 Hi everyone, it's MJ and I'm busy going through the 2017 paper that was written in September. And if we look at the very first question, we can see that this relates to chapter one. And the big trick with these questions is to score all five marks. These are one of the easiest questions, but you don't want to be too overconfident and make silly mistakes. Let's go through this question nice and slowly and you should be getting five marks on this question. The question says the following, the number of cans of fizzy drinks consumed by teenagers each day is the subject of an empirical study. The following data has been recorded. So we've got cans per day and the number of teenagers that drink that number of cans. Assume that no teenager drinks more than five cans per day. Okay, that's great. And now what we need to do in part one of the question is calculate the mean, the median and the mode for the sample. And this should be the three easiest marks that you will ever get in a stats paper. Question two is going to be a little bit tricky. So let's quickly go through number one and we'll then jump to part number two. So this is our data over here. And I mean one way of thinking about it is that we have 25 zeros, we've got 30 ones, we've got 26 twos, 23s, 14 fours and 10 fives, which means if we're going to calculate the mean, what we need to do is the following. Okay, we first remember the mean is equal to the sum of the X's divided by n. So we first need to add up all of these numbers and we've got 25 plus 30 plus 26 plus 20 plus 14 plus 10. So we see that that is equal to 125. Now what we need to do is add up all of these. So we've got 25 times zero. So that's going to equal to zero. We've then got one times 30. So we're going to have 30 plus two times 26. So that is 52 plus three times 20. That is 60 plus four times 14, which is 56 plus five times 10, which is going to be equal to 50. And then what that's going to give us 248 divided by 125. And we're going to see that the mean is equal to 1.984. Okay, now let me just explain where we're getting 30, 52, 60, 56, 50 from. Or remember, these are the number of cans per day times by the number of teenagers. This is just a frequency diagram representing how the data is. So that's why in chapter one, we did look at the ways of presenting data. And this is a more compact form rather than saying, you know, 0, 0, 0, 0, 0, 25 times and then 1, 1, 1, 1, 1, 1, 30 times and 2, 2, 2, 2, 2, 2, 26 times and so forth, so forth, so forth. So you must be comfortable with dealing with data in these various formats. And I think that's where they try and make it a little bit tricky is presenting the information in a different format. But anyway, that is the mean that is a formula you should like it should be ingrained into the back of your head. So that should be good. We also needed to work out the median. Okay, so how would we work out the median? Remember, median is the middle value. So what is the middle value? Okay, what we're going to be doing is if we take 125 and we divide it by two, we're going to see that around the 63rd position is going to be our median value. So yeah, that's going to be our 63rd. If we had to order it, fortunately, this data is ordered from smallest to largest. And what that means is if we had to almost draw like a cumulative, we would see that there's 25, then there is 55 and then 26 plus 55 is equal to 81. So we can see that the 63 is going to be lying here. So therefore, we know that our median is equal to two. So our median is therefore going to be equal to two cans. So 63rd position and we've got two cans. Okay, and then the mode is the most easiest of them all. And that is just which one has the highest frequency. And here we can see 30 is the biggest number between 25, 30, 26, 20, 14, 10. So bam, the mode is going to be equal to one. Don't write 30, don't be silly, don't write 30. Remember, these are the values that we're interested in. And that's just the frequency of how often they're occurring. So we're looking at 30, but then we're going to be choosing the number one. Just like, yeah, this was our position of 81, and we chose the number two. Okay, so what we have, let's write it out here. We have the mean is equal to 1.984, the median is equal to two. And the mode is equal to one, okay. We need this information now for part two, which says, comment on the symmetry, okay? Symmetry, and this is why you need to learn a little bit of the jargon. But I think everyone should be comfortable with symmetry. Comment on the symmetry of the observed data using your answers to part one. That's very, very important that you're using your answers to part one. And without making any further further calculations. Because if you had to draw this out, okay, and this is where this question is a little bit tricky. If you kind of had to draw this out, you would get a little bit of a curve like that. You can see 25, 30, 26, 20, 14, 10. That's kind of the shape that you're getting, but, but, but, but. If you had said, this is positively skewed, you would get, even though the data theoretically is, you would probably get marked wrong. Because what the question's asking you to do is it wants you to use your answers to part I. So it wants you to almost be blind to the actual data. And say, well, if you had to make an answer just on this, what would you say? And look, we know that in order for something to be symmetrical, if the mean and the median are similar or very close, we know that that's gonna be the case. And we can see that 1.984 is close to two, okay? Where it does get interesting is the mode is one, okay? And we would expect the mode to also be two, to also be two if symmetrical, okay? However, if we come and we look, we can see that two, sorry, I have like scribbled everywhere here, 26 was the second higher. So because our mode is one, but the second highest, the second highest is two, then we can kind of see we have two for the mode, two for the mean, two for the median, and we can basically say that it is symmetrical. So we're gonna get a mark for saying that it's symmetrical, and we're gonna get marks for giving our reasons. And our two reasons are the mean and the median are close to each other. And we're also saying that the mode, even though it is not technically two, it is close to being two. And this is where you wanted to get, where you wanna get your marks is, is you wanna explain. You could say more, like I guess in a better way by saying, there is not enough evidence, this is if you like done hypothesis testing. There's not enough evidence to support asymmetry. And I think if you write that, they will be very, very impressed with you. Because there, what you're also doing is you're carrying on the philosophy of statistics by saying, you're not saying what something is, you're just saying what it isn't, because there's a lack of evidence to support what that isn't is. I hope that made sense. But there we go, rather say, yeah, that. Not enough evidence to support asymmetry because of the fact that the mean and the median are close to each other and the mode, even though it is not technically two, the second highest or most occurring value was two, which are. Means that there's not enough evidence to support asymmetry. And there we go, that is your typical exam question for this chapter, for this course. Please let me know if you've got any comments and I'll do my best to help you out if there's any sort of confusion. I'll see you guys for the rest of the course. Keep well, cheers.