### Elementary Statistics: Organizing Quantitative Data, The Popular Displays

this is organizing quantitative data the popular displays and first the first item is a histogram this is constructed by drawing rectangles for each class data the height of each rectangle is the frequency or relative frequency of the class the width of each rectangle the same and the and the same and the rectangles touch each other there we go now we’ve got a frequency distribution this is stated values either individually or by groups of intervals lot of their corresponding frequencies or counts companies laid in a bar chart show this maybe I have ages and that goes from 0 5 10 15 n s goes to 4 and this one goes to nine this one goes to 14 and this one goes to 19 then we have frequency it is 5 in this range to anas range 12 minutes range in three in this range that’s an example of frequency distribution now we can also display that with a histogram construct about draw rectangles for each class date and we’ll see more of this in a little bit now I’m just like before we had a frequency distribution you can have a relative frequency now the relative frequency you’re going to some these together five plus seven x plus two is seven plus 12 is 19 then 22 then for our relative frequency will take each one of these frequencies / that total and then multiply it times 100 so this one will be 2 / 22 x 100 so will be 12 / 22 x 100 and this will be 33 / 22 x 100 now let me do this I’ve / 22 x 100 and I’ll round it though I don’t know one decimal place 22.7 percent 2/22 x 109 into one decimal place to be 9.1 percent and then 12 / 22 x 100 would give us 50 54.5 % 3 / 22 x 100 would give us a thirteen point six percent now typically you like these add up to one hundred percent you know like this point seven plus one is point eight it gives us 14 19 so I do the nine carry the one so I really want one of these to be great or like thirteen point seven percent now technically we’ll leave it this way but if you’re in the workplace well somebody when you take this to a meeting they’ll sit down a DS together and they’ll Wars a missing point one percent it’s like it’s rounding leave me alone but he’ll they’ll do that so now we’re also have something called a cumulative frequency so let’s go ahead and talk about that now cumulative frequency it adds up that current frequency plus all the ones above it so this one’s five this one’s too plus all the ones above it which is five so I gives us seven then this one is 12 plus all the ones above it so two plus five that gives us 1 19 and then the last frequency which is 3 plus 12 plus 2 plus 5 and that gives some 22 so that’s your cumulative frequency now you can also have your cumulative relative frequency where you add up all the the relative frequencies plus all the ones above it so this one’s twenty two point seven percent this next one will be nine point one plus twenty two point seven percent

and I’m not very good decimal splits in CS point 8 130 130 one-point-eight percent let me add up in my calculor 9.1 plus been a long day and create videos all day haha then our next one will add up the current percent plus all the ones above it now all the ones above would be this one plus this one which we are guides 31.8 so to save space I add up the 31.8 because that’s already the ones above it okay so 54.5 plus 31.8 gives us a tee 6.3 and again in theory if i take this the current percent and all the ones above it that should be the eighty six point three percent which should give us a hundred percent Machar’s gives us ninety-nine point nine percent because of rounding but that’s a cumulative relative frequency if any of the problems we refer to that so that’s how you get all of your different different frequencies let’s see just to briefly show that in Excel okay so if i have my ages here now to get i’m going to put a single quote otherwise it’ll think it’s a numerical measure and just soon as I find my 024 so do 0 dash four and then five dash nine we’re going to see that there is a reason why you’d want to even separate these and separate columns but I won’t do it yet 10 to 14 and 15 to 19 okay so we got our frequency here and we got five to 12 3 okay so let’s find our relative frequency well for the relative frequency I need to add these together so I’m click down in this cell here I want my home tab and then I will click the autosum and it’ll guess what you want do the sum on and that’s fine b2 through b5 is what I’m wanting so I click this check mark here or press enter now for relative frequency here i’m going to equals and i’ll do be too because that’s where that number is the number 5 / and i want to / b 6 but specifically i’m gonna put a dollar sign be dollar sign 6 around that then x 100 we’ll talk more about that later but for now that’s our formula okay so now if I click this cell i just created put my mouse in the lower right-hand corner till it becomes a plus click my left mouse button and hold down and drag it down then it’ll on build those and i see i got 22.7 9.1 54.5 13.6 so the same same values now if i look at each individual formula here you notice this says be three which is this cell / b6 let me press escape to get out of that let me click this one click in here be 4 / be six now so is it as a filled it down it’ll just itself based upon what row its up but by putting the dollar signs before the B and dollar sign before the six that’ll hard-code it so it’ll always point to this no matter how much how much I fill it fill it down okay our cumulative frequency I’m going to do an equals and it’s on the current frequency plus all the ones above it so I do five here this one I can do and if I don’t want a hard coded by the way I do equals B to not point to the five this one I’ll do an equals B 3 + b2 so I have the current frequency plus all the ones above it this one I’ll do equals B 4 plus B 3 plus B to the current one puts all the ones above it now there’s a very quick way to do this but and then equals and then b4 b5 sorry plus B 4 plus B 3 + b2 now I specifically show this so you can see the exact formula for cumulative frequency now press Enter

ok so now from wanting cumulative relative frequency all I add the current percent puts all the ones above it so this one’s bc to this one will be be at C 3 plus c2 so I’m adding the current for a relative frequency plus all the ones above it this will be equals C 4 plus C 3 plus c2 current crimpers relative frequency plus all the ones above it and last one will be C 5 plus C 4 plus C 3 plus c2 then I’ll give me my my how to press ENTER now notice I didn’t have any kind of rounding issues here because I’m in excel in it it keeps all the decimal places for me but that’s how you build all of your different frequencies the excel now it also says a frequency distribution besides listening data values on the corresponding frequencies can be displayed in a bar chart and we’ll see more of that here in a little bit now we got something called our lower class limit this is smallest numbers can be long to the different classes so let’s take a look at this my lower class limits LCLs and my upper class limits UCL’s so for my example above my lcls would be 0 5 10 and 15 my upper class limits would be 49 14 and 19 like that and actually I meant to draw it over here my pin to work there goes ok so my lcls and my UCL’s so what was it 0 5 10 and 15 and there are benefits to listen II separately that’s 49 now you notice this went from zero to five to 10 the went by it’s increasing by 5’s isn’t it so over here i’ll increase by five so i don’t need to go and look 0 4 plus 5 is 9 9 plus 5 14 14 plus 5 is 19 that’s called class width so on this one our class width is equal to 5 okay well if I get back there there’s my powerpoint so lower class limits upper class limits casada actually I’m put that on there there we go now we got open into dissipations this is the first class has no or lower class limit or the last class has no upper class limit so maybe you want just greater than 15 maybe there’s so many oddball characters out here past 15 that the list them all would be ridiculous so we just do greater than 15 now this is the way your your class classes go from smallest to largest now that isn’t written in stone I you could go from largest to smallest but this way I think actually makes more sense now we have gap this is difference between lower class limit of one class and the upper class limit of the class right above it or below it depend upon data being ascending or descending here oops our gap is dif tring where this one ends and this one begins so this goes from four to five so that’s gap of one then you got half the gap this is multiply the gap by point five so half the gap o point five times one gives us 0 point five okay we got an upper class boundary UCB and we also got a lower class boundary lcb and I auto-reverse these actually right there okay so let’s take a look at a lower class boundary in our upper class boundary with their upper class boundary we add half the gap so half a gap is 2.5 so this would be 4.5 now this would be 9.5 we adding half the gap but the fact that i know the class width is 5 i can easily come up with us four point five plus five is 9.5 plus another 5 gives us 14.5 plus another 5 gives us 19.5 now over here lower class boundary we subtract half the gap so 5 minus 0 point 5 gives us 4.5 and add 5 that give

me 9.5 14.5 and you probably see a pattern here right this one’s in ins in 4.5 this one begins in 4.5 this ends a 9.5 that’s it begins in 9.5 now going backwards i’m going to subtract point five so this will be negative 0.5 that’s what we’re saying here lower class boundary is lower class limit minus half the gap upper class boundary is upper class limit plus half a gap now this you know these dangers over here well if your value is 4.2 it’s not in any one of these classes that’s the problem with it but if we include the boundaries then you see how this encompasses all of our values now there is a problem with that you know if he values 4.5 where does it fall as fall here or fall here well the places we use the boundaries that really doesn’t matter one of the places is where we have charts bar graphs like this and you see how we’re 11 ends the other one begins so 4.5 actually is in both places if we look at a bar graph whether the bars right together okay next one is the class midpoint while our class midpoint is the lower class limit plus the next lower class limit divided by two so we get 0 plus 5/2 that gives us 2.5 next one would be 5 plus 10 divided by 2 that’s 15 divided by two which is 7.5 notice how i went from 2.5 to 7.5 so i can keep adding 5 the class width over and over to get the rest so this is 12.5 and this would be 17.5 this is what we talked about class with difference between two consecutive lower class limits or boundaries or two consecutive upper class limits or boundaries excuse me this one from zero to five five to ten see how they’re all increasing by five so doesn’t matter which one you look at that’s your class with okay constructing a frequency distribution this is just kind of a general guideline it says decided on a number of classes in and this will be between 5 and 20 it doesn’t do any good to have one class and have everybody fall into that for example let’s say I come up with hours watching TV for a week and see this will be 0 to 20 and this one be 21 now see how this one from 0 to 20 1 i’m adding 21 all right that’s class with so i had 21 here and that would give me 41 okay and then I’m my frequency now here I just have two classes and say 50 beep fall here one person falls here you notice how it’s too bunched up here we have no clue of the real breakdown because we had too few classes now if you have too many then won’t mean anything like over here if I have hours and there’s one two three four five and so forth and then if my frequency here’s 11 12 1 there the the class width is so little that we can’t see st. kind of any kind of pattern so it’s usually between 5 and 20 second class calculates class with highest value minus lowest value then divided by N and round up to a convenient number when I say round up to a convenient number adding by 5s is pretty easy now adding by 21s isn’t so easy so if you come up with you know after you’ve done that calculation you come up with 3.17 well I could definitely take it up to four or maybe if you just want to add five five would be there it’s not a real precise procedure you just kind of tweak it those you go on now three choose the lower class limit for the beginning class using the lowest data value or convenient number and grab a drink here ok and then keep adding class worth over and over to get another together LCLs of other classes figure at UCL from these usually a gap of 1 or 0 point 1 or 0

point 0 1 assumed for example if you have one decimal place in your data then use point one since it has one decimal then go through your data and put tally marks an appropriate class and add them all at the end let’s give you an idea how would you do this by hand which isn’t a good way ok so I got 70 80 85 95 100 60 60 to 75 80 391 and we won’t build a distribution decided a number of classes hmm let’s say n is equal to 50 k are small towards the small end calculate the class with highest value minus lowest value then divide by n so highest value is 100 minus my lowest value which would be 60 I guess / in this one comes out to nice but that’s some 40 to 50 5 which is eight it’s not that hard to a date over and over so I could probably live with that if I don’t want to do eight go up to 10 maybe that’d be better go to 10 sometimes you don’t know when to when you’re doing it by hand till you actually get started okay choose a LCL for the begin class using low state of value or convenient number our lowest data class is 60 so you see these are my values so it started 60 here that’s my first lower class limit keep adding class with over and over to get the other use LCLs so my class width is 10 so this would be 70 80 90 and 100 now there is no decimal places in this so I’m assume a gap of 1 so this is starts at 70 this one only in at 69 and have my class width is 10 then this will be 79 80 999 and 109 mmm that was um step 5 then step 6 go through and put tally marks okay my first data point seventy that fall here then 80 would fall here 85 and fall here 95 would fall here 100 fall here 60 would fall here 62 and fall here 75 fall here 83 would fall here 91 would fall here so then our frequency so be two to three two one so I shows our breakdown so these are our classes and then these frequencies go along with it again this is not set in stone maybe eight would have been better to go by instead of 10 sometimes you don’t really know until after your creative as you’re looking at like it’s not quite what does what and look at so the new tweakin maybe make about fives okay relative frequency distribution we’ve talked about replace frequency the Fallen formula class frequence class frequency divided by sum of all frequencies we did that um at the very beginning here and people to frequency we talked about let’s take a look at creating a histogram using technology a little bit easier than way we’ve been doing okay we’ll use our data points here they’re not very exciting but the work okay so our steps a put data in column a and it says let’s say it is an a.1 325 ours won’t be but I get there here it is okay 7085 or 80 85 95 105 a hundred sixty 60 to 75 83 + 91 so ours is nay 13 10 it then says put your upper class limits your bins in the column B and C

ours was if I went with that same same data i just created ours 109 99 89 79-69 so those are upper class limits well to create this histogram you first have to choose your file you come down here to your options then choose add-ins and this point this comes up you choose manage and Excel add-ins you click the Go button there and you want to choose the first two analysis toolpak analysis toolpak vba so we check them and then hook okay and under the data tab you’ll have an option here data analysis so then it says okay click data tab data analysis we actually got one that says histogram here this is on every version of Excel so you click histogram and click OK now for our input range this is where we put it in column a so in our case will have a 1 colon a 10 and our bin range been is upper class limits which was in be won through b5 now I actually went went and one example went from smallest to largest and it works they don’t have to be in the same certain order now down here if you choose chart output and then click OK this will give you our frequency distribution it also give you the histogram the bar graph like that now people may not be used to what you see here see how this is the only your upper class limits well if you know that your gap is one the first one ends at 69 and next one starts at 70 so let’s be 70 to 79 then this one will be 80 through 89 90 through 99 100 threw 109 and going backwards knowing our class width is 10 this would be 60 to 69 then your frequency is going to be two to three two one this is what more people are used to seeing they’re not used to the bin bin values you see they’re very quick way to get a histogram now that’s he steps right here let’s say I want to just create a histogram and I don’t want to think that hard on it I don’t want to put the upper class limit center so mid Leto’s all I got disease well if I choose my data tab choose data analysis choose histogram click OK I’m going to get rid of the bin range blank those out and up here we won’t put in a 1 colon a 10 chart output select it and click OK now it’ll pick what it thinks the best bin values are which may or may not be what you want but if you want a very quick and dirty to create the data then that’s how you do it and I am I show you examples here of the the values you can set you know for the input range the bin range chart output I usually you want to select now we can also create a histogram on our graphing calculator so let’s follow these steps okay so this says put data into l1 summer press my stat button I do in turn edit and our data was 70 so 70 inner 80 inter 85 inner 95 in our 100 inner 60 enter 62 enter 75 inner 83 enter 91 enter then it says okay stat button edit then now we’re going to do a second y equals and actually before I do that and press y equals if you got something left over from algebra go ahead and press it clear to clear that out okay now we’ll do the second y equals then it says choose on our then interim plot one so go ahead and press ENTER on plot one and

we want to choose on the third graph type l1 and one so the on my flashing cursor is on on so I’m going to press ENTER one time down or 0 to type and it says choose the third graph type well that’s one looks like the bar graph of our histogram so I’m going to right arrow over put my flashing cursor on a little bar graph and press ENTER and then it says L 1 and 1 well it should already be set to l1 and one if it isn’t you can down arrow to this and do second and then one put that one there a frequency should always just be the number one now unfortunately when you come down here it’s an alpha mode so you would have to press your alpha key one time and then pressure one if it did wasn’t it one wasn’t already there okay then it tells us press window button suppress window that’s weird okay I’m gonna exit out and we press mode and if you got on the New York to at40 buttons look like mine down here at the very bottom on the second page see where it says that Diagnostics that is off ah I guess this is gonna be a little bit different no matter what on anyone wait way under a second y equals I’m in here once you get done that you do a second mode exit out I guess then press my window okay that says X min is a lower class limit of the first class so we’re lower class limit well we actually started ours at 60 didn’t we so let’s go into 61st lower class limit then the x max is lower class limit of the class beyond the last class that’s weird okay but this will go from 60 to 100 the next one would be 110 so put it in 110 and our xscale xscale is class with which is 10 y min 0 to put 0 in wimax your guess now left this pretty vague because you almost have to kind of guess yeah I I don’t guess here because I know that my highest one frequency wise is three but you don’t know that before you go into it so you could pick something you know leave it as 10 and then says graph button so if i press the graph I get here and I see oh that’s way down there so I can go back to my window and why why max instead of doing is 10 I can change it to five in press graph and I’ll come back here and see okay well it’s still down a little bit let me change it to six so go back to my window go to ymax put six actually I’m for is what I’m thinking of is that was five wasn’t it so for then press graph you can see okay I see a little bit better of it now now if you press trace it’ll put your cursor right here and it’ll tell you for this one n is equal to and it goes from 60 to max is less than 70 60 less than 70 is 69 if I do my right arrow key so yeah this is 70 and then max is less than eighty seventy seventy-nine and it also gives you the end here’s in is three in is two and then in is one now when you’re all done creating that on the calculator then you want to sit your calculator back so press y equals plot once highlight so I do an au pair o put my flashing cursor on plot one and push it on it then down arrow and prove push zoom choosie standard 61 so press six okay now that’s a lot that’s actually the first time I’ve ever created that with those instructions worked out pretty good actually most of the times we just want do a quick and dirty with a histogram now I definitely recommend for my class that if you’re creating the histogram music cell it is so easy but on your ti 384 if I want to click create

a quick and dirty go enter your numbers in go into stat going to edit put all your numbers in then do your second y equals I do enter on one okay we won on so I press ENTER on that down arrow right arrow over to the bar graph make sure it’s selected and we get that site to then do zoom and up arrow and I got a lot of extra junk here you won’t have all these let’s go in the newer ones and we want to zoom stat so I choose that it’ll get me its best guess on what I want so very nice may not give you the class with Slyke you want but you know give you give you something decent now just to make sure that that actually updated me press stat going to edit we change some numbers I put 52 in here 41 okay so we got smaller values okay so let me go back to second y equals in Iran plot 1 and everything is already set up supper perfect so press zoom and I want to choose a ninth one so zoom stat that one and again it’ll give us the bar graph based upon that and repress a trace it’ll tell me that this goes from 41 the less than 50 5.75 you see the class width and the the upper class and lower class may not be that nice now right out of that right arrow here and you see all the different values anyway again to clear it out I press y equals very quick way to turn plot one off as you do an up arrow press ENTER on it and then dinner and then I choose oom and these standards set it back to where it was anyway that’s how create a create a histogram on your ta three TI 84 okay stem and leaf and I think yeah use the religious to the left of the rightmost digit to form the stem each rightmost digit form the leaf example hundred forty-seven would be would have 14 is damn and seven is the leaf now these give you the steps and you can refer to them stems of the digits the left of the rightmost digit let leaves or the rightmost digit right stems in vertical column in ascending order no duplicates right each leaf to the right of these vertical from smallest to largest these can repeat well it’s great one of those and actually I get some ready to say this page it is organizing quantitative data popular displays 801 okay so we’re going to create a stem-and-leaf so I’m given some data here I got 91 92 92 85-83 82 82 71 73 75 now excluding our rightmost digit this what’s left over here we’re going to put down in this column from smallest to largest so I got seven eight nine then the rightmost digit is what we put in this column now here’s um let me do the 70s first 71 73 and 75 so put 135 so going backwards this will be 71 73 75 so give you that now for our 80s we have 82 82 83 85 so we’ve got to 235 so 2 235 now we don’t repeat the values over on this side but we do repeat them over here our leaves now 90s we got one to two so again 91 92 92 this allows you to see the distribution of the data now if it’s a bell curve more or less if it’s kind of uniform this is could almost be considered uniform how high up these go then it says that the data is bunched up

can split into stem into two or more can repeat this referred to split stem so instead of having seven here I could have seven and seven and this first one would be 71 through 74 try it again okay and then our next seven would be 75 through 79 so you split up into two separate pieces there we go now dot plot you don’t see these a whole lot but place each observation horizontally and increase in order in place dot above each observation each time is observed well going across here first one you see one dot here next when you see two dots and three dots and maybe this is down to two dots this is real popular in exercise equipment beyond that I don’t think I’ve ever really seen graphs to that sort not real popular in business and our last one shape shape of a distribution if we look at our our bar graph histogram or if we look at our stem and cell then you can look at and see what how the data is ranged now if it’s straight across that’s uniform or symmetric if it forms more or less a bell-shaped that symmetric now the skewed right the skewed left their opposite would think when I’m looking at this skewed right example looks like me all the data is on the left side so that’s good right now in this one the skewed left looks like all my data is on the right side or that’s it right on this I’ll come I all my data is on the left side here is skewed right over here all my data is on the right side and it’s skewed left so it’s opposite what you might think I go to all the trouble to create a bar bar graph histogram and let’s say this is what it looks like well if I connect together the middle parts of them like that now form something likes like that which is more or less bell-shaped so that’s how you can tell a shape of your distribution now these aren’t that easy to create by hand but in Excel they’re very simple to great so again get use that or you worse case created in TI three TI 84 and that’s into that section so let me bring up for a quarter stop that you

## Recent Comments