in this video we're going to talk about how to calculate the mean of a group frequency table so what we have here is the grades of students and you can see the grade intervals of 40 to 49 50 to 59 and so forth 60 to 69 will be a d 70 to 79 is a c this is a b that's an a anything less than a 60 it would be an f in most schools the second column contains the frequency so this tells us the number of students who scored in this range for instance there are six students who got a d on the exam eight students who had a score between 80 and 89 so with this data how can we calculate the average test score of the students in this class now it's important to understand that we cannot compute an exact mean for the grades in this class and the reason why that's the case is because we just don't know the individual data values in the data set so therefore we can estimate the mean now we know that the mean is basically the sum of all the data values divided by the number of data items in the set but for a group frequency table the mean is going to be something similar but that equation will be modified a bit it's going to be the sum of the product of the frequency values times the midpoint because we can't really use an individual score here since we have an interval so the best ideal individual score would be the midpoint of each grade interval so it's going to be the sum of the frequency times the midpoint divided by the sum of the frequency values and so that's the formula that we need to use in order to calculate the mean so let's take this one step at a time let's calculate the midpoint values first the midpoint is going to be the sum of the lower boundary of the interval plus the upper boundary of the interval divided by 2. so in the grade interval of 40 to 49 the lower boundary will be 40 the upper boundary is 49 divided by 2. so we get 89 divided by 2 which is 44.5 so that's the midpoint for this interval now let's do the same for the next one so the second midpoint is going to be 50 plus 59 divided by 2. and so that's 109 divided by 2 which is 54.5 at this point we can see a pattern the next one is going to be 64.5 and then 74.5 and then 84.5 the last one is different so it's 90 plus 100 which is 190 divided by two this is going to be 95. now our next step is to take the sum of the frequency column so this will give us this particular value which will help us to calculate the mean so it's going to be three plus five plus six plus nine plus eight plus seven so this tells us how many students are in the class and so there's a total of 38 students in this class now the next thing we need to calculate is the sum of the frequency time the midpoint values so we're going to do is multiply these two now it might be easier to basically make an extra column and then sum up those values okay that line is not straight so let's call this f times m so what we're going to do is multiply the frequency by the midpoint so 3 times 44.5 and that's going to be 133.5 and then multiply these two values so that's 5 times 54.5 which is 272.5 and then 6 times 64.5 so that's going to be 387 and then 9 times 74.5 which is 670.5 and then 8 times 84.5 that's 676 and finally 7 times 95 is 665. now let's take the sum of this column so go ahead and add those numbers using the calculator so the total that i have is 2804.5 so now we can calculate the mean the mean is going to be the sum of the fm values divided by the sum of the frequency values so this right here basically all of this correlates to this number so that's 2804.5 and then divided by the sum of the frequency values which we have here and that's 38. now before we divide it where do you think the mean is going to be in which interval will it be in this interval this one this one which one notice that the highest frequency occurs here the chances are the mean is going to be between 70 and 79 because most students scored in that range so if we take 2804.5 divided by 38 this gives us a mean value of 73.8 and that makes sense since most students they had a c on the test now here is another question for you in which interval can the median and the mode be located in what would you say now the mode occurs where the frequency is the highest and the frequency is the highest in this interval so the mode exists between 70 and 79 now what about the median in order to figure that out we need another column so we're going to call this cumulative frequency we're gonna start with this number three and then we're gonna add five and three together which will give us eight then we're gonna take this number add six to it that's gonna be fourteen and then we're gonna add nine to fourteen which gives us 23 and then 23 plus 8 is 31 and then 31 plus 7 is 38 so we get this number because there are 38 students and half of 38 is 19. so the middle number is basically the median 19 is somewhere between 14 and 23. so after 14 we're going to be done with this interval 23 is in this interval between 70 and 79. that interval starts with the number 15 and it ends in 23 and so 20 is in that interval so the median is somewhere between 70 and 79 as well now let's work on another example for the sake of practice so here we have the weights of different students in a class and what i want you to do is i want you to use this group data to calculate the mean and determine which interval contains the median and the mode so go ahead and work on this example so let's begin by calculating the cumulative frequency so this is going to be 6 6 plus 8 is 14 and then 14 plus 12 is 26 26 plus 7 is 33 and 33 plus 3 is 36. and if we basically take the sum of the frequency values that should give us 36 as well which we can put down here now let's determine the midpoint values so it's going to be the upper boundary plus the lower boundary divided by 2 or vice versa so the upper boundary is 139 the lower boundary is 120 divided by 2. so that's 259 divided by 2 so the midpoint for the first interval is 129.5 now let's do the same for the second interval so let's add 140 and 159 and then divide by two and so this is going to be 149.5 so we can see that the midpoint is increasing by 20. so the next one is going to be 169.5 and 189.5 now the midpoint between 200 and 220 is 210 if you add these two you're going to get 420 420 divided by 2 is 210. now let's take the frequency and multiply by the midpoint so 6 times 129.5 that's 777. if we multiply 8 by 149.5 that's 1196 and then 12 times 169.5 this is going to be 2034 and then 7 times 189.5 that's 13 26.5 and then 3 times 210 is 630. so now let's take the sum of this column so the sum is going to be let's use a different color 5963.5 so at this point we can calculate the mean so it's the sum of the frequency times the midpoint values divided by the sum of the frequency values so we have this already that's 5963.5 and the sum of the frequency values is 36. now before we calculate the mean which interval do you think the mean is going to be now most of the numbers are between 160 and 179 and the left side of the data is heavier than the right side of the data so it's either going to be in this interval or in this interval one of those two so if we take 59 63.5 divided by 36 that's going to give us 165.7 approximately so that's the mean which turns out to be in this interval but notice that it's closer to 160 than it is to 179 and this is due to the fact that the left side of the data is heavier than the right side of the data now which interval contains the median and the mode so the mode is basically the interval with the highest frequency which is this one again 160 to 179 now the median half of 36 is 18. keep in mind 36 is the number of students that we're measuring the weight of so half of that is 18. in which interval is the 18th student located in now this interval's for the first six students so if we were to write a range it would be from zero to six or technically from one to six now this range or that interval covers a range of seven to fourteen you start with the next number and you stop here now the interval from 160 to 179 it starts from 15 and it ends in 26 so 18 is in this interval which means that the median is also located in this interval as well