in this video we're going to calculate the standard deviation of a set of numbers now there's two formulas you need to be aware of the first one is the population standard deviation now this formula is represented by the letter sigma that's the standard deviation it's equal to the sum of all the differences between every point in the data set and the population mean the population mean is mu which is this symbol here and then you need to square it divided by n which is all of the numbers in the set and then you got to take the square root of the whole result so that's the population standard deviation the next formula is the sample standard deviation so let's say if you have just a sample of a population not the entire population if you just have a sample data out of the entire data then you want to use this formula s which is the standard deviation is equal to sigma the sum of all of the differences between every point and the mean that's the sample mean in the other equation we had the population mean represented by mu but this is the sample mean which is basically the average of all the data points in the set and then you have to square it but it's going to be divided by n minus 1 as opposed to n and so that's how you calculate the standard deviation of the sample now let's work on an example let's say if we have two set of numbers four five and six and also three five and seven which one has a greater standard deviation let's use the population standard deviation formula but if we had to guess which set of numbers has the greater standard deviation is it the one on the left or the one on the right what would you say we need to understand the basic idea of standard deviation you need to know what it measures standard deviation tells you how far apart the numbers are related to each other so the more spread out they are the greater the standard deviation four five and six are closer to each other than three five and seven and you could tell if you plot them on a number line let's put five in the middle so 4 5 and 6 here they are on a number line now in contrast let's put the same numbers on this number line we're going to have 3 5 and 7. so if you look at the the red points the points in red are further apart the points in blue they're very close together so therefore four five and six has a lower standard deviation in three five and seven so sigma is low and here the sigma value is high now go ahead and calculate the standard deviation for this for the set of numbers three five and seven so what's the first thing that we should do the first thing that we should do is calculate the mean to find the mean it's going to be the sum of all the numbers divided by 3. now because the three numbers are evenly spaced apart the mean is going gonna be the middle number five three plus five is eight eight plus seven is fifteen fifteen divided by three is five so that's the mean now what should we do next now that we have the mean now think of the formula it's going to be a sigma of every point minus the mean squared divided by n and then all of this is within the square root so here's how to use the equation first we're going to use the first point 3 subtract it by the mean and then squared next we're going to take the second point 5 subtract it from the mean squared and then it's going to be 7 minus 5 squared so each of these three points you're going to plug into x sub i and then you're going to square the differences between each of those values and the sigma represents sum so you're going to add every difference that you get or you can add the square of every difference that you get and now let's divide it by n so n is the number of numbers that we have in this set there are three numbers inside so n is 3 and then we're going to take the square root of the entire thing three minus five is negative two negative two squared is four five minus five is zero seven minus five is two two squared is four four plus four is eight so we have the square root of eight divided by three and at this point we're going to use the calculator 8 divided by 3 is about 2.67 and if you take the square root of that you're going to get 1.63 so that's the standard deviation for 3 5 and 7. now let's calculate the standard deviation for the other set of numbers four five and six so why don't you go ahead and pause the video and try this example calculate the standard deviation using the same formula so let's go ahead and begin let's calculate the population mean it's going to be 4 plus 5 plus 6 divided by the number of numbers that we have which is 3. four plus six is ten ten plus five is fifteen and we know that fifteen divided by three is five so once again any time the numbers are evenly spread apart the mean is going to be the middle number so now we can calculate the standard deviation so sigma is going to equal the square root but before we do that let's calculate the differences so the first difference that we have the first number is going to be 4 and we're going to subtract it from the mean and then square it the next number is 5 subtract it from the mean and then square it and then after that the last number is six this is going to be six minus five squared now it's divided by n and let's not forget to take the square root of the entire thing four minus five is negative one negative one squared is simply one five minus five is zero six minus five is one and it's all divided by three one plus one is two so we have the square root of two divided by 3. now 2 divided by 3 as a decimal is about 0.67 and the square root of 0.67 is 0.816 so as you can see the standard deviation is less because these numbers are closer to each other they're not far apart from the mean in the other example three five and seven they're further apart from the mean which is five three is two units away from five four is only one unit away from five and that's why the standard deviation is so much less now let's go back to the first example we said that the population standard deviation is approximately 1.63 so given this information how can you calculate the variance v a r i a-n-c-e how can we find the variance the variance is simply the square of the standard deviation so 1.63 squared is equal to now keep in mind this is a rounded answer i don't remember what the exact answer was but once you square it it's about 2.66 so that's how you can calculate the variance the formula for variance is basically the sum of all the square differences between every point and the population mean divided by n it's basically the same formula without the square root symbol well that's it for this video so now you know how to calculate the population standard deviation and also the sample standard deviation even though we did just one of them the process is the same of finding the other one the only difference is you have n minus one instead of n you also know how to calculate the variance as well so that concludes this video by 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