Statistics Formulas and Equations

in this video I'm going to go over some basic statistics formulas that you need to know for those of you who might be taking this course soon so let's go over the basics let's say we have the numbers two 355 and then 8 9 10 and 12 if we want to determine the median of this data set the median is simply the middle number so the middle number is seven that's going to be the median the mode is the number that occurs most frequently so the mode is five the number five has frequency of two in this data set now now if you want to calculate the range the range is difference between the highest number or the maximum and the lowest number which is the minimum so two has the lowest value that's the minimum 12 is the maximum so the range is going to be 12 - 2 which is 10 the number five is the first quartile the median is the second quartile and nine is the third quartile the first quartile is the 25th percentile the third quartile is the 75th percentile and the second quartile is the 50th percentile now there's something called the IQR inter quartile range and this is the difference between the third quti and the first quti so that's another formula that you want to add to the list now the next equation you need to be familiar with is the average also known as the arithmetic mean or in this case the sample mean the sample mean is the sum of all the data points divided by so if we were to add up the numbers two plus 3 + 5 and so forth and we have total of nine numbers in this data set if we plug this in this will give us the average so the sum of those nine numbers is 61 and if we divide it by 9 we'll get 6.7 repeating so this is the average or the arithmetic mean of the data set now if you want to find the arithmetic mean of two numbers it's simply the sum of those two numbers divided by two if you want to calculate the geometric mean of two numbers it's the square root of the product of those two numbers so it's the square root of * now if you want to calculate the harmonic mean it's 2 / the reciprocal of and it's two divided by the sum of the reciprocal of and so it looks like the reciprocal of the arithmetic mean formula but instead of and you have the reciprocal of those data points now if you rearrange this equation by multiplying the top and bottom by AB this equation becomes 2 * / plus so it's twice the product of and divid by the sum of and so that's how you could find the harmonic mean of two numbers now if you want to find the root mean square of two numbers it's going to be the sum of the squares of those numbers divid by two very similar to this equation but you're squaring and now if you were to pick any two numbers and if you were to calculate these four values you'll find that the harmonic mean is less than the geometric mean Which is less than than the arithmetic mean and that's less than the root mean Square the root mean square is the highest of these four it turns out that the geometric mean is equal to the square root of the product of the arithmetic mean and the harmonic mean now let's go back to the arithmetic mean so so we said that the arithmetic mean is the sum of all the data points divided by so what that means is you take each data point X1 + X2 + X3 you add up all of them and this can go all the way to to the and then you divide by the number of data points or the size of the sample that you have and that'll help you to calculate the arithmetic mean so that's the expanded formula of the arithmetic mean if you want to calculate the geometric mean you could use this formula it's the product of all of the data points all the way to to the and then it's raised to the 1 / so let's say if we have two data points X1 and x2 in this case is 2 so if we replace X1 with and X2 with we get that the geometric mean of two numbers is the square root of and where this has an index number of two but if you have more than two numbers then you want to use this formula to calculate the geometric mean of multiple numbers next is something called the weighted mean to calculate the weighted mean you need to take the sum of the products of WX divided by the sum of the weights or the sum of so it's going to be W1 X1 plus W2 X2 plus W3 X3 and this is going to continue to nxn and all of this is divided by W1 plus W2 + W3 all the way to WN now for those of you who want to see some example problems on how to use these formulas feel free to check out the links in the description section below I'm going to be post more content where you can find examples on how you can employ these formulas the next Formula is the harmonic mean so it's over the sum of the reciprocal of the data points so 1 /x in its expanded form it looks like this it's over 1 / X1 + 1 /x2 + 1x3 and so forth so that's how you can calculate the harmonic mean if you have multiple numbers now to calculate the root mean Square it's going to be the sum of the squares of the data points ID by but all within square root so you need to square each data point and add them up and then divide it by the size of the sample and then take the square root of that entire result so that's how you can calculate the root mean square of data set now you need to be able to make distinction between the sample and the population what you need to understand is that the sample is subset of the population it's just small part of it so whenever you see xar this represents the sample mean if you see lowercase this is the sample size now whenever you see the Greek letter mu this has to do with the population mean so notice the difference between the two that's the sample mean this is the population mean if you see capital that is the size of the population not the sample size which is different from the population size for instance you may have 10,000 residents in city but your tests might be on 300 out of that population of 10,000 so your sample might be 300 but the population capital could be lot bigger so just know that lowercase and capital they're not the same they're different now represents the standard deviation of the sample so here's how you calculate the sample standard deviation it's the difference between the data point and the sample mean you need to square that difference and take the sum of all of those Square differences and then you need to divide it by minus one and then take the square root of that result so that's the formula you need to calculate the standard deviation of the sample now to calculate the standard deviation of the population instead of using it's going we're going to use Theta so the formula is very similar it's going to be the square root of the sum of the Square differences between each data point but instead of the sample mean we are going to use the population mean and instead of the sample size we're going to use the population size and it's going to be capital and not minus one so that's the difference between the sample standard deviation and the population standard deviation now the next Formula you want to be familiar with is variance to calculate the variance of the sample is simply the square of this equation so it's everything that you see there but without the square root so that's how you can calculate the variance of the sample to calulate the variance of the population it's the square of the population standard deviation so everything that you see above but without the square root symbol so that will help you to calculate the population variance and this is how you spell it CA now the next equation we're going to talk about is the coefficient of variation to calculate the coefficient of variation for the sample it's simply the sample standard deviation / the sample Mean Time 100% now to calculate the coefficient of variation for the population it's going to be the population mean divided by actually take that back it's the population standard deviation divided by the population mean times 100% so in both cases the coefficient of variation is the standard deviation ID the mean time 100% so it's the ratio between the standard deviation and the mean and it's typically reported at as percentage so if you have coefficient of variation of 0.25 what this means is that the standard deviation is 25% the value of the mean now the last formula that I'm going to go over in this video is something called the mean absolute deviation the mean absolute deviation it's going to be the sum of the absolute differences between each data point and the sample mean and all of this is divided by so it's kind of similar to how you would calculate the standard deviation the only difference is you're not taking the square of these differences and you're not taking the square root of the entire result other than that the process is somewhat similar now there's many other formulas in statistics that you're going to learn as you go through the course but just want to give you some basic formulas that you may encounter in the beginning of Statistics so that's it for this video and for those of you who want to see how to put this in action feel free to check out the links in the description section below I'm going to be posst more content there that you can take look at so feel free to check that out when you get chance