A LEVEL MATHS: Standard Deviation and Linear Interpolation Exam Practice LESSON 18

hello and welcome back to another one of these a-level maths uh revision lessons so today we're going through again statistics and again we're breaking up the topics really really nice and small so we don't get overwhelmed um in this instance I guess we're going to go through interpolation and standard deviation again one of the trickier topics in kind of as statistics and it does come back in a22 as well and so what is interpolation well if you think about the word it kind of looks like extrapolation right and when you extrapolate it means you go outside that's what extra means outside the data range in order to predict things so you might continue a line of best fit for example so interpolation means inside so we're actually going to go inside these values and you're going to see exactly why well right now actually so what we want to do here is we want to add another column for cumulative frequency okay so we'll start with that so again that's going to be 7 then 17 then 32 then 51 and then 60. okay and this will just help us work out the median and work out what's going on exactly so to work out the median normally what we would do is we take the number of data that we have which in this case is 16 we half it so we're looking for the 30th value and again this is just to explain exactly how or why we need interpolation so if we look at the 30th value it's going to be somewhere in this grouping here right because we have 17 up to this point then we have 32 up to this point so somewhere in this boundary within this class is the median now the problem is I can't say that the median is for example 165. because I don't know if it's Midway through here okay however we don't have the original data so we can't actually work out the exact medium we have to work out an estimate for it and it turns out that the best way to do an estimate is to assume a kind of normal distribution inside of the class classes itself so there's a few people at 160 a few people at 170 but most people are towards the middle okay it's not exactly exactly in the middle but towards the middle and the way we do that is first things first you want to identify which class the um uh medium would be in which is 160 to 170 so it's in the class 160. 170. and now here's where we're going to start so we start off with 160 centimeters for our median so all you do is you write the lower class width or class boundary so lower so I'm going to basically write out a formula for you then what you need to do is add on the extra stuff that we have so when we go up to here we go up to 17 right but we want to get to 30. so we have 13 left over so basically that tells us we want to go 13 spaces into 160 to 170 13 data points into this class okay and now we're going to estimate what those data points look like as in what they're going to be so we're going to Plus 13. and again the where I got that is from the left over amount so all you're doing in this case is you're taking 30 and you're subtracting the cumulative frequency up to that point so again just as a quick uh quick go again so the 30 comes from the 30th value we're up to 17 here and next is 32 so we've got 17 values that's fine we need 13 more values okay which is somewhere inside this class width here and what we do is we divide it by the frequency of that class 15. again that's the frequency of the class we're talking about the brackets and then what we do is we times it by the class width so in this case the class width here is 107 160 to 170 so it's going to be a class width of 10 right 170 minus uh 160. so let's think about what this formula is actually telling me okay because this might seem a bit weird what we're saying is we go up to the lowest class boundary okay so 160 centimeters and then we're adding on the fraction of whatever's left over right so we have 13 left over that is going to be 13 over 15 into this boundary 13 15 into this class we're looking for 13 15 of this class width okay so that's where this comes from 13th 15th of the class width and we can work that out and again you can use a calculator of course and we're going to give it to the nearest whole number because all of our data are to the nearest whole number so we have 160 Plus 13 over 15. again if you're not too sure about that it's totally fine we're going to do a few more questions and you should hopefully start seeing the pattern so we have 168.666666 so we're going to round that to three significant figures so first of all I'm going to write 168 0.6 recurring so that is equal to 169 centimeters to three significant figures again remember to use your um units okay so what about an estimate for the mean well the mean is a slightly trickier Beast okay in order to work out the mean here's the formula we're going to be using now that we are in a levels it is the sum so remember this kind of symbol here capital sigma is some of the frequencies times the midpoints and we divide it by the sum of the frequencies okay now because I have a bit of foresight or again you can call this end by the way so if you don't want to call it the sum of the frequencies you can call it n okay number of data it's completely fine okay now I'm going to work these out separately because okay I'm spoiler warning right with standard deviation we need to work out this again anyway so it's a good kind of um it's a good practice or a good habit to have for you to work these out separately because in every single so this is like an actual exam question right interpolation then mean then standard deviation they do it all the time okay all the damn time so what you want to do is when you work out the mean work out this separately and then you don't have to work it out twice it saves you a few seconds in the exam okay so sum of f of x what is that going to equal well the frequencies and again this is just the sum of all of them so what I'm going to do is I'm going to take the frequency times my class width plus frequency times by sorry midpoint times plus frequency times like this so the midpoints of all these are fairly self-explanatory but we can add like a little column here 145 if you're not sure how to work out a midpoint you can just look for the middle number like I am but that doesn't always work sometimes the class boundaries are very strange all you do is you add the two numbers and divide them by two so 180 plus 200 is 300 380 and divided by two and that'll give you your answer so that's completely wrong that's a hundred ninety that's what I get for not reading the last one yeah so it's a good thing I checked by doing 180 plus 200 380 divided by 2 is 190 not 185 lesson correct yeah yeah okay so we have seven times 145 . if you're thinking this is tedious it is 10 times 155. now I think there are some of you who are kind of wizards with a calculator and might be wondering why I'm not doing the um we're in the calculator so keep in mind doing it all on the calculator using the method it's not the best practice because keep in mind if you've done anything wrong you'll get zero marks at the moment I'm at least getting method marks so for those of you those of you that don't know on this standard Casio calculator you have a which one is it a table um function on your calculator just beginning to menu you can enter all of the data into a table and it will calculate the mean and standard deviation and everything else for you but keep in mind some some mark schemes will say methods entirely done on a calculator are not accepted so you get zero or if you've made a small mistake anywhere and believe me it's very easy to do you'll get zero because you have no method marks because you've done no method so you can do it but you do it at your own risk and again I'd recommend checking a mark scheme for your relevant exam board if you're unsure if you're allowed to do it or not again many examples do this topic so work this out and again I'm the reason why I worked out the long way is because I know for a fact that I'm going to use this in the next part that just comes from experience so what we can do is seven times 145 again I can only tap this a lot faster when I'm on an actual calculator but and that gives me a thousand sorry 10 075. I can read I promise so my mean is just going to be 10075. over 60. an answer so let's call it 168 to three significant figures so I'll write out the full not the full number going which is equal to 168 centimeters to three Sig big perfect again someone starts telling you their height and starts giving you like three decimal answers then um you should run away okay standard deviation so this is a new one that you probably haven't really come across um before so the standard deviation is basically a measure of how far the data is away from the mean on average okay so if you look here our our mean is 168 so it's let's say in this class right now the the standard deviation is a measure of spread this is an important thing you need to know okay because again you're going to be using um use spread and you use average in order to kind of compare data sets right so you need to know what things are measures of spread so you have the range which we don't use anymore in a level because it's babies interquartile range which is used for box plots and now we have standard deviation which is another measure of spread the smaller the standard of the standard deviation the smaller the spread think about the word right deviation means how far it's how far you're away from the normal right and standard means average so standardized testing for example is average testing it's when you grade to a curve to an average so standard deviation is a measure of how far away data points are from the mean on average okay so the more spread out the data the bigger the standard deviation the less spread out the data the lower the standard deviation okay bigger number more spread now to work out the standard deviation here is the formula it's given a lowercase Sigma as a letter and again not all examples actually give you the formula and there's a huge mistake that people make they say these two things are the same they're not okay sum of f of x is what we worked out here sum of f of x squared is not this number squared okay it's completely different the sum of f of x squared would be 7 times 145 squared you squared just 145. okay so that's a very key thing so we need to actually work this out separately it's going to give us a different number to just 10 075 squared so be really careful with that they're not the same thing again lots of people make that mistake and unfortunately I actually deleted my so if you aren't an idiot like I am you can actually just go back through your calculator and square the answers and as in square the X values in fact I I'm pretty sure you can actually get it back so actually no because I pressed answer divide right oh there we are there we are super as often the calculator does do that I was hoping it did so squared squared cool as you can see it doesn't give me this squared uh never mind I put that comma in the wrong place 1.7 million cool so now we can actually put this into our formula so sum of f x squared is one seven zero two five seven five over 60 minus 10 075 over 60. left again super easy to make a mistake okay so let's try and be smart about this answer whoops okay that gives me 13.4 to 360 figures oh okay hopefully that makes some amount of sense you can technically give this unit the units would be the same as centimeters because what they're basically saying so if you want you can give it units of centimeters is that on average so most of the data points are within 13.4 centimeters of 168 centimeters that's where most of the data is and all of it but most of it okay so hopefully that makes sense that took us um quite some time so we'll go through another example again we'll do it nice and slow there's no rush we can just keep taking our time and again once you start getting good at this you'll be able to just hammer it out very very quickly so again you lineage of interpolation to estimate the median of the coded time so I'm going to add a little cumulative frequency and again to make sure we're doing it right it says there's a random sample of 50 employees which means our cumulative frequency should eventually add up 250 and plus 14 to be 39 it'd be 43 and 50. cool so it looks like we've done it right okay linear interpolation so again let's go down okay and we're going to do n over 2. 50 over 2. we're looking for the 25th value so the 25th value is going to be within so 1 10 29 so it's going to be within this class here 10 to 15 okay so we know it's in that class so what we do again if you remember the formula we take the lower class boundary 10 we Plus what's left so we want to get to 25 and we only managed to get through 10. so 25 minus 10 gives me 15. and we divide it by the frequency inside that class which is 19. so we're 15 19 of the way inside of that um class and the class width is 5. and we get 13.9 I'm going to do it to uh 3sf okay and the reason why I'm doing that is it's time in minutes we we can't give decimal for minutes that's fine so we don't normally go to like two or three decimal places so I'm going to leave it like that okay and that's the median of the coded times perfect Part B estimate the standard deviation of the coded times okay so standard deviation you know what we need to do we have to do sum of f x and then some of f x squared as well so again if you want we can write out the midpoints 2.5 is 40 divided by 2 is 20. and 25 plus 40 is 65 divided by 2 is 32 and a half okay so let's go again might be tedious might be mind-numbing but you only have to do this once in uh each exam paper so hopefully that gives you some level of comfort I suppose I put the frequency second that kind of bugs me should I change it no I'm not going to change it now 7.5 times I think that was nine Plus 12.5 times 19. 20 times 14. 32.5 times 7. what does that equal and so technically speaking if you want to be lazy which I do I can also work out some of f of x squared without writing out everything by just doing on the calculator which which you know would be fine as well 2.5 times one I mean have mercy on me right okay that is 815 perfect now again to work out some of f of x squared all I'm doing is squaring the now I need to be careful because it was the second number before but because I'm an idiot and I wrote it first it's actually going to be the first number so yeah be super careful which one is actually your midpoint you need to be squaring the midpoint nothing else be really careful or just stick to an actual convention when you say it unlike me okay and then again we've used the standard deviation formula but I'll write it out one more time for you guys so sum of f of x squared Over N minus sum of f x n squared so if we sub that in now we get square root the answer button is going to be what that is n is 50. minus oops bracket ah crap bracket faction some f of x 815 over 50. lowest bracket all squared and you should get 7.988 so we could call that 8.0 actually no no no we we will need to make sure it's uh accurate actually okay oh so for part C the coded data was calculated using this formula so this one here so what they've done what they've done is they've taken every single piece of data every single one and they've subtracted 20 from it and then halved it so all of the information we just worked out now was actually for of the wrong thing so to speak so they've subtracted 20 and halved it so if I rearrange this right so I get 2X I can get back to the real data by doing this okay which is exactly what I want to do I want to find out what the standard deviation and median of T is now here's where things get a bit more interesting okay now the median and mean you can just sub straight into this formula okay so median and mean substrate in okay so once you have your equation for uncoding the data all you need to do so again this is for the median and if the question said mean then we'll just do the mean the same way okay don't worry about it then the um median for T is just going to be 2 times the median X 13.9 Plus 20. okay just like that 47.8 nice and easy now this works for the mean as well now here's the key thing sorry about the standard deviation it's only affected by times or divide okay now I'm trying to give you an example if I add 20 to all piece all data values right the average distance from each other doesn't change right so let's say we have one two and now I'm going to add 10 okay now hopefully you can see that the mean and the median will change right the median is now 13 and it was 3 and the mean is going to be 13 instead of 3. so it does change hence we need to include it in our calculation now the standard deviation let me ask you this okay it's a measure of spread has the data become more spread out if I do 5 minus 1 here I get four if I do 15 minus 11 I get 4. the standard deviation is not affected by adding or subtracting and now if instead I times the by 2 then the spread does change right I'll get two four six eight ten now the range is 10 minus two eight instead of four it's doubled so all you need to do is actually do the the timesing so in this case since they halved my data I need to double the standard deviation that's it I don't need to add the 20 because adding and subtracting numbers does not affect standard deviation it doesn't affect spread it only affects the mean and median the average Okay so in this question is going to be 7.99 times 2. which would be 14.98 [Music] no 15.98 okay hopefully wait up am I being stupid no 15.9 is correct cool I hope that makes sense hopefully next question next two questions are a bit more fairly similar however check this out sometimes if you pray hard enough the gods of Edexcel and off call will listen to you and they will sometimes give you these values in the um exam in which is you don't have to work them out and it makes these bits a bit faster and to be honest I personally think they should give you it because there's no real reason not to have it okay so let's again back to it we want a commute cumulative frequency column so we have 4 23 64. 90 99 100 which again matches the 100 people here so we know it correct and again it's important to check that because there are people that have actually made mistakes in this and myself included the thing is if you make a mistake that's fine if you can spot it so make sure you check if they've given you the actual total if they have then you're in luck you can just quickly double check it and you'll be fine but make sure you do double check use linear interpolation we're going to do the exact same thing we did before all you do is for the medium we do this time a hundred over two we're looking for the 50th value which we can see is in this class here which is in the 20 to 29 class there's actually something else we need to do okay now when I look at distance that is continuous data right it doesn't have um like half numbers in it so we need to actually make sure we add these boundaries there has to be an overlap we cannot allow it to not have an overlap okay so you always just take half away so these are our new class boundaries okay class widths technically you don't need to do this one but I'm going to because it'll make our life a bit easier later on okay now you might see why I actually spotted this right away well not right away but when I did this because again 50th values in this class the first part of the linear interpolation formula is taking the lower class boundary it ain't 20. it's 19.5 so then you pass how many we had left so we got up to 23 we want to get to 50 so that's 27 left over and the frequency inside that class is 41 and you times that by the class width so 29.5 minus 19.5 which is 10. so again if you look a couple things changed here right the class width here is 10 but here it's 29 minus 20 which some people would write as 9 when if it's including 29 should be 10. so yeah be really careful of that if you don't have continuous data it's just like in the histogram video don't have continuous um sorry if you have gaps between your boundaries and you have continuous data make sure you add those extra class widths so 19.5 Plus open bracket 27 over 41. multiplied by 10 other than the least efficient way possible and you get 26.1 to three significant figures is that a reasonable thing yeah it's miles we don't want to go too deep into miles okay estimate the mean height well again with the mean by the way the mean is sometimes given the whatever the variable is which they didn't give us one so maybe X or something with a little hat on it just FYI it's the sum of FX Over N so 26 51. over a hundred so 26.51 miles yeah so it didn't round it so I'm not gonna like that and then lastly estimate standard deviation again so can you see how quickly the start quickly uh how quick these questions can be once they give you the values okay so again in more of the recent exam papers they've given you the values which is an absolute blessing but again they don't they don't have to which is you know fun so again this is sum of f of x squared Over N which is a hundred minus the sum of f of x over n squared so if you notice all you do is you take you're just taking away the mean squared right so what will that give you let's have a look see so if you wanted you can actually write minus this answer squared for this bit so I'll show you okay and technically speaking I could just do minus answer squared because this thing here the sum of f x Over N squared is just the mean squared and that gives me the right answer but you don't have to do it that way if you don't want to and that gives me 11.1 so the mean is 26.51 Miles with an average deviation of 11 miles and that's it so there's a six marker in a couple of minutes I don't think we took that long I think we took about four minutes three or four minutes so you can really scrape together these marks so that what about this one this one becomes slightly slightly different um hopefully you can see what the difference is it's not grouped data okay this is just the times okay so you need to be able to do standard deviation and mean and everything uh when you just have like a normal regular run-of-the-mill data okay not grouped and it's continuous so in the meantime taken I'm hoping this is um fairly self-explanatory all you need to do is for the mean what you're doing into doing the sum of just X in this case over N I want to work out the sum of X because again I can see the standard deviation bit just as a bit of a word of warning you're always if you have mean you're going to work out the standard deviation afterwards so practice all we need to do is add up all of these numbers so 45.2 plus 46.9 plus 46.1 plus 46.2 45.4 35.1 also for those of you that want to use like just the table Method Keep in mind that entering something into a table generally takes about as long as just working it out but I almost lost my place there for a second I was going to cry um 46.4 and 45.7 good Lord and I didn't show you the answer it's 551.8 um while we're here I'm actually going to work out the sum of x squared which you need for the standard deviation because uh I really don't want to enter this again so keep in mind it's not just this squared okay it's not going to be the same thing so essentially if you add up a bunch of it's like Pythagoras Theorem right if you square and then add numbers you get a different value so if you just add them and then Square them afterwards like Square the answer so please keep that in mind when you're doing this you can't just Square the value like the overall number it will not be correct okay again think of Pythagoras a squared plus b squared is not equal to a plus b all squared that gives you 25 380.38 okay so the mean we just do that and we just divide it by the number we have sorry the number of data points we have which is 12 in this case so yeah I think this would be the first topic that might give you a slight challenge that we've done in the um statistics so far again if you guys think I should like speed it up then let me know um because I know you guys have got exams approaching uh 45 point I'm gonna call nine eight because I don't want to round it too much um but I think taking the slower approach is actually better because at least then you're getting through it but this is probably the first slightly difficult topic I do mean slightly um but I can still catch people out so again the lesson after next so the next lesson is regression which again people find very tricky the lesson after that is when we're starting to probability specifically with the binomial distribution which people find extremely tricky and then we're going to do hypothesis testing which people find even worse so standard deviation if you don't have any kind of classes it looks pretty much identical if you don't have a class with a class boundary so a midpoint then you just don't have the midpoint in here so it's basically the same equation okay so subbing those in if it's our answer square root uh I've done a while left over there 25 380.38 over board whoops 14. minus 551 8. 16.1 wait wait yeah 16.1 not bad scared myself there done easy now we have oh hang on a minute yeah it's over 12 not over 14. I don't know where 14 came from please okay well that made a big difference then at 0.751 a huge difference okay now we have to work out the quartiles so again with the quartiles or I mean hopefully you're okay with doing this the median is the middle value so you do n over 2 which is 1202 so you're looking for the sixth value right one two three four five six so 45.1 uh the lower quartile so we call that q1 is n over four which is going to be the third value one two three so technically we'd need to put this in order first but this is essentially what you do right you just you have the data in order and then you count them along okay so ordering that data bit tedious not gonna lie but we can um do it okay so depending on whether or not you want me to kind of do all of that so