and the first thing we do in Chapter 2 is we get away from how to collect the data this is the ways we collect the data and we move on into how to organize it and eventually how to summarize it so we're going to talk about what's called a frequency distribution right now are there any questions before we get going on this hey if I asked you to look at this classroom and to count the number of brown head haired people and the number of blackhaired people and the number of blond-haired people and put it in a list this is probably how you do it you'd probably do blonde brown black and count them up and put them right here 7 15 12 you'd probably do something like that wouldn't you this is one of the simplest versions of a frequency distribution that I can think of this is a frequency distribution we're going to make it a little more specific for our our reasoning but this is the idea you have what are called classes just groups and you have counts frequencies a frequency is how often something occurs seven blond people that blonde people occurred seven times brown haired people 15 times so these are our classes and these are our frequencies this is a frequency distribution we're going to talk about that right now you know I think the way that I'd like to do this with you is we'll actually create a frequency distribution right now and as we do it I'll be giving you the definitions and things like that and show you how so first thing you need to know about frequency distributions is there's something called a class whip I need to Define that before we actually start making this up so frequency distributions what we have is a list of values these are our values our classes and the corresponding frequencies the list of values with corresponding frequencies so we're going to talk about a class with what we're going to be doing is creating frequency distributions that are based on numbers in in a second what I'm going to do in this class is I'm going to make a frequency distribution based on our ages um so we can kind of have the idea of how you make make one of these things up so first thing the class width this is going to be the difference between two lower class limits I haven't even defined that yet but I'm going to this is the difference between two lower class limits okay well what the world is a lower class limit then lower class limit classes is not like this class it's like the group like the group of of data that you're going to try to collect from like in here do we have anyone under 18 in here okay so 18 is going to be our starting sport under here in this class our first group is probably going to be 18 to 20 year olds and that'll be our first group or 18 to 21 year olds that'll be our first class the class is just the first group or the second group or the third group those are all classes the lower class limit is where the class starts so if I have a group of 18 to 21 year olds 18 would be my lower class limit does that make sense to you it's where the class starts so we'll say here uh the smallest number belonging to a class well that there is an upper class limit also and that's going to be well clearly the highest value belonging to a class so upper class limit this last thing before we do before we get into this upper upper class okay I'd like the give you your steps right here on how to do this the first thing that you need to do when you're creating your own frequency distribution like I'm about to do is determine the number of classes this means like those groups we were just talking about you see you don't want to have too few groups that you can't differentiate between them or too many groups that it it's just too too cumbersome you can't tell Trends so in in this class I'm going to separate us just looking around I'm thinking I'm going to want eight classes here so I'm going to have eight classes so for us in this one example I'm G to make eight I think that two more than that would would not show Trends and I think less than that like do you ever want to have a frequency distribution with one class do you think with one class everybody would be in that class would that show you anything not so much would you want a class for every single person you you wouldn't see any Trends you wouldn't see that so here we have we have a certain number of classes that you're going to determine by looking at your population that you're sing okay the second thing you got to do is you have to figure out your class width this is the important part here's how you in general find a class width what you do is you take your highest value in your popul in your your sample and you take your lowest value in your sample and you find the difference so that that range and then you divide that by the number of classes so I'm going to say here the max value subtract the Min value this is for your sample and then you're going to divide by the number of classes okay we're going to do that in here hopefully no one's worried too much about their age U are there any 17 year olds in here 16 15 okay good 18 18 okay good uh do we have anyone over well I know I am how about [Music] over 50 45 40 40 do you mind no 44 44 Okay so 44 is our max value minus 18 is our minimum value and we're going to divide by the number of classes how many classes I pick here eight okay we're divide by eight so we do that uh someone with a calculator out there what is 45 44 minus 18 how much we get or without a calculator just do it don't everyone talk at once let's go here people how much 26 you said I believe you divided by eight wait a second Mr Leonard eight doesn't go into 26 evenly no chances are this is probably not going to be even in fact we get let's see uh two 3.25 yeah your class width has to be a whole number so here's what you're going to do no matter what the decimal is if you have a decimal you are going to round upwards if you round downwards you're not going to have enough CL enough room for everybody in your sample you you'll have eight classes but it'll stop Too Short you have to make another class if you want eight classes you need to round up that'll cover everybody do you do you see why you can't round up you got to round up I know even though it's 3.2 you go well that rounds to three well yeah in like algebra stuff it does when statistics when we need a certain number of classes we're going to round that up it's going to go to four so we we are going to round up so this is going to become four our class width is going to be four for our example so we're about ready to start making this thing here's how a typical frequency distribution looks we're going to have what we're looking at in our case it's age here and we're going to list the frequencies here so we determine the number of classes I determine that in your book it'll tell you what to use or on your test it'll tell you use eight classes or use six classes so it will tell you that but you do need to know how to do this stuff you'll take the max value minus the Min value that gives you a range for your sample you divide that by the number of classes you round it up and that tells you your class width here's how you find your class width you must have a starting point this is uh step three and typically we're going to start with the minimum value that seems obvious that'll include everybody or you could if if if you want to do this you can start with a value just below your minimum value so I can start with 17 if I wanted to but typically you start with your lowest value and so in our case we had 18 so over here we finally get to start filling this thing out I'm going to put 18 right here the next step is the most important step because everyone can count we can all count frequencies but making up the classes that's the hard part where people get messed up so pay close attention this if you quite don't if you don't quite get it go back on this lecture online and watch it again until you get this down what you're going to do is after you start with the smallest value you're going to create the classes using the class web here's the key the class width is not the difference between 18 to this number this would be the upper class limit the class width is the difference between this number and this number number so we're not going to do this we're not going to go 18 19 20 21 22 we're not going to do that mm we're going to do 18 19 20 21 22 goes here why because this distinguishes classes and if you have classes which are four years apart think about it 18y olds 19 year olds 20 year olds and 21 year olds four years are going to fit in this class do you see that I can't include 22 here that would be a class with of five it would include the 22 year olds you see the point there that would actually be a class with of five so be careful this is where people make a mistake in the test they go 22 and then they're off a sequential year for every class and then and they go wait a minute I have way too many classes this has zero well that's because you did wrong okay so this is the way we do this we go four here and then we we're going to create eight classes from here on out so 18 to 22 22 to what hopefully 20 remember class withth is four and then again to and then how many classes do we need perfect and then what and then what 1 2 3 4 5 6 7even 8 you know what that's going to work out perfect for us because we have 44 in our group right look at our last class is going to include we too many I guess I could happen since you round up you might have an extra class in there which we do we're going to have 0 46 through 47 40 9 so we do this then you create our classes going this way so we're just going to go to one less than our next next uh lower class limit so this is going to be our 21 our 25 R 29 33 37 41 45 and then lastly we're going to have 49 but that's the way we do this we create lower we create all the lower class limits first and finally we're just going to tally up the results we're going to start on this next time you have all the information to make these now but I I really do want to go through this and do this again and count up all the people and make sure that we have that down how many people understood what we talked about today okay again why we might have an extra class is because you round you do round up um that's necessary to include everybody but I guess you can get an extra class on you do have some homework no no no not oh it's oh it's just reverse it it sounds a lot better that's what makes just cuz I want it to okay uh you're going to do by the way this is also on the website which I gave you over there uh go ahead and look at please go on that that way I get more views and feel cooler can you do that for me just kidding uh go to section 1.5 and do page 34 to 35 numbers are 1 2 5 through 25 that assignment is also on the website if you go under and look at your class homework assignment that'll be there as well all right guys that'll be du on Monday have a great day I'll see you on Monday okay so if you do remember last time we were going through our frequency distributions and I have given you definitions for this can you remember what our class width was from last time four four all right and the way we figured out our class width from last time is we took the maximum value for our sample minus the minimum value of our sample we divided by the number of classes we wanted and we round it up as we found from last time this is an approximate way to get your classes but you're never going to be too low you're going to have a class for every person at least maybe I think we have one extra actually we're going to have zero one but that's okay so our class width in our case was four the lower class limit that was the smallest value in each class so we're going to list those out in just a second and the upper class limit is the largest value in each class the class midpoint we'll get to those after we fill out our our chart there our frequency distribution now the question I have for you is this as far as the class width goes does the class width measure 4 units this way from lower to Upper or does it measure this way from lower to lowerer so we're not going to go 18 to 22 this way we're going to go 18 to 22 this way and I gave you the reason why last time we want four years spaced here not five if you go to 22 here there's five years between the lowers and we want only four that's why our class width is four so we'll make this 22 the next one I think I gave you of course we going it by four so what's the next one6 and then 30 then4 mhm perfect2 and 46 okay and I made the other ones up the next part up very quickly last time but what you're doing is you're just going to one step below your next lower class limit so from 18 to 22 we need the number what right here we don't want 22 cuz 22 then you would count 22 year olds in two different classes that's not a good thing so we're going to go to 21 you're right and then you can count up by four for all these limits also so for 21 this one's going to be and it should be one less than our next lower class limit okay do you feel okay on getting the classes made up so we get the maximum uh level for our sample minus the minimum value we divide by the number of classes we want we start at some starting spot either the lowest value of our sample or something just below that we make up our classes by using the class width in conjunction with that starting spot make up all your lower class limits first then we fill the upper class limits in couple of things before we actually do the frequency the class midpoints the midpoint is just the middle value of those classes it's it's nothing tricky to find this you're just going to average the upper and lower class limits we haven't talked about average but we're talking about the the average that you all know as average where you add two things together and you divide by two that's our our average so when we find the middle the class midpoint we're talking about the the value right in between the middle of our class so we're going to take the upper class limit minus the lower class limit I'm sorry plus lower class limit and we'll divide that by two and that'll give us the midpoint for each class just want to have an idea on the boundaries the class boundaries are really only used to these two are really only used to make up a histogram which we're going to talk about in just a little while class boundaries are the very middle between one upper class limit and the next lower class limit so we're going to discover these in just a second but that's our definition so in order to find a class boundary what you're going to take is an upper the lower class limit minus the upper class limit or plus upper class limit and jez I'm saying everything wrong today h my goodness rephrase that for you we're just going to average these ones as well so so the midpoint is the average of these two the lower and the upper for the same class the boundary is the average of these two right in between the the two classes so for us we'll say that the boundary is used to separate classes without gaps because we can't have gaps on a histogram you'll see what I'm talking about in just a minute um are we giving it to us or we have to come up with our own well you see your class width is determined by by your maximum value minus your minimum value and the number of classes that that you need so if I tell you five classes with certain number so the number of classes that we need we're going to be given that yes yeah I'll give you that I for this one I just made up I want eight classes all so for you guys it'll be like in your homework or I'll give it to you on a test okay on a test I would give you like a list of data like I don't know 20 pieces of data I'd say I want you to make up a histogram or a frequency distribution which comes from it uh I'm sorry histogram comes from the frequency distribution so I'd give you the data I'd say I want six classes figured out you take the maximum value minus minimum value divide by six round up that gives your class width does that make sense to you if you forgot how to do that remember go online you can refresh that all you want um the whole lesson's on there already so okay the great question any other questions about any of these things so far we're going to go through in a minute and identify all of our lower class limits upper class limits and all this stuff as well so class boundaries last thing we have to Define this is used to separate our classes without gaps you see on a histogram it's like a touching bar chart we can't have any gaps in there okay so let's go through and let's identify what all these numbers are because I've given you the definitions but we really haven't even discussed what they they actually are for this example can you tell me the first lower class limit that we have very good okay what's the next lower class limit that we have good how many lower class limits should we have because we have eight classes that's right so 22 and then we go on up to 26 Etc we should have eight of them that goes all the way to 46 how about the upper class limits what are the upper class limits here one's the first one and then mhm and again we're getting all the way up to 49 there should be eight of them the class midpoints that's the average of the upper and lower class limits for each class can you tell me what is the class midpoint for our very first class how much is that do it on your paper if you have to you're taking the Upper Plus the lower dividing by two you're averaging that nice glasses by the way I like them kind of mine that's why I like them I got new ones new glasses they look identical same see I can't I'm the creature of habit you can't really change if your things are going well which I hope that these are going well can't change it did you figure it out as I was rambling use your calculator if you have to your averaging averaging a couple numbers what numbers are we adding together up here for our first class midpoint can you tell me that great so we're we're going to have a class midpoint for every single class so we look here we go okay that's my upper class limit that's my lower let's add them together let's divide by two that tells me what's right in the middle of those two numbers and what number is that I'm I'm sorry I heard rustling and mumbling what was it 19.5 19.5 okay I believe did anybody else get 19.5 do you know how 19.5 is being found you some people know okay if you're not quite sure how you're finding the class midpoint watch how the CL class midpoint is found for each class you're taking the upper CL stands for class limit in our case the upper class limit for our very first class is 21 so we are taking 21 we are adding to it what are we adding to it oh that's the lower class limit that's this right here you're adding 18 and then you're dividing by two why are you dividing by two why are you dividing by two so how you find an average of two things is you average of any number of things you add them all together and you divide by the number of things you added so here we have two classes you're adding them you're dividing by the number two and then this number this value is going to give you 19.5 you with me now what's the next midpoint can you find that I'll give you a little hint well firstly you understand you can add these two together and divide by two right Nodge your head if you're with me on that that's how you find that class midpoint or you can be a little smarter than that what's our class width just add four to that number you're going to come with class midpoint okay this is kind of neat all these numbers are multiples of well you add four to that they're not multiples of four but you keep adding four to it here you keep adding four to it therefore the midpoint you're going to keep adding four to it you don't have to repeat all that work you don't reinvent the wheel just add four to it so someone who has added these two and divided by two tell me what you got 23. and guess what if I add four to this number I get 23.5 so I'm going to find out midpoints right there are you with me can you tell me just very quickly what's the next midpoint without doing any math that's kind of nice how about the next midpoint 31.5 uh someone on the right hand side of the room how about the next midpoint my right righties next midpoint perfect how many midp points are we going to have 39.5 we'll have 40 what's the next one and lastly we'll have 47 45 and those are the values that are right in between each class raise your hand if you're okay on the lower class limits upper class limits and the class midpoints feel okay about that good very good remember you don't have to do the work for every one of these you just have to know the first one and then know your class width if you know your class width it becomes very very easy here so we should have eight lower class limits eight upper class limits eight class midpoints let's talk about class boundaries and then we're going to fill this thing out it's going to look beautiful I'll tell you what else we can do with this the class boundaries are used to separate your classes without gaps here's how you do it what you do is and by the way I know it's a little confusing where what class boundaries actually do but they're not not used in any computation they are literally for one thing and one thing only what you going to do is be able to make a histogram up with this information that's it it doesn't tell you where things go like where you're going to tally up numbers that's that's what your classes do the class boundaries really are just to separate classes so you don't see any gaps are you with me on this okay so class boundaries they're the value that's right in between your upper class limit for one class and the sequential lower class limit for the next class you can average them add them just like you did the class midpoints and divide by two or you can just kind of think about that what's right between 21 and 22 right say it louder 21.5 yeah it's 21.5 do you know what you're going to get if you add 21 and 22 and divide by two you get 21.5 someone in the middle of the room here can you tell me the next class boundary can you tell me that say Lou yeah you can still use that class wood thing by the way or you can do the math if you really really really want to but you can average these two or just add four or just think about 25 26 what's right in the middle 25.5 next ones we can find all of them if we'd like you're going to have 29.5 I hope is that right yes all right what's the next one 1 2 3 four five six seven how many should we have eight 49 49.5 does 49.5 make sense to you how we get that there's not another class but you see that we're we're just going up by four every time now I know there's eight here and there's eight here and there's eight here and right now we have eight here but there's actually one more there's actually one more notice that these class boundaries are really like slicing up a loaf of bread okay so we have this this loaf of bread that goes from8 uh just before well 18 to 49 and we're slicing it up here and here and here and here and here and here and here if you have a loaf of bread and you slice it you're going to get one more piece of bread for however many slices you make does that make sense get a loaf of bread you slice it once you have two slices true slice it three times you're going to have four slices we've sliced it using these classes but that means that we're going to have this little extra one to begin with because we have to start somewhere so can you tell me what the very first class boundar is going to be yeah that's right notice how in this case the class boundaries they really don't make a whole lot of sense for 17.5 do they well if you consider to have 0 five of a year that doesn't make a whole lot of sense it makes sense in this case though if we are considering these numbers to be your age without decimals cuz I mean if you're over the age of like I don't know seven you stop saying I'm like six and 3/4 you know don't you I don't say that I am blank age3 obviously I just say I'm blank age uh but you say I'm 22 are you 22 Yeah Are you seriously I should work an affair uh so if you say I'm 22 you go I'm 22.4 or 22.6 if we're just counting the the integer of our age or the whole number of our age then 17.5 works right because you're either 17 or 18 you're not going to count any extra people with 17.5 that's what I'm saying the 17y olds are excluded automatically you with me on that so it's it's kind of a a weird thing to say but it's used really just to separate some classes from each other that's it it's not mathematically being used in any way up here so we do have a first one we're going to have one more class boundary then we have classes because we have to start somewhere we have to end somewhere okie dokie now let's go ahead and fill this thing out some more we've already made our classes using our class width and our lower class limits and our starting spot and a range of numbers being subtracted divided by the number of classes that we want that's how we got all this junk now we're going to fill it out by asking people how old they are so how many people are between the ages of 18 and 21 raise your hand 18 to 21 keep them up because I got to count to you 25 okay so we have 25 how about 22 to 25 awesome how about 26 to 29 30 to 33 Okay 30 4 to 37 38 to 4 42 to 45 and 46 49 I don't think we have anybody can you add yourself on maybe count the number of people count that all up and we'll see 32 uh I'm like 23 I'm just really would actually be better if I was like 80 and just never aged would that be sweet that'd be better just Immortal people I mean yeah I did but I could be lying but no to do that count of myself in 18 to 21 year olds too so anyway um so we have all these frequencies for each class we have it so you know just by looking at it where the majority of people are right because this I mean typically this is what's going to happen in most of your classes we're going to have people scattered but they're going to generally be right here and we'll have some people over here but most people are going to be here and that's what frequency distributions can show is Trends and patterns if we just had everyone's age down on a piece of paper it'd be hard to kind of see a lot of Trends and patterns especially if there's lots and lots of data here we only have maybe 40 something 40 something pieces of data you start having like a thousand pieces of data that you start seeing trends when you do this but not if you just have them listed out it's really hard to see so this is one way to we can organize our data are you okay on getting your frequency distribution okay what do we do with this stuff well one thing that we can do I'm going to show you some extensions of this concept one thing we can do is create what's called a relative frequency distribution from this thing are there any questions on this this stuff first so a relative frequency distribution we're going to add some uh some columns here I will say this is going to be relative frequency relative frequency distribution here's how this one works what it does it compares the frequency for each class to the total number of data items you collected that's how relative frequency works this one just gives you the number of people relative frequency is going to give you write this down please a percentage this is going to be a percentage it's going to compare the class frequency to the total number of your of items in your sample how could we find right now the total number number of items that we have just collected without having me count every person in this room that's really boring yeah if you C if every person had a spot which we made sure of right every person has a spot because that's how we created our frequency distribution to make sure that would happen so if every person has a spot if you add all these up it's going to give you your total count for your sample okie dokie okay on that one I said that twice today already old school bring it back oky doie so add them all up if we add them we get the 25 plus 10 plus 4+ 2+ 1 plus 1 how much is that I got 43 we're going to use a a letter for that that's a lowercase n it's 43 that's how many items we collected you can also signify it this way have you ever seen this symbol before it's like a Greek letter Sigma mean some means you're adding up all the frequencies that also is the same thing some of the frequencies or n in some cases it's called so we want to relate the class count to our n or to the sum of our frequencies we're going to create a percentage here's how you do it if you want to find the percent of the people who are in this bracket you're going to take the number your frequency and divide it by the total class count or the total nend sum of frequencies so class frequency divided by the sum of the frequencies can someone out there with a calculator you should all have a calculator at this point someone out there with a calculator can you do the 25 / 43 and tell me what you get to the third decimal place 581 581 like that did anyone else get 0581 yeah did you all get 0 581 I hope you have a calculator to follow along because part of this class is calculator I mean you are going to be using one like every day so bring a calculator so 5 what's 581 mean 581 sounds like a full number rolls off your tone what it mean 58% of the people are that that class right there 18 21 perfect how'd you get 58% we can change proportions which is like a decimal into a percentage just by moving in two spots right so if I say 581 maybe for your boss you don't say 0 581 cuz he might have done what I asked what you did when I asked you go I don't know so maybe we go okay instead of0 581 we could write this as 58.1 but you better put a percent if you're going to do that either way it's fine with me I really don't care you just need to know that when we say 0 581 what we're suggesting here is that 58.1% of our sample was between the ages of 18 and 20 21 that's what we said does that seem reasonable for our class sure yeah most people 58 most is above 50% for you just so you know if anything ever says most it means above 50% most people are between 18 and 20 21 no problem 58% let's do the rest of them on your own take out your calculator figure out the percentages for the rest of our classes guys if I'm walking around and you don't know what you're doing now is the time to ask me when I'm here to help you right now so if you're like a little lost [Music] so okay let's get back up here if you're struggling a little bit on finding this what we're doing for each individual class you're taking the frequency divided by your total count that's your n or your sum of frequencies which is pointed to the 43 so for this one we did 25 div 43 now that did give us 581 we're just changing that to a percentage because most people are better with percentages you don't have to if you don't want to that I really don't care for the next one we're going to take What number divided by what what's 10 ID 43 2 23 23 what you have to round correctly in here so learn your rounding rules you look at the decimal place past where you're trying to round so if I said three decimal places I think on on your calculator it would have been uh 2 32556 yes no no what is it it's 23 25 58 58 okay pretty close come on now so we're trying to round here right we look just at the digit to the right not all this crap we don't care about that just at this digit if that's five or more you better round this one up if this is less than five you leave this one alone and then you stop your your your number so in our case we are 233 if you route incorrectly in this class it's going to kill you and the reason is is because we deal with very small numbers and very precise numbers a lot of the times we'll be at the fourth fifth sometimes even sixth decimal place when you're working with equations if you round incorrectly on just a little portion of it and we use that number over again then we use that number over again we use that number over again do you see how your error is going to multiply do you see that it's like an exponential error after that that's not a good thing you're going to be way off and in this class it's a really a precise class so here we need to learn how to round if if you don't really know you're not quite sure come and see me I'll help you out with that so in our case we had I'll I'll make make it also a percentage so we rounded to 233 we'll make it 23.3% how many people got 23.3% good all right now you've already done the rest of one I'm sure so what's our four divided by 43 9 what now 3 9 9 thanks do you want us to put no I just m how about the two which one okay how about four the one how about for the zero Z I think I can figure that one out again and zero by the way how much should this add up to better right if you add that and it comes out like 95% or like 120% you probably made a m Stak somewhere so that's a way that that you can check your work your relative frequency should add to 100% that's what it should do if it doesn't you got an error somewhere now because we're rounding could you technically be over or under a little teeny bit Yeah you could because you're rounding but it's not going to be much it's not going to be like over a percent so if it is then you have an error there how people are with me so far relative frequency good so relative means as compared to the whole that's what relative means it's a percentage how much over under would it be wrong probably not over a percent okay could um if we were over like 3% if it was like 103% there would be something wrong either that you have a lot of classes that could I guess that could happen around okay so let's move on from relative frequency distributions there's one of them when we have to talk about before our histograms and some of our graphing here it's called a cumulative frequency distribution you ever heard that word cumulative you you have a something that's cumulative right your GPA you have the GPA cumulative GPA is not just for a semester is it it's for what whole college experience yeah it's exactly right so cultive means you're kind of adding to it as you go right you take a semester that's your GPA that's also your cultive but the next semester you take not only do you have a GPA for that individual semester you combine it with the first one are are you following me along because I'm going to make I'm draw an analogy here after you take another semester it's not just that one but it's this one combined with the other two and one after that is combined with other three you just com keep on combining them and combining them that's how a cumulative frequency distribution works as well it just keeps on adding to it class by class so cumulative frequency distribution this keeps on adding as you go class by class I'll say add sequential classes together let's try this for a second so we have uh we have our very first class how many people in our very first class so if we're going to make cumulative frequency distribution what you keep asking yourself is how many people do you have right now including all the previous classes so after our first class check it out after our first class how many people do we have total after this right at this class now now we're going to go down to the next class how many people do we have total after this class remember it's cumulative so it includes the first class how do you get to 35 great so we're just adding up everything before that's what makes it cumulative as you keep going down you just add everything up above it so this is going to be 35 that's how many people are this class or above how many people are going to be after our third class over here so you're not adding these two are you you're adding the this column your frequencies cul the frequency means you're adding the frequency as you go so here we had only 25 people here we have 35 people here we have how many again how about after this one how about after this one this one 42 guess what is this a coincidence those are the same do we added up all the people in a different way we added up all the people right when you get down to the end these should be exactly the same number if they're not well you have a mistake there cumulative frequency will end at the total number that you collected that's it that's what end do you feel okay with all of our class width lower class limit upper class limit class midpoint class boundary making up our frequency distribution and now with relative and cumulative frequency now raise your hand and feel okay with that that's good that's great you'll get some practice tonight when you go home and do some of that homework by the way you're going to go home and do some of that homework all right what's Monday without homework right right I know right it's actually a decent Monday okay we are in a second going to make this more graphical because a lot of people have trouble just looking at numbers and understanding really what we're trying to get them to grasp so especially in in real life we like to use graphs uh pie charts or or histograms bar charts things like that to make our data more visible because people can look at a graph and go oh I see but when you look at the number a lot of people aren't going to really grasp that concept you get what I'm saying so we're going to I'm going to show you how to make up a few graphs here before I do that I have to talk about the word normal I don't mean normal like guy is not normal okay that's not the type of normal we're talking about we're talking about data normal data normal means that data goes up to a center point and goes back down this data right here is not normal you're going to see that more in our graph than than uh than in this data but this data can you see how most of the data is centered right here at the very beginning of our frequency and then it really Trails off here do you see that there's no Trail off up here it's starts highest and it goes lowest you see that this means it's not normal normal data would look like this if we were to graph it on a like a bar chart or something it would have a rise Peak and a fall that's a normal distribution this one is not going to be a normal distribution we'll see that when when we graph it so when they ask you what normal means normal means that your data Rises to a peak and then falls down again that's what normally is okay so let's try making up a couple graphs we're going to do a histogram first if you're not familiar with the histogram you know what a bar chart is Right bar chart has a horizontal and a vertical axis and you just make little bars a histogram is simply a bar chart where the bars are touching there's no Space Between the Bars which is why we had to have the class bounders in the first place so we had our bars touching so histogram what we're going to say is this is just a touching bar chart touching bar chart not t talking about touching like hallmart touching you know that right thank you for my Pro not like that it's like you know the bar you get that I was just kidding by the way that was a joke told you it was funny it was it was funny you got to take my word for it of course we're going to have two axes the horizontal and the vertical it's not really an XY because it doesn't go anywhere else it's only the quadrant one if you were going to consider it that way on the horizontal axis we are going to put our classes you use either the midpoints or the boundaries it's your choice by the way you don't have unless they specify which one you're going to use it doesn't matter you use the midpoints or the class boundaries on the vertical is going to go your frequency or your relative frequency a bigger bar there okay so let's try this we're going to use either class midpoints or boundaries I'll show you both cases and show you that it really doesn't make that much of a difference what you use uh some people can choose to leave a little Gap here I don't I just make it If This Were a straight line i' make it uh right against this vertical a here so we look at our first class what we're going to do is make up how many classes do we have to get our eight bars they have to be equidistant so for instance if I'm going to make this one that far apart can't make the next one like that that would just look silly so we're going to use the same spacing all the way through so we have our eight we're going to have our eight bars here here's the difference between the class midpoints and the class boundaries do you guys have the midpoints still on your papers if not go back to them here can you tell me all my class midpoints the first class midpoint here's how you use your you know maybe show your boundaries first and I'll show you big points show me your because you said the 19.5 right let's do the boundaries first um the first class boundary was 19.5 here's how you would show your class boundaries you do 19.5 here what's the next class boundary was it 17 17.5 the first one good oh you said the midpoint huh yeah yeah and I said midpoint first and ask for boundaries see I screwed you everyone up my bad 17.5 thanks for that correction what's the next class down how about the next one we and these should be on your paper I just happen to erase them and 49.5 is our last one then we go back to our data and we find out for each class how many people were there so can you tell me how many people were between 17.5 and 21.5 as it relates to our class that's just talking about this class right here it's a little bit of overlap but not much uh we just need those that 0 five to make sure that our our Gap there's no gaps between our our bars there so how many people are we talking about here how many 25 okay so 25 people are between these ages this relates to our first class so we're going to have a bar that looks like this and we just keep on going how much should our next class be 10 so drop all the way down here to 10 looks like the next class was four so we're going to drop down to four do that then we had two we had one we had zero so we're not going to put anything there there's there's no bar there we're going to go back up to one if you followed that nuder head see see if I still have you with it here so we're just putting this one in a picture same information in there same exact information we are using boundaries instead of Upper and Lower Class limits because otherwise we'd have a gap here and we want to show that we're not missing any people that's why we don't have any gaps so if you're a person between these ages you were counted you have no no missing pieces here yes Does it show the data a little bit better for the common person yeah yes absolutely instead of looking at the numbers you go oh wow most your people are here aren't they you go yeah we have a a strong drop off strong drop off do you see how it doesn't fit the normal curve a normal curve would have look it look at the board everybody okay some of you are looking look up here a normal curve would do this have little bars a little bit bigger little bigger biggest bars in the middle then it would go back down to little bars that's how normal would look a normal curve would be this superimposed on that this is definitely not normal this curve goes like that let me get let me get the sound effect right right this one go that's how normal sounds see pretty it's prettier this is not that pretty just Fizzles out so that is our that's our histogram that's the most basic histogram it has to do with our frequencies can you change it to include midpoints instead of boundaries yes and it's so easy you don't change anything all you do is you ready for it erase this for to me I like the midpoints better it looks cleaner you put the midpoints you're using the number that represents your your data class the best so here you would use what was our first midpoint can you tell me okay and then you'd have the 23.5 and so on and so on and so on so youd have 27.5 am I getting them right so far next one should be 35.5 so you can use class midpoints to make up your your histogram as well either one it really doesn't matter what if we were to make a relative frequency this is a basic frequency distribution standard frequency distribution if we were to make a relative frequency distribution the graph wouldn't change but this information would change from frequencies to relative frequencies so you'd say oh okay this was 58% so maybe you'd say like oh this would be like 60% and then you go down from there you say this would be like 50% 40% 30% everything would look identical that's about right right everything exactly the same only here you wouldn't have frequencies anymore you would have relative frequencies you just have percentages looks the same doesn't it just has those percentages off to the left hand side notice that the information didn't change it's still the same information it's just how you represent it on your graph still okay with me okay the last thing that we can do in our last minute is you can make up a cumulative frequency distribution and that is just this information put on that graph so I'll change it one last time the graph will change in this case we're just putting this information over there so for our first class we'll use the midpoint still we started with how many people 2 okay so we have 25 R from that can someone out there tell me the next bar that I'm going to make am I going to go back down to 10 if I'm making a cumulative frequency distribution go 35 I'm now plotting these numbers over here I'm doing these bars because we're doing a cumulative frequency distribution so we go next up we go to 35 next one would be what not much higher then we have 42 then 42 again I think right yeah 43 43 Oops why people do this one is so you can see where most of the growth is uh because notice how it really really Plateau over on the side of it it says after this part was serious growth it well as far as our our ages here and then nothing we really didn't have that many people over here that's you can see that with the the frequency distribution as represented on a histogram so at this point do you feel okay on making up all these things you can do a frequency distribution you can do relative frequency distribution cumulative you can represent on a graph how many bull everything we talked about today good deal all right