many there are. The mean is two. If you order the data, so it will be zero, two, two, two and four. The middle value. What is your median? It's two. Two. So the answer for the question. It's one, number one. Number one. It will be number one. Yes. The data set is symmetric. The mode is zero, two. And if they would have asked about the mode, the number that appears more than the other numbers will be equals to two. Yes, you are right. Given the symbols, A, B, C and D, those are the symbol, the population. I'm not going to say that. Mu, S, X bar and sigma. Identify the symbol that represents a parameter as a measure of a population. It's A. A is a parameter. And what does it represent? It's a population. This is the mean. Population mean is a parameter. What is B? Simple mean. What is B? It's not simple mean. Nope. What is B? It's a mean. Nope. What is B? It's not division. What's, is it a sample or population? Sample standard deviation. It's sample standard deviation. So is this a parameter or a statistic? This is what we call A. Statistic. It's a statistic. What is C? What does it mean? Which mean? Sample mean. So this is sample, sample mean. And if it's a sample mean, it is A. Statistic. Statistic. And D. Population. It's a population standard deviation. So if it's a population standard deviation, therefore it is A. Population. A parameter. So which option is the right one? A and D. A and C. C and D and A and C. Is A and D. The answer is option B. When we did describing the table, especially numerical data in terms of tables and charts, and we spoke about a stem and leaf plot, and I said when we deal with descriptive statistics, we will look at how the questions are posed when they give you a stem and leaf plot. Now here we have a stem and leaf diagram for days to maturity data. And we need to find out which one of the following statement is incorrect. So number one is asking the range. Let's calculate the range. What is the range? Your highest minus lowest. What is the highest value of this data set? It's 86. It's 86 minus your lowest value. What is the range of the data set? The range is 50. What is the largest fifth number? It's 80. So we start counting from the last one and we go one, two, three, four, five. And that is our largest fifth number should be 80. There are 32 numbers here. Quickly count them. Are they 32? I got 33. 33. 33. There are the three. What is the mode of this data set? There's no zero on this data set. What is the mode of this data set? There's no mode. What is the mode of this data set? 70. It's 70 because 70 appears one, two, three. The others are just one, two, this one, two, one, two. 51 is 51, 51, 55, 55. The only number that appears more than the other number is 70. The median of this data set, yes? Can I just ask on that, if 70 would have been twice, would have this them and leave been a multi mode? Let's say this was one as well. If I convert that to one, there are numbers that appear more than the others, which is 51 in this instance, 55, 70, and 71 because I converted that to one. That would have been your mode, which is multi mode. Thank you. Let's find the median. Remember, because we've got so many values, 33, let's use n plus one divided by two. 33 plus one divided by two gives us 34 divided by two, which is 34 divided by two gives us 17. You can count from the beginning or you can count from the back and get to 17. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 18, 14, 15, 16. 64. 17, therefore that is the correct answer. The median is 64. Okay, so based on the question that I skipped in the beginning, which is this one, which I thought is going to take us forever. If you can do this on your calculator and calculate all these values, you should be able to answer the values for the next questions that I'm going to skip, especially based on that data set like this one because it's going to take us forever to calculate the variance and the standard deviation. We are only left with 40 minutes, especially because the data set has 50 records. So it's going to take us forever and it's going to also take you forever to calculate the variance and the standard deviation of this data set if you use the formula. Remember that the formula for the standard deviation, we cannot do it right now in class. So sorry, the standard deviation, or not the standard deviation, the variance x, y is the sum of your observation, y minus your mean squared divided by n minus 1. So it means you have to calculate the mean of this data set, find the mean, then take this value, subtract from the mean, square the answer. Take that value, subtract from the mean, take the answer. It will take you forever to do this if you get this kind of question in the exam, but they will not give you in the exam. Otherwise, because this is one of... I'm not going to say it now. I was going to say it, but I'm going to skip it. Since this data set is very long, it's easy to use your calculator to capture the information and calculate the mean, the standard deviation, and the variance. And that is why I'm saying let's take this and which is the question that I said. Let's discuss it on WhatsApp. Yes. I think last month or so, I attended a lecture class online. I think it shows us the also other way of calculating a mass standard deviation using Excel. Are you familiar with it? Yes, I'm familiar with Excel, but then it's going to take you forever in the exam to use Excel. That is your calculator. As you can see that it's easy quickly like that. So with Excel, remember you will get your data set. Sometimes you will get it as a picture and you still have to type it down. Type it down, type it down, type it down. So here you type it once on your calculator and you know you're done with it. And usually one question after the other, they follow each other and you are able to do all your exercise. So use your calculators. Follow the steps that we did. But when we do some of the calculations now, you will see how we do them again on the calculator because the data set will be small. But I want you to take this as a practice exercise because there is a purpose why I'm doing it that way as well. And it will look familiar when you do in your assignment as well. So that at least I would have helped you with something in your life other than just explaining some concepts. Okay, so once you have done that activity that I just showed you earlier, you should be able to come here and answer this question. Exercise six. So exercise six is also one of your exercise that you will do on your own at your own spare time and discuss the answer on WhatsApp. Okay, so let's do this one because this is the data set is very small. So I'll start with the sharp calculators. I'll capture the data. The problem with this kind of a calculator it always pops up. So let's start with the other one. I will use the sharp later on. Let's start with the case. So since my calculator is a reset, I must go to mode one. Oh, not one. Mode state two. Mode state two. One. So cashier, you press mode and then you press two and then press one, four, one minus var. So we press one and then we capture the data. So we'll say one, five, nine equals, 170 equals up until we capture 256 equal. So let's go one, five. So those who with the sharp calculator you can also go on and calculate 159 equals 170 equals 172 192 93 99 201 16 capture the draw 216 117 230 235 256 and I should have 14 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. I should have 15. I have 14 somewhere. I didn't capture the other book. So let's see which one am I missing? Go up to the beginning. 159, 170, 172, 173, 192, 193, 199, 201, 201, 201. 201, 217 equals 230 equals 235 equals 256 equal. I still skip one. How many 201s? I think you skipped 216 201. 216 I don't think it's skipping. I'm getting the same error from my side even though I added 230. 9 is 15. 256 and I have 15 of them. So you just need to check your your your numbers. Let's just double check them. Even though you're doing it on the calculator, it's always good to double check. Okay. So let's capture on the cashier, on the shop. I need to put my calculator to state mode. Mode 1, 0. And the first one was 159. 159 not 0, 9. M plus. I need to restart this because I put 15 159 M plus. 170 M plus. 173 M plus. 1 I skip 172. 172 M plus. I skip 172. M plus. 192. So with the cashier, with the shop calculator, it is going to be very difficult to see which values you captured right and which one we didn't. So you will have to repeat the step twice. If you're not sure about what values you've captured. I'm on 5. So 1, 2, 3, 4, 5. So I must do 193. 193 M plus. 199 M plus. 201 M plus. 201 M plus. 201 M plus. 216 M plus. 217 M plus. M plus, two, three, zero, M plus, two, three, five, M plus, two, five, six, M plus, 15. Yes. So now we have both data sets on so we can calculate the MC. Okay. So now let's answer the question. We didn't even have to do all the things that I just said you must do because if I looked at the question, it would have been much easier to do. So the first one says the number of observations they want to know how many if we count them. You can't calculate how many they are because you already know how many they are. It's 15 of them. So that should be correct. Otherwise, if you are already done with calculating, you can just go alpha and then go n and say equal. It will give you the number of how many they are. On the case show, you will do the same as well. Sorry. To calculate how many they are. So you will know that they are 15 because you've calculated them, or you will go and say shift set, and you will use the var, which is four, and there is your n, which is one, and that will give you how many they are, they are 15. The next question was the sum of all observations. So it means they are looking for the summation x. On the case show on the sharp calculator, your sum x is here at the bottom on button number on the dot or the decimal on the, it's sum x. So you just press alpha dot equal, not that, alpha, alpha dot, what's up, alpha dot should give us the sum of x equal, and that is how much they are, 30, 1, 5. If you add all of them plus, plus, plus, plus, plus, plus, it should give you 30, 3,015. On the case show, calculating how many they are on the case show, say shift set, and you go to the sum, which is 3, and there is your sum x, which is 2, and equal 30, 1, 5. So the incorrect one here, it is that one value. What is the range, highest minus lower? So you will say the highest value here, the data is protected in this instance. You will just say 256 minus 159, and that should give us 97, that would have been correct. The mean on your calculator, on the case show, shift set, you go to var again, which is 4. Our mean is 2, press 2, and we press equal, the mean is 201. You can do the same with your sharp calculator, alpha, and then we press 4, and press equal, 201. That's how easy it is and straightforward. Yes? Yes? With this question 6, regarding the mean, the range, and the number of observations, I know that you're trying to teach us how to use a calculator, but have used it, I mean, just by observing it, does it mean you have to always use a calculator or if there's something? No, no, there is a short way. You can use your normal calculator, I just wanted to show you the other alternative way. In this case, in this question, they also had the standard deviation, that you can just use. So if they would have asked about the standard deviation or the variance here, then you have already all your data captured, you just answer that question. And I think I have that kind of a question, which is not this one, this one. So on question 17, same data set, calculate the standard deviation. Easy to do. Standard deviation, remember, alpha 5 equals 26.72, 26.7, which would have been this answer. So on the SHAP calculator, since you start your values, you say alpha, and then you press button number 5, and that will give you your answers. On the Casio calculator, on a Casio, you will say shift 1, and you will press 4, and you will press 4 again, equal. And that will give you the same answer. And for the coefficient of variation, you will do the same. You will calculate the mean divided by the, oh, sorry, the standard deviation divided by the mean is the same data, as you can see that different questions from different, have the same, like you need to do different calculations on the same data. Going back to where we were, we were on exercise number 6. We already answered 6 and 17, all in one. I think that is what is going to happen as well in your assignment questions as well. You will find the table, which you can answer question number one, and question number two, and question number three, at the same time with the same information. So if you have stored your data onto your calculator, you should be able to answer all of them, without redoing all the steps. Okay, moving on to the next one. Next question, which one of the following statement is correct with regards to coefficient of variation? Remember, coefficient of variation tells you the variability, the relative variability to the mean. It gives you the relative variability to the mean. It is always represented in percentages. And it is your standard deviation divided by your mean multiplied by a hundred. So based on that, which one of the following statement is correct? Is coefficient of variation measures of location or central tendency? Is that correct? Am I alone here? I think the first one is correct, supposed to be a measure of variation. It should be incorrect, so this is not the right answer. It should be a measure of variation here. Coefficient of variation is the difference between the largest and the smallest value? Number two is also incorrect because this is defining what the range is. Number three, obviously they have the perceptions of what cylindrical is like when they've never been here, and maybe you don't have the video to see, but when they come here they like, wow. Of course it is. Okay, new tip. Okay, is number three correct? Is coefficient of variation, is it the sample standard deviation squared? I think that one is also incorrect. This is incorrect. The sample standard deviation squared is not a coefficient of variation. Number four seems to be correct. It is number four correct or incorrect, so you're saying it seems to be correct. It is the sample standard deviation expressed as a percentage of the mean. So it is a sample standard deviation as a percentage of the mean, which is almost exactly the same because it is x times, so we can say it is s times x times 100. Oh, x divided by x times 100. You can just multiply x by 100. Okay, so this is correct. It is a mean squared. It is not. Next question. Consider the following data set. You are given the data set which statement is incorrect. What is the median? You need to order the data from lowest to highest. What is the first quota? What is the third quota? Remember, the median is the same as the second quota. The median is the same as the second quota. What is the mean? And is the distribution symmetric? So in terms of this distribution symmetric, use the answer for the mean and the median. Don't try and use the quotas and all that because then you're going to have to do a lot of calculation. What is my maximum minus my minimum, my quota three? And is it less or equal to my quota? So the difference between quota one and the smallest one. So those are very confusing, but you can use the median and the mean to find out whether the thing is symmetrical. So, order the data. I'm going to also order it from my side. You have time. Okay, so let's do this. Quatal one, let's first find the position. Quatal two position and plus one divided by two. How many they are? One, two, three, four, five, six, seven, eight. One, two, three, four, five, six, seven, eight. They are eight. Eight plus one divided by two, which is nine divided by two, which is 4.5. It's on the 4.5 position. One, two, three, four, five is between, are you guys here? Yes. Yes, we are here. It's between seven and nine? Nine, yes. So going to eight, seven plus nine divided by how many they are? Equal to eight. It's equals to eight. So the median is correct. First quarter and plus one divided by four and plus one is nine divided by four. The answer is 255. And what do we do when it's 2.25? We estimate that the value is on? On two. On two. So we're going to count one, two. First quarter. Four. So that is correct. Third quarter. We find the position by three times n plus one divided by four. I'm not going to repeat all of them again. So it will be three times nine divided by four. It's 6.75. And we're going to estimate that quarter. It's seven. It's at position seven. At what? Two, three, four, five, six, seven. It's on. All right. Number two boys. It's correct. Calculate the mean of the data set. And all of them divide by how many they are. 9.5 is correct. 9.5. Because it is the sum of all the venues divide by how many they are. Which is the sum of all of them is 76 divide by eight. We'll get 9.5, which means that is correct. Is the mean and the median the same? No. The median is eight. The mean is 9.5. Therefore, the data is not symmetric or the distribution is not symmetric. Consider the sample set. Consider a simple data set of one, two, three, four, five, and six. Which one of the following statement is correct? You must pay attention when you answer the question. That says position. That says value. And that says value. Interquartile range. Remember, we find it by saying the value of Q3 minus the value of Q1. So position for Q1 is n plus one divided by four. How many they are? They are six. One divided by four, which is seven divided by four. What is the position? 5.75. That means it's 1.7. The first question is position. And we're looking for the correct answer. So this is not correct because it is on 1.75. The position is 1.75. If we were looking for the value, we were going to estimate and round it up to say this and position two. Something like that. But because we're not looking for the value, the position, we use the same as what you have when you calculated the position. Quarter two, we need to find the position, which is n plus one divided by two. So this will be seven divided by two, which is on position 3.5. So go find the value on position 3.5. 3.5 is located between two values, right? Yes. So the value will be? 3 plus four divided by two. 3 plus four divided by two, which gives us? 3.5. 3.5. That is, that should be the answer for quarter two. This is incorrect. Number three, the median is four. Remember, quarter two is the same as? Quarter two is the same as? The middle value, which is the median. So if we find that the median or quarter two is 3.5, can the median be 3.5 of this data set as well? Yes. So the median is also 3.5. Therefore, this is incorrect. We need to go find the value of third quarter. So first we need to find the position three times n plus one divided by four. So which is three times seven divided by four. And the answer of the position? 5.25. What is the value? The value will be located at position five, which is five. And this gives us 5.2, which is there. The value, so this answer should have been only five. So we should get third quarter to be five. So now we need to go back to the question that we answered in number one. We were looking for the position there. Now we need to find the value because we need to substitute the value into the Quartal range. So with Q three will be equals to five and Q one. Our position, we find it at 1.75, which means we're going to estimate it at position two. So that will be five minus two, which gives us three. Which means the answer that we're looking for is option five. Let's look at the last example. Okay, so this is another question that you will need to answer with once you have done that activity. Oh, actually you don't even have to do that activity. You can answer this question. It's easy. So we can do with, we can do this question. So second quarter, same as what we did n plus one divided by two. We need to find that the second Quartal is calculated as the range between 25 and 26 observation. We cannot, we cannot assume that unless if we find the position first. So the position we find it at. There are 50 observations plus one divided by two, which is 51 divided by two. What is 51 divided by two is 25.5. So is we looking for the incorrect one is a correct. The second Quartal is calculated as the average between 25th and 26 observations. So our quarter, our position is at 25.5. So it means we're going to get the value from position. The value that we will get, it will be the average of position 25 and 26. So it means this is correct. Number B, B says the position of the first Quartal is dating. So you need to go calculate the position of the first Quartal, which is n plus one divided by four. So 50 plus one divided by four, which is 51 divided by four, which is equals to 12.75. Which is 12.775. So this should be 12.75. This is the position. So therefore it means that it is incorrect. The range of the data set, it's your highest minus lowest, your highest value is 20.2 minus your lowest value is 6.2. It's 14. Which is 14. So that should be correct. N plus one divided by four equals three times. Okay. Number G, the position of the third Quartal is 38. The position is 38. So we're going to say three times n plus one divided by four. So that will be three times 51 divided by four. When you're 38, you're taking D incorrect. It is 38.25. The position is 38.25. Which makes D also incorrect. Unless if they are estimating already when they're looking for that position on this answer. It should be the position that we will find the value will be 18 because it's 12.75. And this one is 38.25. It will be 38. Okay. So number E, the position of the second Quartal is 12.5. Let's look at the last one. The second Quartal position is second Quartal. We did calculate that. We said it is 21 divided by two. 51 divided by two. Which was 25.5. So this is the incorrect one. Okay. So it means on this option, the position they already estimated them or rounded them off. To say this is 38 and that is 18. Which makes E the incorrect answer. Okay. Like with any other activities that we do, because we left with two minutes. There is activity 12, which should be easy for you to answer this question. There's activity 13, 14, 15, 16, 17, 8. We did already 18, now 17. So 18, 19, 20, 21, 22, 24, 20. So all of them. Until 25, you can do them on your own. You have the WhatsApp group or my UNISA to discuss. I have done my bit. Now it's your time to learn and ask questions and help one another as well. So where did I stop? So that when I give you, when I, so from question 12. Until 25. Plus the practice exercise. You can do them on. You can do them on WhatsApp. If you want us to give you input. You just need to ask or show us how you did some of the questions and then we can have the discussion. Otherwise, then thank you for coming. Going to stop right here and recap. We did all the calculation. Remember using your calculator. It's something that you need to practice. You cannot do your assignment now and then forget about it and only come back again when you go write the exam. You need to constantly practice, practice, practice so that you get used to using your calculator. It's easy. Once you learn the steps, go back to the recording, practice the steps as I have done them. Once you, once you you've learned the steps, you will see that it will be easy, easy, easy to do. You can also even just every time you see a question with tables and they're asking about standard deviation. You will automatically just go in and start punching in and storing the data onto your calculator. It's easy. Otherwise, thank you for coming. If there are any questions, feel free to ask. I am going to stop the recording. Because I don't share what we discussed after this. This is my signature of the recording. Any questions? Sorry, Lizzie. Just to make sure I'm at the right space. You see the recordings I can find.