Mean Absolute Deviation Statistics

let's talk about the mean absolute deviation the mean absolute deviation measures the average distance of all the values from the mean now let's talk about how to calculate it let's say we have the numbers 7 11 19 22 and 29 in this first example let's use table to calculate the mean absolute deviation so in the First Column I'm going to put all of the values and then I'm going to put the mean which we need to calculate soon and then we're going to take the difference of each data point with the mean and then after that we're going to take the absolute difference of each data point with the mean so the first thing we need to do is we need to calculate the arithmetic mean of these six numbers so it's going to be the sum of all the values divid so it's 7 + 11 + 14 + 19 + 22 + 29 and we're going to divide this by where is six and we could put these numbers here too so we have 7 11 14 19 I'm going to have to make this longer and then 2229 7 + 11 + 14 + 19 + 22 + 29 that gives us sum of 102 and 102 / 6 gives us the arithmetic mean which is 17 now once we have the mean what we're going to do is we're going to subtract each data point from the mean so 7 - 17 is -10 11 - 17 is -6 14 - 17 is 3 19 - 17 is 2 and then and this is 5 and then 12 now in the last column we're going to take the absolute value of this column so all the negative numbers will become positive and then we're going to take the sum of this column so 10 + 6 + 3 + 2 + 5 + 12 is now to calculate the mean absolute deviation it's going to be the sum of the absolute value of the difference between each data point and the mean and then we're going to divide that by so this value we already have which is 38 and is 6 so it's going to be 38 / 6 now we could do this mentally 38 over 6 is 36 over 6 + 2 over 6 36 + 2 is 38 36 over 6 is 6 6 2 over 6 reduces to A3 so 1/3 is3 repeating so this is 6.3 repeating so that is the mean absolute value deviation for this particular problem and this is the formula that you need in order to get the answer now let's work on another example that illustrates how we could use this formula without the use of the data table so let's say we have the numbers 5 12 16 and 18 feel free to pause the video if you want to and go ahead and calculate the mean absolute deviation for this set of numbers so using the formula we're going to take the sum of the absolute difference between each data point and the mean and divided so the first thing we need to do is calculate the mean so let's take the sum of the five numbers and we're going to divide it by five 5 + 9 + 12 + 16 + 18 that's equal to 60 and 60 ID 5 is 12 so 12 is the arithmetic mean in this problem now using this formula we are going to take the absolute difference of each number with the mean so the absolute value of 5 - 12 and then 9 - 12 and then we have 12 - 12 16 - 12 and then 18 -2 and let's divide this by in this case is 5 so 5 - 12 is -7 and the absolute value of -7 is POS 7 9 - 12 is -3 the absolute value of -3 is POS 3 12 - 12 is 0 16 - 12 is 4 and 18 - 12 is 6 and then we're going to divide this by five now 7 + 3 is 10 4 + 6 is 10 10 + 10 is 20 and 20 ID 5 is 4 so this is the mean absolute deviation for this set of numbers now let's understand this visually so what this number means is that on average all of the values have an average distance of four units away from the mean so let's draw number line and I'm going to put the mean in the middle we know the mean is 12 we have number at nine and another one at five we have number at 16 and at 18 so this point which is directly on the mean it has difference of zero between itself and the mean this number is three units away from the mean and this one is seven units away from the mean 16 is four units away from the average of 12 and 18 is six units away so each of these numbers they represent absolute deviations from the mean and what we've done is we've taken the average aage of these five numbers to get the mean absolute deviation because we summed up 7 + 3 which is 10 4 + 6 which is 10 so that gives us sum of 20 divided by the five deviations that we have here and so we get mean absolute deviation or an average absolute deviation of four so on average each number is about four units away from the mean on average