 Hi everyone, it's MJ and welcome back to the course on stats. Now in the previous videos we looked at the sample mean and we looked at its expected value and its variance and we started doing the same for the sample variance but it was quite a long video when we looked at just say the expected value. We did all this crazy maths and we went a lot in depth to try and answer the question why we use 1 divided by n minus 1 instead of 1 over n and I couldn't help it but I gave a little history lesson as well and what that did is it kind of ran out of time for for that video that video started getting a little bit too long so what we're going to be doing in this video is we're going to be continuing but this time we're going to be looking at the variance of the sample variance so I like that I like that we're looking at the variance of variance it's quite a cool little title and we're also going to throw in what is the distribution of the sample variance it's quite interesting so a quick little recap we saw that s squared is a random variable which means it has a so distribution and we saw that the expected value of s squared is equal to sigma squared when we use something known as the basal coefficient of 1 divided by n minus 1 so we're using that so that the expected value of our sample variance is equal to that of our population variance but now what we're going to be doing is looking at this what is the variance of s squared I mean if we had to try and do it the old-fashioned way we would see that we do run into a little bit of mathematical problems if we had to go this way fortunately fortunately there is a beautiful shortcut which I'm going to be showing with you guys and I'm going to show you that the answer to this the variance of the sample variance is equal to the following result and I mean this is kind of what we want we want that as n increases so we will have more information so we'll be more confident so we will expect the variance of the variance to decrease as our sample size increases but now the big question is well how how did I just get to that you can't just state that well in a weird way we can just state that well what we're going to be doing in this video is because we need two results the one result is that the sample variance and the sample mean are independent this happens when we use a normal population or when we're sampling from a normal population but what also happens is when we sample from a normal population we get the following result and this result is very important if you want to try and prove it be my guest it is above the actuarial syllabus it's not impossible I mean you do need to use those moment generating functions and a little bit of your intelligence to do so but in the actual science we are just given this result which is lovely because it makes our life a lot more easier the reason why it makes our life so much more easier is because look at this we you should be you should be recognizing this this is the good old chi-squared distribution and I should just copied and pasted the results from our probability distribution course just they don't have to go all the way back and find it and remember chi-squared distribution is the special case of the gamma where alpha is equal to v divided by 2 and lambda is equal to a half and we know that the gamma function of a half is equal to the square root of pi but don't worry don't worry we don't we don't need to know any of this crazy maths for this part we don't even know any of that craziness all I'm interested in is this this is what is the awesome part and that is the average is equal to our degrees of freedom and the variance is equal to two times this degrees of freedom degrees of freedom is given by that symbol over there so let's use that let's use that of our chi-squared and we see chi-squared n minus one where n minus one is well n is equal to the total observations which means that the mu in this situation is going to be equal to n minus one and sigma squared is equal to two times n minus one so why is that pretty cool um I think hold on let's maybe explain chi-squared a little bit more just just in case people are like hold on what what exactly is this chi-squared thing because it is a weird symbol it is a weird symbol a little bit of the history behind it Carl Pearson absolutely loved this thing it was like his little baby if I had to quickly just draw it out for you it has these two following shapes the the shape changes as your degrees of freedom change we will see that it is positively skewed when our n is small and as n gets bigger it becomes more symmetrical okay but like I said the results that we are after is that the mu is equal to the degrees of freedom and sigma squared is equal to two times those degrees of freedom and that's what we're doing we're applying that to this distribution here which is using n minus one and what this is going to do is going to make a lot of maths easier because remember how we did this whole long expected uh formula over here um if they ask you in the exam and it's like worth 10 marks then you do it this way but if it's only for like say two marks and they give you the extra information that you are sampling from a normal population then you can use the following you know that the expected value and you use this result over here you can say that the expected value of n minus one uh s squared divided by sigma squared is going to be equal to n minus one we know that because we know what the expected values is of our chi-squared distribution that's where that's where that magic's coming in um and I mean if we had to work this out what we would see is that n minus one divided by sigma squared expected value of s squared is equal to n minus one where we can now divide this on both sides and we're going to get one divide this on both sides and we see expected value of sigma squared is equal to sigma squared so what we've just done is all of that maths in like two lines um and we can do exactly the same or we can use that same approach where it now comes to the variance so if we take the variance of n minus one s squared divided by sigma squared we know that it is equal to two n minus one how do we know that like again like I'm going to say again we're getting that from the chi-squared properties okay so let's do it what we know about variance is when we take something out of the variance we have to square it uh this term is already squared so we're going to have one over sigma uh to the power of four look at that um and then we're also going to have n minus one squared uh times the variance of s squared equal to two n minus one if we multiply both sides by sigma four we're going to have um you can just throw sigma four on to that side and then let's divide so we will have something like this the variance of sigma squared we make it the subject of our formula it's going to be equal to n minus one times sigma to the power of four divided by n minus one squared which we then cancel out those terms and we have two times sigma cubed divided by n minus one bam which is done that is that is the result that we were looking for and you can see the maths is not that difficult so like I say in the example they ask you to prove this and it's only with two marks please don't go do all the mathematics like we did in the previous video so why is this so important why do we need to know what the variance of sigma squared is well remember what I spoke about in the video about the sample mean we know that the following from the central limit theorem it is equal to uh all is distributed approximately normally but this is in the situation where our variance is known we can use the sample uh variance when this little guy over here is unknown and what we're going to see is x bar minus mu over s over n we can get this t distribution now the reason why we can get the t distribution is if we remember from our probability distribution chapter t over n is kind of like a weird blend of the normal um distribution and our little chi squared distribution so those two things combined give us the sky over here but we're going to talk more about student t in the next video what I just wanted to do in this video was show you the result that the variance of the sample variance is equal to that anyway thanks so much for watching and please let me know if you have any questions keep well cheers for more content study advice and exam questions enroll in statistics by mj link in the description below