Statistical problems come in 2 forms. Finding the mean, median, mode and range for some data and extracting certain aspects of information given to us in a statistical chart or diagram. The problems involve basic number work but algebra may be required so it is important to apply the basic number problem-solving framework.
Statistics is one of the three key areas of math, including mechanics and regular. Pure maths statistics is definitely one of the more controversial topics among students and quite a few students don't like it, primarily because it's fairly difficult to establish a set method to do a lot of statistics questions. And uniquely among math, it's one that doesn't follow set methods being used over and over again requires a degree of interpretation on mathematicians. So again, a topic for which we are going to need plenty of practice. Now, we're going to be looking at this through the lens of the problem solving framework that I already set out in other videos previously in this miniseries, I suppose.
And in those videos, we sort of go over them in greater detail. Now statistics, you have to go a little bit off the wall because most of the information and statistics is deliberately contained in very small amounts of information. So it's more about being able to take every new question with a degree of flexibility as opposed to using the same method. As usual though, we'll go over these, starting primarily with reading the question, highlight info and tick it off as it's used. Because usually union statistics, once something is used once, you may not need to use it again.
However, in statistics, usually doing the same things to the data, but in different ways, depending on the question what do I need to do? What will I need? And usually that's fairly apparent because the question will normally then ask you fairly straight off the things you need to do. And we'll go over, actually the two main kinds of statistics questions. What don't I need?
Usually the frivolous information, of which there can be quite a lot actually, in statistics in particular, when you begin to do things like say, solving standard deviations, you're going to need to set up tables with lots of information and not all of it may be useful. What techniques should I use? For the most part, in statistics, it's largely going to be a case of interpreting graphs. In algebra, however, with the advent of more and more powerful graphic calculators, you might need to actually use your calculator and some of these questions and learn how to use those. So that's a useful thing to bear in mind for the future.
Is my answer correct? This is always the bugbear of quite a few students, but the very least it's fairly straightforward to look at quite a few averages or things like that and see if there are sensible value, whereas things get a little bit less clear and we're doing things like statistical variance or standard deviation and have done everything. Usually there is a unit that can happen in statistics. Sometimes it's unusual, sometimes the unit for a question is pounds, other times the unit is number of buses. So you just need to pay attention to the question and see if you got everything correctly.
Now, fortunately for us, statistics usually splits problems into two main categories, at least at this level. The first one is finding averages. Now, in common parlance, whenever we say the average of something, we're usually talking about the mean, because there's actually four different averages that can be found for each piece of data. And already we're getting into what I meant earlier. When talking about statistics, you're going to be doing the same things as in finding an average, but in different ways.
The mean mode, median and range are all things you have to be comfortable with when doing statistics. The mean being what we normally consider an average, adding everything up and dividing by the total. The mode, which is the most common result, the median, which is the result in the middle if you arrange everything in order and remove outliers. And the range, which is the difference between the greatest value and the smallest value, you have to be comfortable with all four of these techniques. And so we're going to be using them in a bit.
And as well as that, a part of the application process. And statistics is understanding when one average is appropriate and when it is not. The second one is interpreting graphs and other information. And normally this is in the form of things like cumulative frequency graphs, pie charts, bar chart. Histograms one advantage the stats has is that it's trying to display real life information, which means that unlike in some other more abstract pieces of math, like in geometry or trigonometry, usually statistical information is designed to be as well laid out and as readable as possible.
And so a lot of the graphs that you'll be considering are things which just about anyone could and should be able to understand. We're going to apply these pretty much immediately to a couple of exam questions, but we need to bear these in mind as we do all these problems here, in particular, this average one. So I really enjoy this question. This came from an actual exam and I'll read out here. So eight boys and eight girls is a class run 100 metres.
Okay, now I'm going to underline the information as we go because we're going to be highlighting eight boys, eight girls, class run 100 metres. Time is taken to the nearest second. That could be useful for each girl are and then we've got the information. 15, 2020, 418, 1921, 26 29 seconds. The mean of the boys' time is 25 seconds.
The range of the boys times is 14 seconds. Thomas says the boys in our class are faster than the girls. Is he correct? So it's actually quite a common style for statistics questions where we lay out some information. We expect you to do some kind of manipulation to it and then get to perhaps what might not be an obvious answer.
Unlike algebra, it's not usually obvious or after, because clearly, we're after some kind of yes or no here. If we go over, what we actually need to get is we need to find out what faster means in this context, because again, faster could mean lots of different things. Because if we take the boys' times, the mean is 25 seconds. That seems to be fairly useful. So the mean is 25 seconds.
I'm going to write that as X bar. That's a common way of writing it. So X is just a piece of information. The bar is average, mean is 25 seconds. And we know that the range of the boys time is 14 seconds.
So we don't have every individual piece of data, but we know that the smallest piece of data is going to be probably 18, and the largest piece of data is going to be something in the region of 32. If you take both of these numbers, add them together and divide by two, you get 50, which gives us the mean. So this gives us the range of values we're likely to have when we're talking about this information. Now, we can't really extract the median for this bit of information. And in this you'd have to just assume the X bar here is also going to be similar to the median.
Now already this sort of illustrates the kind of questions we're going to be getting statistics, because we can strike out the information we don't really need now. Actually, the class run, even the fact there's really eight boys and eight girls doesn't even really matter. We're really interested in this set of times and then another set of times we can't really see. And the term faster really does mean something weird here because the fact that Thomas said boys implies it's boys in total. So the mean of the boys times is 25 seconds and the largest value of the boys is 32 seconds, which is actually quite a bit larger than what we get the girls here, which is 29.
But to figure out the average, the mean for the girls, we're going to have to figure out the mean of this information. To do that, we're going to have to add up all of this information here for a start, okay? And then use it to figure out the mean. Okay. So looking at it, obviously you have something like a calculator if you're doing this in the exam, but looking here at us on like 40, 60, 75.
Adding onto that, we're then going to have 9900 and 17 plus 57. So that's going to be 174.
We know that there's going to be eight of these girls. So then that's going to lead us to a relatively sensible value of about 22 seconds. We're going to round that to the nearest second. Okay. So the main result is going to be a little bit higher for the boys.
The largest result is larger for the boys, and then we can see that even the smallest result is also larger for the boys, so we can make a reasonable assertion that the boys are faster than the girls. In the averages that we've selected, we can say for certain given the median. But this is what it's like having to work with these quantities and statistics. You have to look at all of them individually and decide whether or not it's true. This is what we want to disadvantage statistics, though, and this is an unclear question.
Faster is something you have to interpret yourself and look at these four different data values. So this last one I wanted to go through is this stuff on graphs. And again, this has a lot of interpretation on the part of the person doing the question. So this chart shows a firm's profit for each of three years. Give two reasons why this graph is misleading and how much money was made across the three years.
This is a bar chart of some kind, or at least an attempt at a bar chart to show all the profits. Like, there's two ways this is misleading. I'll actually try and give you a second. If you can think of it, pause the video. So the first one is the fact that we can see that the width of each year is different.
Now, although it's a subtle trick, this is the kind of thing which makes the human brain go, okay, that implies that 2014 had vastly larger profits in 2013, to 2013 looked smaller. But of course, we're only really interested in the Y axis. The fact that these years are all just one year means they should all be the same width. And similarly, this is the kind of thing you might want to do to your shareholders. It shows the kind of profits that you're getting.
But the data is actually started at £2000 and not zero thousand pounds. So in fact, this also should have stretched down here. And again, misleads the bureaucracy actually makes 2012, not as small as it may be, necessarily looks in the first place. And it gives us a better idea of what it's like overall. So if we really wanted to redraw this, we probably just have to erase these parts here and then these parts here to give us a much more better representation.
If you want to look at how much money was made in total, cumulative frequency graphs, basically add things on to the side here and we're going to have to look at these three values and add them on ourselves. So we get 30 plus 35 plus ten. So we're going to get 65 plus ten, which is 75. So the total amount of three years is £75,000. Now, one thing to be careful of when looking at a question like this is I've written thousands.
And one thing which quite a few students may have actually missed is the fact that this is listed in thousands. So when looking at a graph, I find it's always useful to get your final answer and then really double cheque with the information in the graph. Are these years relevant anyway? Not especially in this case. If we were asked to make it across per year we might have to divide to turn this into a mean of £25,000.
And the fact this is listed here in thousands of pounds means we need 1000 after here. Okay. That's a very brief overview of the kind of problems and statistics. It's a very massive field but it shows the two major kinds of problems that you guys would be seeing when you're doing these questions. Here's why I'd ask you guys maybe to do at home, is to think about times where you've seen questions, particularly like this one here which has these averages which maybe aren't quite the same. If you have any questions, please do put it in the chat below and otherwise hopefully enjoy the video and it helps you with your consult.
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