[Music] in this video we're going to learn about cumulative frequency diagrams let's have a look at a regular frequency table so one like this this one is about the test marks that some students have scored in a test in an exam it's really common for them to ask you to complete a cumulative frequency table for a cumulative frequency table the first entry will always be the same as the regular frequency table if we look at the first entry in the regular frequency table we can see it's from 0 to 20 marks with a frequency of six so we can just copy this over into the cumulative frequency table as well from 0 to 20 marks with a frequency of six but here's where it changes so we're now going to look at the next entry so marks from 20 to 40 with a frequency of 22 but because this frequency is cumulative we're going to include all of the previous frequencies as well this means we want to include all of the people from the the first group and all of the people from the second group if we're going to get all of those people then we need the marks that are from 0 to 40 now so for our next entry the test Mark goes from 0 to 40 and we need to add together both of those frequencies or we could just add on 22 to the last one which was 6 so 6 + 22 gives a cumulative frequency of 28 now if we move on to the third entry we want to include this group from 40 to 60 but also both of the previous two groups so if we're going to get all of those people the marks must now go from 0 to 60 and then we need to add together all of the previous frequencies as well so we need the 6 plus the 22 plus the 36 to do this quickly we could just add the new frequency 36 onto our previous total so 28 + 36 gives you accumulative frequency of 64 we then just repeat this process all the way to the end of the table so for the next group from 60 to 80 with have a frequency of 45 our test Mark now needs to go from zero all the way up to 80 and we can just add on this new free frequency of 45 64 + 45 is 109 and for the final group from 80 to 100 the test Mark now needs to go from zero all the way up to 100 so we're now going to get all of the people we can add on this new frequency of 11 109 + 11 is 120 so this is the completed cumulative frequency table you'll then almost certainly be asked to use this information to draw a cumulative frequency diagram so let's take some axes and we're going to put the test marks along the bottom so they need to go from 0 to 100 and we can label that as test Mark then we need to do the cumulative frequency and this always goes on the vertical axis you can see the cumulative frequency goes up to 120 so we need to make sure we go up to at least 120 and we'll label this cumulative frequency now we need to mark on some points for each of the entries in the cumulative frequency table so looking at the first entry here from 0 to 20 it has a cumulative frequency of six we need to plot this in a very particular place though since this entry is saying that up to the point of 20 marks we have six people we're going to plot this at the end point of that interval so at 20 we're going to plot a cumulative frequency of six and you always do this for cumulative frequency you always plot at the end point of each of the intervals so we go along the bottom to the test Mark of 20 and plot a cumulative frequency of six which would be somewhere like this then we move on to the next group from 0 to 40 with a cumulative frequency of 28 this 28 tells us there are 28 people who got up to 40 marks so at the end of that interval 40 we're going to plot a cumulative frequency of 28 so we go to 40 and 28 and put a cross there we then continue this with the next group so an end point of 60 with a cumulative frequency of 64 that goes here and the next one an end point of 80 with a cumulative frequency of 109 that goes here and for the final group a test Mark of 100 with a cumulative frequency of 120 and that goes here there's one extra cross that we ought to put on is as well if we look at the regular frequency table we can see the lowest point of the first interval is this number here and since nobody's scoring below Z in this test we can be sure the cumulative frequency up to the point of Z is zero so we can put across it 0 0 as well just be wary that that first group won't always start at zero but whatever value is there is where you can plot a cumulative frequency of zero we then join these crosses together you have a choice at this point you could use straight lines so connect them up from left to right like this with a straight line going from one cross to the next or alternatively you could do the same thing but use a smooth curve something that looks like this both methods will get you all of the marks in the exam I would personally recommend sticking with straight lines just because I think it's easier to do so this is a completed cumulative frequency diagram let's just review how we made it well first of all we worked out the cumulative frequencies by adding on the previous frequency each time then we scaled the axes making sure the cumulative frequency was on the vertical axis then we plotted all of the points making sure to plot each cumulative frequency at the end of each of the intervals and finally we connected them up using a straight line or a smooth curve sometimes you may be given a cumulative frequency diagram that's already been drawn like this one here and then you'll be asked some questions about it let's run through some of the types of questions you could be asked about it this diagram is about some students who were late to school and you can see we've recorded the number of minutes that they were late the first thing that they could ask you about a diagram like this is to calculate the median so we're going to work out the median number of minutes late to answer this we first of all need to work out what the total cumulative frequency is to find this you look for the highest point on the graph so that's this one here and then just read across left at the cumulative frequency value and you can see this one is 40 this means that in total there are 40 students in this diagram that relate to school if the total frequency is 40 but we're after the median we want the one that's in the middle this will be halfway up the cumulative frequency half of 40 is 20 so what we need to do is go up to 20 on the cumulative frequency axis then we draw a horizontal line from this value of 20 until we hit the cumulative frequency curve and then go down to the bottom and read off the number of minutes late so for this one I read off the value of 6 and A2 so the median number of minutes late was 6.5 the next thing they could ask you is to calculate the inter quartile range to do this we need both the up quartile and the lower quartile let's start with the lower quartile so the lower quartile is one quar of the way through the data if the total cumulative frequency is 40 we can find the lower quartile at 1 qu of 40 which is 10 so we need to go up to 10 on the cumulative frequency we then read across from 10 until we hit the curve and then come down and read off the number of minutes so the lower quartile here comes in at 2 minutes now we'll do the upper quartile this is 3/4 of the way for through the data so since there are 40 in total for the cumulative frequency 3/4 of 40 is 30 so if we go up to 30 then go across once we hit the curve we go down and we can read off this value so the upper quartal is 25 minutes then to find the inter quartile range you do the upper quartile subtract the lower quartile so the inter quartile range is 25 minutes take away 2 minutes which is 23 minutes both of these are really common follow-up questions to cumulative frequency diagrams now let's do one more so this time we're told that students who are more than 5 minutes late are going to receive a detention and we need to work out the percentage of the students who received the detention this time rather than starting with accumulative frequency and working out a time we're given a time and we're going to work out the frequency so in the question it mentions students that are more than 5 minutes late receive a detention so if we locate 5 minutes late on the bottom axis that's here and then go up to the curve and then across we can read off a cumulative frequency of 18 this means we would estimate that there are 18 students whose lateness was below 5 minutes therefore they're not going to get a detention but we want the students that are above 5 minutes because they are getting a detention since there are 40 students in total and we know there are 18 below 5 minutes we can do 40 takeway 18 and get 22 so we would estimate that there are 22 students above 5 minutes therefore 22 students are going to receive it detention this question doesn't say how many students though it says what percentage so let's first of all write this 22 as a fraction of all of the students so there are 22 receiving a detention out of a total of 40 students and to turn this into a percentage we multiply by 100 if you work this out or type it into your calculator you'll get the answer 55% now let's try another example so this time we have a diagram about the time spent waiting for a meal at a restaurant and this is in minutes first of all we're going to work out the median again to work out the medium we first of all need need to work out what the total cumulative frequency is so we find the highest point on the diagram read across to the left and you can see this is 80 on this type of question you need to be particularly careful the scale goes up to 100 but the cumulative frequency only goes up to 80 so some students mistakenly think the maximum cumulative frequency is 100 but it's not it's 80 in this question now we need to locate the median so if the total cumulative frequency was 80 we need to go up half of this which is 40 we then go across to the cumulative frequency curve and then go down and read off this value I think this value is about 46 so the medium would be 46 minutes now we're going to practice the inter quartile range once again so we'll do the upper quartile and the lower quartile let's start with the lower quartile if the total cumulative frequency was 80 need to got one quar of this 1 qu of 80 is 20 go across and read down and we can read this lower quartile at 34 minutes now we'll do the upper quartile so this is 3/4 of the cumulative frequency so 3/4 of 80 is 60 so we go across from 60 and then down and read off this value and this time it's 55 minutes then to find the inter quartile range we do the upper quartile 55 subtract the lower quti 34 and this gives you the answer 21 minutes and for one final question one person from the restaurant is selected at random and we've been asked to work out the probability the person weighted between 50 and 70 minutes once again this is a question where we're given the time and we need to work out the cumulative frequency so let's do this for 50 and 70 so let's start at 50 on the time axis go up to the diagram and then across and 50 corresponds to a cumulative frequency of 49 this means that 49 people waited up to 50 minutes now let's do the same for 70 so up to the curve and then across and we read off 77 so 77 people waited for up to 70 minutes now we've been asked to work out the number of people waiting between 50 and 70 minutes well if we know there were 49 up to 50 and then 77 up to 70 we can just subtract these to work out the number in between so if we do 77 take away 49 we end up with 28 people so we would estimate there are 28 people between 50 and 70 minutes and we need to give this one as a probability so we need to give that as a fraction of the total there were 80 in total so the answer is going to be 28 over 80 thank you for watching this video I hope you found it useful check out the one I think you should watch next subscribe so you don't miss out on future videos and now go and try the exam questions in the video's description