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# Population standard deviation

## Video transcript

let's say that you're curious about studying the dimensions of the cars that happen to sit in the parking lot and so you measure their legs and so let's just make the computation simple let's say that there are spot there are five cars in the pocket parking lot the entire size of the population that we care about is five and you go and measure their lengths one car is four meters long the other car another car is four point two meters long another car is five meters long the fourth car is four point three meters long and then let's say the fifth car is five point five meters long so let's come up with some parameters for this population so the first one that you might want to figure out is a is a measure of central tendency and probably the most popular one is the arithmetic mean so let's calculate that first so we're going to do that for the population so we're going to use going to use mu so what is the arithmetic mean here well we just have to add all of these data points up and divide by five and I'll just get the calculator out just so it's a little bit quicker and so this is going to be this is going to be four plus four point two plus five plus four point three plus five point five and then I'm going to take that sum and then divide by 5 divided by 5 and I get an error an arithmetic mean for my population of four point six of four point six so that's fine and if we want to put some units there it's four point six meters now that's the central tendency or a measure of central tendency we already we also might be curious about kind of what is how dispersed are the data is the data especially from that central tendency so what would we use well we already have a tool at our disposal the population variance and the population variance is one of many ways of measuring dispersion it has some very neat properties which is it tends the way we've defined it as the mean of the squared distances from the mean tends to be a useful way of doing it so let's just do that let's let's actually calculate the population the population variance for this population right over here well well all we need to do is find the distance from each of these points to our mean to our mean right over here and then take the meet and then square them and then take the mean of those two squared distances so let's do that so it's going to be four minus four point six squared plus four point two minus four point six squared plus five minus four point six squared plus four point three minus four point six squared and then finally losing running out of space plus five point five minus four point six squared and then we're going to divide all of that I'm going to divide all of that by five to get our population variance and so what's that going to give us let's get our calculator out so four minus four point six squared that's negative point six squared negative point six squared is going to be the exact same thing as point six squared so let me write that zero point six squared plus four point two minus four point six is negative four point six a negative point four but when we square it the negative is going to disappear so it's going to be plus point four I'll just write point four squared and then we have five minus four point six that's 0.4 so plus 0.4 squared four point three minus four point six that's negative point three the negative goes away when you scare it so it's going to be plus 0.3 squared plus 0.3 squared and then finally five point five minus four point six is going to be 0.9 so plus 0.9 squared point nine squared and that and then we will divide by the number of data points we have and we get 0.316 or if we want to write it we could write Z this is going to be 0.316 now let me ask you what is a mildly interesting question what would be the unit's what would be the units for this population variance we since we're carrying we're happening happen to care about units in this video well up here this is 4 meters minus 4.6 meters 4.2 meters minus 4.6 meters so these are all meters these are measurements in meters we saw it up here so these are all measurements and meters when you subtract them you will get meters but when you square them you get meters squared plus meters squared plus meters squared plus meters squared plus meter squared and then you're just dividing that by a eunice a unitless count of the number of data points you have so the units here are going to be square meters and so you might say hey that's kind of a weird unit if we're trying to figure out if we're trying to visualize or think about how dispersed we are from the mean if we if when I visualize it I visualize dispersion or how varied they are in terms of meters not meters squared so what could we do and the big hint is comes out of just even the notation for variance it's this Sigma symbol squared so why don't we just take the square root of it so why don't we just take why don't we just take the square root of our variance which we will denote with just a sigma makes a lot of sense and in this case what's it going to be it's going to be the square root of 0.316 and then what are the unit's going to be it's going to be just meters and we end up with so let me take the square root of 0.316 and I get zero point five six I'll just round to the nearest thousandth zero point five six - so it's approximately zero point five six two meters so you might be saying Sal what do we call this thing that we just did the square root of the variance and here we're dealing with the population where you haven't thought about sampling yet the square root of the population variance what do we call this thing right over here and this is a very familiar term often times when you take an exam this is calculated for the scores on the exam this is our population we just in a new color I'm using that yellow a little bit too much this is the population population standard standard deviation it is a measure of how much the the the data is varying from the mean in general the larger this value that means that the data is more varied from the population mean the smaller it's less varied and these are all somewhat arbitrary these are all somewhat arbitrary definitions of how we've defined variance we could have taken things to the fourth power we could have done other things we could have taken not taking them to a power but taking the absolute value here the reason why we do it this way is it has neat statistical properties as we try to build on it but that's the population standard deviation which gives us nice units meters meters in the next video we'll think about the sample standard deviation