in this video i will talk about graphs of frequency distributions we will discuss how to construct frequency histograms frequency polygons relative frequency histograms and ogives so first let's talk about a frequency histogram it's a bar graph that represents the frequency distribution the horizontal scale is quantitative and the measu and measures the data values the vertical scale measures the frequencies of the classes the bars must touch consecutive bars must touch so the general shape or basically you've probably seen histograms before looks something like this they got the data values along the x-axis and the frequencies along the y-axis because consecutive bars of a histogram must touch bars must begin and end at class boundaries instead of class limits the numbers that separate classes without forming gaps between them so let's look at this example we're going to draw a frequency histogram from this frequency distribution we talked about this frequency distribution in my last video so what we're going to do is we're going to determine the class boundaries so we know the distance from the upper limit of the first class which is 190 and the lim um from the upper limit sorry the distance from the upper limit of the first class to the lower limit of the second class is one so the upper limit was 190 the lower class limit for the next class was 191 so that's a difference of one so what we're going to do is we're going to cut that one in half which is 0.5 half the distance is 0.5 so your first class boundary will be 154.5 so basically you go one back or half back and then your first class upper boundary will be 190.5 so you go half up okay so you go half back from the lower limit half up for the upper limit so these are your class boundaries as you can see in the second column all we did was go half back half up so i got it for each class and then you can draw your your um frequency histogram so if we look at the first class let's look at the second graph these are the same types of graphs except the vertical axis horizontal axis is different so if we look at the second one we're looking at the class boundaries the first class boundary was 154.5 to 190.5 and it has a frequency of three so when we go from 154 to 190.5 right here we draw our bar up to three so right here goes up to three for the second class we had a frequency of two so we'll draw the bar up to two now there are two ways you can make a histogram you can either do it with the class boundaries or you can do it with the midpoints we know the midpoint of the first class was 170 2.5 the middle of this first class is 172.5 i don't have it on this slide but that's what we did in our last video this that's the middle of this class so we can draw draw the bar there 172.5 it has a frequency of three the next one was two zero eight point five it has a frequency of two so as you can see they they have the same shape different horizontal axis you want to put a broken axis here to indicate you this is where your graph starts you never started at zero so as you can see this is a visual before we just had tables or charts and we just looked at the numbers but here you have a graph and from this graph you can see that two-thirds of the adults are paying more than and fifty cents 262.50 out-of-pocket prescription medicine okay all right so now let's look at a frequency polygon and it's a line graph that emphasizes the continuous change in frequencies so we're going to do a frequency polygon based off this data here and to construct it we're going to use the same horizontal and vertical scales that we used in the histogram labeled with the class midpoints so our vertical axis will look like this it will have these numbers those are the class midpoints and when you draw a frequency polygon it should start on the x-axis and it should end on the x-axis so should end on the whole horizontal axis start and end on that axis so what we're going to do is since our first midpoint was 172.5 we need to go back and if you recall our class width for this particular problem was 36. so what we're going to do is we're going to go back 36 and that gives me 136.5 this was our midpoint for the last interval that we had but we need to go up 36 so when we go up 36 this gives me 42.4 sorry 424.5 so when we draw our graph we're going to draw our numbers at the midpoint so for the first class for this first class we have a frequency of three so at our midpoint 172.5 we're going to go up to three the second class had a midpoint of 208.5 had a frequency of two so then we draw our graph down to two okay but you have to put a line going back to hit the horizontal axis and you when you get to this last class you have to draw a line that hits the horizontal axis to end it so this is a frequency polygon a relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram the only difference is your vertical scale will have relative frequencies and not frequencies okay so let's go back this is our table that we had our expanded frequency distribution table and this time we're going to look at these numbers here these will be on our vertical axis so that's what we're going to graph we're still going to have the same class boundaries but our relative frequencies will go along the vertical axis so when we go from 144.5 to 190.5 that was our first class and as you can see we have a relative frequency of that first class to be 0.1 so we'll draw our graph up to 0.1 for the second class we have a relative frequency of 0.07 when we go when we draw next uh graph my next bar it will go up to zero point zero seven and so on so as you can see it basically has the same shape as the frequency histogram except a different vertical axis the last graph i'm going to discuss is the cumulative frequency graph or orgive it's a line graph that displays the cumulative frequency of each class at its upper class boundary the upper boundaries are marked on the horizontal axis the cumulative frequencies are marked on the vertical axis so if you look at this graph this is the general shape you have your data values along the uh x and your cumulative frequency along the y-axis or the horizontal axis you're going to put your upper boundaries on the horizontal axis actually so what you're going to do is you're going to specify the horizontal and vertical scales once again the horizontal scales consist of the upper class boundaries and the vertical scale will consist of your cumulative frequencies you're going to plot points that represent the upper class boundaries and their corresponding cumulative frequencies and then you're going to connect the points in order from left to right the graph should start at the lower boundary of the first class and it's going to be at zero and then you should end at the upper boundary of the last class and when you draw your line up to that point that should be your sample size so if we go back to this problem remember when we did this problem we had um a sample size of 30. so when you add up your frequencies it should be 30. so when we end at this graph we should end up at 30. all right so let's look at this graph this is the ogive remember i said that the graph should start at the lower boundary of the first class and it should be at zero the lower about boundary of the first class is 154.5 so that's where we're going to start our graph at 154.5 and it's going to be at a zero as you can see the rest of these numbers are the upper class boundaries if you go back to the table these are my upper class boundaries that's what i'm graphing along the horizontal axis okay so at 190.5 we should be at 3. 190.5 we should be at 3. now remember this is cumulative so when we go to 226.5 we got to add the 2 and the 3. therefore we should be at five and we are 226.5 we're at five when we go to 262.5 that's this number here we need to add all this up so now we should be at 10. we're adding all three of these numbers so if you look at the graph 262.5 we're at 10. so we keep doing this q until we get to the very last class boundary the very last class boundary is 406.5 and at this point we would have added up all the numbers and when we add up all the numbers we should be at 30. so from this graph you can see that a total of 10 adults had expenses of 262.50 or less okay because this is cumulative so remember this was 10 here so at this point 10 adults had expenses of 262.50 or less at this point also the greatest increase in cumulative frequency occurs between 298.5 and 334.5 so it looks like this is the greatest increase right in here graph cast kind of shoots up okay that's the end of this video