AQA GCSE Maths FULL SUMMARY Topic by Topic Part 6: Statistics

hey guys just to let you know this lesson is in collaboration with flip Academy a free online platform for all of your GCSE a-level IB and cbse revision out now on web at www.flip.academy and also as an app for Apple IOS and Android it really is better than all the other GCSE platforms out there it covers all the topics in great depth and has an app that you can download your courses on and use offline hello everyone and welcome to flip Academy today we're going to be looking at the statistics topic of the AQA GCSE math specification and the first one is infer properties of populations or distributions from a sample whilst knowing the limitations of sampling so first of all we should go through what kind of sampling is okay so a sample is essentially a fraction of the total population and when I would say population I mean population is all people so let's say you had people in the UK so about 65 million people okay that's the population okay now the thing is if you wanted to find out something about let's say how many hours or 65 million people in the UK wanted to work as you can imagine that's not the easiest thing to do is it so using the whole population uh the advantages to this are that it's completely representative right which basically means it's accurate because you are literally asking every single person okay which means results are more reliable but as you can imagine there are some this disadvantages right so some cons now the cons are that it's extremely expensive what so really think about 65 million people let's say in your school there's probably around 1 000 students right maybe two thousand let's say two thousand could you imagine asking every single one of them a list of questions right now do that a thousand times that's two million then do that 30 times and you're getting close to what it would be like to ask every single person in the UK it's extremely extremely tricky this also means it's very time consuming now the funny thing is the UK actually does do this there is a census that goes out every five years where it's sent to every single household but again this costs hundreds of millions of pounds so it's very very expensive and time consuming so whenever you're looking at a question you have to think about what you can infer but also know that there are limitations okay it's not going to be 100 reliable if you do sampling which is where you take a fraction of the people there are going to be some outliers are going to be some misdirection so as an example I'm just going to think of something so for example uh if I type something up a bus company uh wants to find out how many people travel on local buses in a town so what they do is they call 100 people 19 people say that they used a local bus within the last week the company concludes 19 of people use local buses okay and then what they may say is either what are the limitations oh why could this be wrong or in accurate so these are the kinds of questions you can expect okay so let's think about it let's think about their method uh first of all I can already see an issue and this is always so this is a little trick it's a bit of a cheat the population the sample size is always going to be too small so any sampling question if they say what's a limitation what's wrong with their method how can they improve it something like this say that the sample size is too small okay because it always is unless you ask literally every single person okay so sample size too small so that right there we'll get you a Mark should they ask you that kind of question okay now some other things let's think about it right let's look at the question they asked they essentially asked if they used a local bus within the last week okay well what if someone used it two weeks ago or eight days ago they it doesn't count on them right so people could have used it more than one week ago so if they want to find 100 people use it regularly but everyone uses it that some people use it every other week they're not going to be counseled in this right there's another thing yes depending on when they fund them some people are at work right so uh people phoned so if we say this was during the day then people phoned didn't work hence they won't use a bus right they're less likely to use a bus and if you do in the evening you phone people during the evenings well then those people aren't on buses so then there's a good chance that they don't use buses either so depending on when this sample was done what time of day or what day of the week this can be some this can bring up issues another thing they like to do is they um sometimes like a some someone will kind of go up to people on the street I'm sure you've seen them on the High Street the people you're trying to avoid eye contact with and they go up and ask people and they try and find information and the problem with this is that if you're in the High Street during the day then chances are the people you're asking aren't employed because everyone employed will be at work right so this could be like either they're retired or they're you know again not working or they work strange kind of jobs not normal nine to fives so again those can be limitations so that's what we should be thinking about whenever you kind of talk about populations and distributions from a sample okay they are never 100 accurate and again well you can always say sample size too small make sample size bigger that's how you make it better and it's always correct so it's fairly easy couple of marks that you can kind of get the next thing is you need to be able to draw a bunch of different types of diagrams here bar charts pie charts and pictograms so bar charts pie charts and pictograms you should know from year seven to nine okay but with frequency tables and also kind of Time series graphs they can be a bit trickier and you also need to know their appropriate uses so time series graph obviously you're going to use it when you have a set of data going along and you have numerical data that's the first thing and it also has to be um discrete data so it can't be grouped okay and they've plotted the first four points and we just have to plot the next few so the Dance 2015 2016 is 113 each of these goes up in two and I know this because each square there's five squares between 100 and 10 120 so there has to be it has to go up in twos so 112 and then 13 would be in the middle for 2016 so if we can try and draw it halfway okay it doesn't need to be a work of art 2017 was a hundred nice even number that should be quite easy for me to do 2018 was 90 and then 82. so 90. and 82 . now if you look at how they've joined up the dots with the time series graph or a scatter graph you're going to join them like I connect the dots okay draw nice straight lines again you would use a ruler in order to get that and that would be too much so things they're looking for here is a you've joined them up with straight lines not a curve or anything and then you've plotted it on the correct so each coordinate is correct basically basically use a graph to estimate the average daily uh time per student in 2020 so if I write the number or the year 2020 where would I expect it to go well if I look it kind of looks like it starts off kind of it Steeps and then it Steeps it steepens and then it kind of gets less steep towards the end so I'd expect it to be something like this so that would be 78 76. now again it's an estimate so they give you a ton of leeway assuming that you don't kind of drop all the way to zero or go up right you have to be going down that's one criteria and the other one is you have to be going down by a reasonable amount right so if you did maybe 70 that's also accepted in this question okay and again this is from the June 2020 paper too next question so uh with the kind of different types of dates that you can have with pie charts that's really good for when you're trying to look at a proportion okay so a percentage bar charts you need to have discrete categories so there can't be any overlap and with frequency tables again you're looking at something similar to this you have numerical data okay pictograms also numerical data but you need discrete groups next one we have to so support it only to screen continuous data so you want histograms and cumulative frequency graphs okay so first of all we have November 2021 uh paper one we have the cumulative frequency masses of 40 necklaces okay so again we're going to have continuous data for this a jeweler buys every necklace with a mask greater than 21 Grams you just got to estimate how many she buys okay so 20 each of these goes up in one so if we have 21 here then what we're going to do is draw a nice straight line a little dotted line going up from 21. read that across and again looks like it's going up in ones so this is 16. so 16 of these necklaces are less than 21 Grams which means 14 minus 16 of them are above 21 Grams so if we do that that's going to be 24. necklaces not bad right now turning this into a box spot so box spots actually come up later in the spec but this is part of the question I thought it'd be quite good to go through now we need to draw a box spot to represent the data so the first thing we're going to do is I'm going to clear that okay with a box plot you have lines at the lowest and highest mass so let's start with that so zero one two three let's draw a line going there and then also at 28 so one two three goes up there cool and the next three lines you have are for the lower quartile the median and the upper quartile okay okay now if we're looking for the lower quartile that is a quarter of the value so all we need to do is we take n and divide it by four where n is the number that we have and we have 40 necklaces so we're doing 40 over 10. so we're looking for the 10th not the tenth looking for the we are looking for the tenth I've just written it the wrong way around the 10th necklace okay so how do we find that we've got a cumulative frequency graph if we go up to the 10th value we're going to draw a nice dotted line all the way along and all the way down and it looks like it has a mass of 18 grams okay so if we draw a line at 18 now so that would be over here and I like to make them slightly smaller to make it look more like a box plot but again it's not completely necessary for the median you take the number of dates that you have and divide it by 2 instead so This Is 40 over 2 which is the 20th necklace they go back up 20 let's read that across and straight down that is going to be 22 grams cool so 21 22 I say it doesn't have to be a work of art but it has to be better than that so if we go like this that looks pretty decent and then for the upper quartile we do three n over four so what that would be is 3 times n so 3 times 40 is 120. over 4 will be the 30th necklace it's going all the way up again 24 grams perfect so 24 is right before here so now don't forget we're not done we have to actually make these boxes so in terms of the marks you get marks for working out all of the quartiles and again this shouldn't be too bad okay q1 is always the number of data divided by four Q2 is the number of days divided by two Q3 is three times the number of data divided by four so basically three quarters of the number half of the number a quarter of the number so and again to read them off since it's cumulative frequency and you're looking for what number it is keep in mind when you're looking for the quartiles you need to order your data they're already in order and have a cumulative frequency graph right the mass goes up going to the right and the number goes up going up so hopefully that's not too bad the next question is all about a histogram so with histograms what do you need well the first thing you need to work out okay is you need to work out the frequency density so some people like to do the to do class width and then frequency density so frequency density all that is is it's the frequency over the class width and so what's uh the class width well the class width is the width of the class so all you do is you take the bigger number and subtract it from the smaller one so five minus zero is five 10 minus 5 is 5. 20 minus 10 is 10 50 minus 20 is 30. so remember with a histogram we don't have to have consistent classes they can all be different and now because this always turns up in a calculator paper we can just divide them so frequency divided by class width so 18 over 5. and we have 3.6 and then just change that top number to at 23. and we get 4.6 uh 17 divided by 10 I don't need to calculate for that 21 divided by 30 would be 0.7 it fully kindly labeled the frequency density uh axis for us sometimes they don't and we've got this so what are the key bits about eight about a histogram so histogram the difference between a histogram and a bar chart is a the type of data right here we have these kind of continuous groups right because of the bar chart we have very distinct groups we are not going to have any spaces between our bars each bar will come off of the next of the previous bar and the last thing is that the width of the bars do not have to be consistent so zero to five we're going up to 3.6 so you have a look at this it looks like it goes in 0.1 that's really nice actually so it goes up to here and that's from zero to five right cool so again you would draw this with a nice straight line all the way down to where five is it's not bad along now my next bar is going to come off of this first bar so 5 to 10 goes up to 4.6 so I can literally draw it coming off here and on over fact 10 is over there again keep in mind I can't do it 100 accurately this is pretty good but yes you do need straight lines okay 10 to 20 is 1.7 so this is going to be a slightly easier for me to do 1.7 will be up to here and lastly 20 to 50 is going to be 0.7 very nice and that is a perfect histogram and look at that that is four marks and I definitely did it faster than four minutes so not too shabby so again key things to remember you need to work out the class width first then you need to do the frequency density take the frequency divided by the class width done and then plot it not too bad for formats I think so box spots quartiles and into quartile range and then it also says analyzing the player distributions of data sets central tendency blah blah blah blah blah let's have a look okay so this is all about inter this looks very complicated but it's not that bad okay so the Box spots we've kind of already gone through it so again just so you know we have the quartiles right quartiles are as follows the lower quartile is again you just take n over four find out what value that is median or again middle quartile velocity Q3 n okay not too bad now in order to compare data sets we like to consider first of all spread okay first we we measure spread is just the humble range okay and that is the biggest value minus the smallest right that's our best friend come across this a million times done it since Primary School easy however there's a big problem with the range okay and it comes from this outliers okay let's say we had a nice data set and it was like two three four five seven hundred right it can happen it can happen now if I told you you know the range here is going to be 700 minus 2 698 right that's not representative so I know what you're going to say well what if I just done get rid of the biggest value okay yeah you're right if you do that then you have four five minus two which is three which makes a lot more sense so that's the first thing that they expect you to be able to know is to get rid of outliers before doing the range okay however there are cases where you don't have access to the raw data right they may give you what the quartiles are right and this isn't so weird so if any of you want to become data scientists in the fir in the future or use statistics at all this is quite important you tend to get data in aggregate which means you have thousands upon thousands of pieces of data are you going to go through them and get rid of every outlier manually okay well if you have a million pieces of data you're going to go through all of that because that'll take you hundreds of years and I'm not even exaggerating so there is something else we can do there is a light at the end of the tunnel now we're big boys and we're in GCSE you can also use something called the interquartile range so into quartile range which I'm going to abbreviate to IQR and it's simply the upper quartile minus the lower quartile why is this better well if you remember how you work out the median well if you remember anything about the median it's not affected by extreme values in other words it's not affected by outliers because you cost them off one at the end right it's true for the other quartiles as well Q3 upper quartile on the low quartile are not affected by outliers which means when you take the interquartile range you are getting a much more accurate measure of spread because it doesn't seem to count outliers right if you have a piece of data that's like 1 million the rest of your numbers are around 100. it automatically gets rid of it for you right so computers are very good at all like sorting numbers from lowest to highest if you use the interquartile range you don't need to worry about outlines because it gets rid of them okay so that is really really important the next thing they like to talk about is central tendency now this is a bit of a funky definition it's a bit of a funky word and this is literally just working out average right so mean median mode right and also say modal class that's the same as the mode right a brief overview the mean is just the sum of all the data over the number of data and the mean is your bread and butter okay it's really really important because this gives you a very accurate value of what the middle value is okay but it has some downsides the major downside is it doesn't take into account outliers in the sense that it gets affected by them quite severely so again if I had like a large set of data and I took the mean if there's one value at a million the rest are at a few hundred my mean is going to be close to a million it's going to be hundreds of thousands which doesn't represent most of the data the median uh the upside and downside is that it's not affected by outliers but at the same time it doesn't exactly take into account extreme values right which sometimes extreme values make sense if you're talking about the wealth of individuals in the world there are people that have a few hundred pounds to their name they're individuals with billions to their name if you want to work out the average income of the planet we should take into account extreme values right that makes sense but also looking at the median can help because maybe we want to get rid of the people earning billions and get rid of the people earning nothing right because again some people earn nothing so again we need to keep those in mind uh the mode is the most common right it's just the most common number and the mode is very rarely used very seldom very very seldom used um something that you might consider for example is if you're talking about shoe sizes so let's say you're a shoe salesman right um you'd like to order the correct number of shoes in different sizes right so if you know that the most common shoe size is an eight for example you know to order much more size eight shoes than any other size but again it doesn't take into account most values right it just takes into account the most frequent value so again if you had shoe sizes you know there's lots and lots of shoe sizes but we have a spike in the middle it doesn't really take into account the average okay that has a lot of sentences a lot of words I hope it makes some amount of sense um but again June 2020 paper 3 hopefully this will pique your interest just simply drawing a box plot but then also as you can see here we're probably going to be comparing so Amari and Ben each play a game here are some information about Amari scores you have the lowest highest blah blah so again I told you that the lower whisker goes where the lowest value is and the highest whiskey goes with the highest value is now the lower quartile uh we're going to I like to make it a bit shorter let's make it look a bit more like a box of course I was at 19 which is here and the median's at 17 this day is really really spread out interesting okay and again perfectly straight lines as always and cool that's it two marks easy peasy to give us all the values not too bad now Ben has another box bar and it says here who had more consistent scores what's consistent mean right consistent in this case means you get this well not in this case consistent means you get the same thing over and over again right let's say if you're consistently late by five minutes it means that most of the time you're late by five minutes so in this case we want to find out who gets the most consistent scores means who gets the same score most frequently right so in this case we're looking at the lowest spread okay because if you have a lower spread it means you're more consistent you're getting the same numbers or the same score in this case more frequently okay so if you look at that I mean just looking at this box plot most of our values are between 13 and 19. that's a big range most of our values here are between 12 and 15. which means Ben is more consistent however work out interquartile ranges again matters very seldomly oh it looks about this and again right it could be a smaller smaller spread so let's work out the spread for Amari which I already know right this spread is going to be equal to in this case the interquartile range so if we have a look at our values here 20 minus 12 is this is for the regular range would be eight this would be nine they look to be similar inconsistency right if we just look at the biggest and the smallest value if we look at them at the majority of the values we should keep in mind that the interquartile range takes into account 75 of the values yeah looks a bit different so the interquartile range is going to be the upper quartile minus the lower quartile so 19 minus 13. and let's look at Ben so we're quite lucky that they do kind of tell us what to do right work out intercourse our range 15 minus 12. three he's a lot more consistent so all we have to do now is write Ben more consistent don't worry about writing like a caveman it's not an English test then more consistent good not too shabby I hope now the last thing I wanted to go through is the mean but I already told you absolutely the mean is just the sum of values divided by the total number of values however you can also have grouped frequency tables okay so what you need to do here well they give you a bit of a hint this is always going to be an estimate so if you ever see the words estimate of the mean it means you can't actually get the real value because we don't have all of the data they've grouped it together so all you need to do for a grouped frequency table if you want to work out the mean is you find the midpoint of each now the midpoint is quite easy to work out so sometimes you can look it look at it look it look at it however if you want to work out the real thing all you do is you add the two values together and divide by two now you're probably not shocked to work out those two and again you're probably not shocked to work out this would be seven probably not shocked to work out that this would be 12. however they can give you some values which aren't as nice right so if you ever need to work it out just do five plus nine okay so again they can give you values that don't divide nicely and what you want to do next is you want to take the frequency and times by the midpoint 6 times 2 is 12 77 and 36. now what is the mean with a group frequency table well the mean is the sum of frequency times midpoint over the number of data so in other words you can do this all in one step on a calculator okay some people like to do that that's completely fine but in this case it'd just be 12 plus 77 I would recommend doing it this way because a they've given you the facilities to do it but B and this is probably more important it's easy to make mistakes when you're typing in a long calculation on a calculator so making it as short as possible is a clever thing to do so again I'll do it in a fraction just make it easier 12 plus 77 plus 36 over 20. 6.25 give your answers a decimal that's a good sign that's three marks I'll find that it's pretty good pretty good playing statistics to describe a population so again they this is very situational which is why I couldn't really find many questions to be exact about this this is more like a skill okay and applying statistics to describe a population what you'll need to do is again what they might have is a list of Ages of men and women and you'll work out the mean of both ages and they'll say on average who is older men or women something like that and those are the kinds of things you need to do you need to work out mean medians and modes uh spreads so range and interquartile range in order to find out stuff about a um population and after that this is pretty uh interesting it's a really big spec point but it's actually really quite easy you need to interpret scattergrass or bivariate data using correlation however you need to know that it doesn't indicate causation so what is causation well okay let's say you had right and let's say it's time of day right and this is like the number of shirts sold right now what you might expect is you might expect that or let's just say this happens right uh early in the day there aren't really many sales so you have like dates like this and then at lunchtime it kind of stagnates and then in the evening it spikes up right and so overall you can see that's your line of best fit overall we can see that as a time of day increases the number of shirts sold so that's a positive correlation right shirts sold increase with the time of day now can we say that as the time of day increases it causes more shirts to be sold no instead the reason why I pick data like this is that as the time of day goes on people start getting off work which is when they would go to buy clothing after work right whereas in the middle of the day they're at work so the actual cause of this is the fact that people are off work not because the time of day goes on it's a very minor point but you should just know that in terms of correlation it doesn't imply causation okay that's if that just recognize that sentence right remember that sentence okay can I say this no because correlation doesn't imply causation okay so in terms of correlation I did speak about it briefly where this is positive if um the line goes up basically so if the data tends to go up it's positive correlation if it goes down it's negative correlation and if it's like this whether there is no line it is no correlation so graph a graph B would be no correlation what about graph a well they give us a few more options strong and weak right so weak correlation would be something along the lines of where it kind of you know it looks like it might be going down but we can't really be sure that's weak correlation with this I can see a pretty much straight line going down so that is a strong correlation because I can see the line extremely clearly and it's negative because it's going down too bad is it for the statistics module so I hope you enjoyed all of these videos in terms of the AQA maths and I'll see you in the next one