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all right welcome back everybody we are moving into chapter two talking about descriptive statistics so um in chapter two we're going to be looking primarily at two main things we're going to be looking at visual representations graphical representations of a lot of the tests that we run the data that we use different ways that we can display it to make it easier to understand and to read and we're also going to be looking at some measurements of data so looking at some of the things you commonly associate with statistics like mean median mode that kind of stuff so that's all coming up in chapter two so let's go ahead and get started with the graphical representations so in section 2.1 we're looking at stem and leaf graphs line graphs and bar graphs so to start off we're talking about a stem and leaf plot this is a way to represent data that is similar to a histogram which we'll talk about a little bit later to draw stem and leaf plot each data value must be broken up into two components the leaf and the stem so basically what we're doing with the stem and leaf plot is we want to be able to write out all the data in a more efficient way so we don't have to write each data point over and over again but then in a way that we can still see what the data is okay so for example here up in the the box still for the number 173 the stem would be 17 the leaf would be three okay so the stem consists of all digits except for the right most digit so like if we had in this example down here we have the number 33 so when we're drawing the stem and leaf plot the stem is the three the leaf is the last three then we have 42 like that and then we have 49 and 49 so instead of just rewriting 49 and 49 again i already have the 40 right here so i'm just going to add a 9 there on this side that covers one of them and then you you have to put duplicates as well so there's another 9 there so by writing it this way now we can read this as 42 49 and 49 okay so three data points all consolidated into one little section here okay so it's very helpful to write it more concisely this way but still see what the data is and it also gives you an idea of where the data is grouped the most so you can see right there's more data here at 4 in the 40s than there is in the 30s okay so that can also be helpful a helpful way of reading that so if you look at the next page they actually did this one for us so this stem and leaf plot shows that most scores fell in the 60s 70s 80s and 90s 8 out of the 31 scores or about 26 percent we're in the 90s or hundreds so a high number of a's okay so if we look and you can see right there's the 33 the 42 49 and 49 53 55 55 etc etc and then if you're looking at the shape of it you can see where the data is spread the most all right so kind of like they said right in here 60s to 90s is where most of the data is so that can give us good visual information as well okay so that's a stem-and-leaf plot then we have a frequency polygon so frequency polygon is a line graph where a point is placed on the midpoint of each interval at height equal to the frequency okay so the frequency is just the number of times it occurs typically the points are connected with straight lines to emphasize the distribution of the data all right so let me get a thicker pen here all right so from 140 to 179 okay so we're drawing a frequency polygon for the cost of textbooks the midpoint is 160 so that's down here the frequency is four so that means that i'm going to plot a point here at four all right then from 180 to 219 the midpoint of that is 200 so they've already labeled all the midpoints here there was only one textbook that fell into that category fell into that range of cost so that one just has a value of one okay and then we keep going so the next uh next group has a midpoint of 240 and that one occurred four times so that's right there next one 280 occurred six times 320 that group occurred 10 times and then 360 that group occurred six times the 400 midpoint that group occurred twice 440 that group occurred three times all right and then we go ahead and connect to the dots as best we can and we always go ahead and just start with zeros on the corners right on the edges that's gonna make it uh make it connect okay and then this one i'm not sure what happened there there's a point right there so we've got that that and finish this off bring it back down to zero all right and there is our frequency polygon okay so the frequency polygon it's called a polygon because it does it is a shape right you can you can look at the connected form of it right it's multi-sided um and then it gives us accurate information about what's going on so you can see where most of the cost of the textbooks are is right up here right so right in the middle of all of these ranges is where most of the textbooks are all right so that's a frequency polygon then we come to bar graphs so bar graphs usually pretty pretty common so everybody usually recognizes a bar graph this is a graph that display displays a rectangular bar for each category where the length of the bar indicates the frequency of that category right so however tall the bar is that's how many data points you have there right there's a horizontal and vertical axis the horizontal axis lists the categories the vertical lists the frequencies so for this example by the end of 2011 facebook had over 146 million users in the united states the table shows three age groups the number of users in each age group and the proportion of users in each age group construct a bar graph using this data all right so let me make some axes here there we go and then we're going to have three groups on the horizontal axis so we're going to have if you look in the table the age groups are going 13 to 25 26 to 44 and 45 to 64. so those are each going to represent their own bar in the bar graph and 26 to 44. and 45 to 64. okay so down here this is our our age group and then up here we're going to have number of users in millions all right so in millions so i'm going to count up let's see what we have here we have a 27 million a 53 million and a 65 million so just go i'll just go by tens so 10 20 30 40 50 60 70. 20 50 60 70. okay then our bar graph we need to draw the rectangle for each category and make sure it has the corresponding height so for this first one the age group of 13 to 25 has 60 over 65 million users so that's going to be somewhere somewhere right in the middle there between 60 and 70. the age group from 26 to 44 has about 53 million so that's going to be somewhere right around here okay that's a little bit above 50 million and the age group from 45 to 64 has about 27 million so it's going to be a little lower than 30. there we go and then we can fill these in all right and that's our bar graph so you can see if you're just visually looking at this and you notice that age is increasing as we go this way but notice the number of users is dropping as we go that way so that can give you some quick information about what is going on in this scenario with this data so that is a bar graph and that brings us to the end of 2.1 so now heading into 2.2 we're looking at histograms frequency polygons and time series graphs so a histogram this is one of the most common commonly used diagrams in statistics so this is similar to a bar graph but for quantitative data histograms have a horizontal and vertical axis where the horizontal axis lists the values in the frequency table and the vertical axis lists the frequencies okay so horizontal is the numbers vertical is how many right how many of each do we have okay uh the rectangular bars in a histogram unlike a bar graph these rectangles do touch each other they do come right up against each other so for this example below are data of 16 randomly selected weights in pounds of models from a model agency construct a histogram from this data so what i'm going to do is i'm actually going to use technology show you how to do this with technology and we'll be looking in just a little bit at an example we're going to do by hand but for this one first we're going to hop onto desmos for this one so desmos great tool for especially for these types of grafts so what we would need to do is we would need to take each of those weights put them into a table in desmos and make sure to put them down the correct number of times so i'm going to switch over now okay so if you look on my upper left hand corner here you can see i have all of the data up here notice this is a 16 element list so it's got all 16 data points in it and then all we need to do once we have that data set is just type histogram of the data set and when we do that turn it on this is what we get so let me zoom in here and there you can see it's kind of small because our scale is a little a little wonky here but these are rectangles right and on the horizontal axis you see right we've got like an 85 down there a hundred um and then up on the vertical axis it's measuring their counts okay so uh and we talked about right the histogram the rectangles touch each other and you might be wondering well these aren't touching each other well the reason they're not touching each other is because they're too far apart numerically if you wanted to we could we could make these rectangles wider so i could give them a width of like 10 like this and it starts grouping the data together a bit more and you can see now they're coming right up against each other but just for the sake of maintaining that individual data i'm just leaving it with a width of one but we can make wider histograms as well okay and there we have it so this is what our data looks like and then you can start looking for patterns looking for trends things like that so there are different ways of creating histograms using different software using calculators but i'll always pretty much use decimals for it so there we go all right whoops so here's the calculator steps if you happen to use ti 8384 now let's look at an example where we're actually going to draw one so the total cost of textbooks for the term was collected from 36 students construct a histogram for this data by hand and by using the calculator so we're going to do it by hand and then we're going to also use desmos just to confirm that we get one that looks similar okay so this data we were looking at earlier right we drew drew a graph for it um it was a frequency polygon but now we're looking to do a histogram so let's start with labeling our axes okay so we're going to go ahead and label using the midpoints okay so the midpoints we're starting at 160 and counting up by 40s so i'm going to start labeling so that's 160. 200 and i'll make my text smaller in just a second to make it fit better 240 280 320 360. 400 and 440. let me shrink that down just a little bit there we go that's pretty good and then shift that over now actually let me sorry about that we're just just playing around with this a little bit there we go that's pretty good okay so now you can see those midpoint values on the vertical axis notice we we go as low as one as high as ten i don't have ten tick marks on here so i'm just gonna count up by twos so that's two four six eight ten okay and now we go ahead and draw rectangles that are centered around these midpoints okay so around the midpoint of 160 i want a a rectangle that has a height of four so let me i actually make this blue and let's fill it in blue as well okay so height of four centered around 160. there we go close enough and then centered around 200 that one has a height of one so we've got a give it a height of one down here so it's pretty pretty small one and they are gonna touch uh next one we have a height of four at centered around 240. so another one that's kind of that same that same height okay and then 280 we have a height of six and theoretically these are all supposed to be the same width but little bit wider not good enough okay and then next one around 320 we have 10. so this is our tallest one and then centered at 360 we have another height of six centered at 400 we have a height of two and then centered at 440 we have a height of three all right and there we go we have our histogram so this gives us the same data that that frequency polygon gave us before but this is a more common method that we have for drawing drawing this out so now i'm going to come back to this in a sec but let's look on desmos as well so here you can see i already put in the data so it's a 36 element list so i put it all in there and then i did a histogram of that data and i set a bin width of 40 because in the problem it broke it down into cost groups of 40. and that is right there and if you compare that to the one that we drew you hopefully notice that you can see it on both there it's exactly the same alright so our histogram looks good okay so uh there's a brief note that they want to just mention for drawing histograms with grouped data we can use the midpoints of each class or the lower limit of each class typically i personally use the midpoint just because i think it centers it a little bit better makes it look a little nicer but it just depends on the situation you might use the left hand endpoint that's okay too then there is a time series plot so a time series plot is obtained by plotting the time in which the variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis so a time series plot is just plotting kind of like a frequency polygon but the horizontal axis is time so as time passes so here's an example of that so we have the data showing closing prices of the dow jones from 1990 to 2007. and then we have a time series plot for the data on the left so you can see here right as time passes we're measuring these values right so we've got all these different values being measured as time passes so that's the only difference between a a time series plot and a frequency polygon is that the horizontal axis is time all right and that brings us into section 2.3 so now we're going to talk about measures of the location of the data so we're going to start off talking about the concept of a percentile so the k percentile denoted p sub k of a data set is a value such that k percent of observations are less than or equal to the value so let's say that again k percent of the observations are less than or equal to that value right so you commonly hear examples revolving around like test scores right and there's an example up here someone scored in the 90th percentile of the exam that does not that does not mean that he got a 90 on the test it means that he got a test score that is higher than 90 90 percent of the other scores okay so i'll say that one more time so if if he scored in the 90th percentile it doesn't mean he got a 90 it means he scored higher than 90 of everyone else okay so you know for if everybody else you know if 90 of people got an f and he got a d then he could still be the 90th percentile because he scored higher than 90 of everybody else regardless of what his actual score is so for this example the graduate record examination so the gre is a test required for admission to a lot of u.s grad schools university of pittsburgh grad school of public health requires a gre score no less than the 70th percentile for admission into their human genetics and ph or ms program so what is this telling us it doesn't mean you have to get a 70 percent on the gre to get into this school it means that you must have a score higher oops you must you write that with proper grammar you must score higher on the gre then 70 percent of all the people that took it okay so it's not about the actual score you got necessarily it's about how you did comparatively okay so that's what we mean when we say percentile so now the concept of a percentile brings us into what we call quartiles so a quartile is actually how we divide up data when we're looking at large sets of data and we want to look at kind of groupings of it we group it into quartiles so they're going to be three quartiles so there's the first quartile which we call q1 has 25 percent of the data below it and 75 percent above it the second quartile or q2 which is also called the median is quite literally the middle it's the middle value whe e fifty percent of the data is below it and fifty percent is above and then there's the third quartile q3 where 75 percent of the data is below it and 25 is above it so these are called quartiles because they split the data into quarters right into four pieces so quartiles divide the data into 25 percent groups okay and the interquartile range oops excuse me the interquartile range which we call iqr is q3 minus q1 so what does the iqr tell us it tells us where the middle 50 of the data is if we want to look at the middle then we look at the interquartile range and we will do some work with the quartiles in this section so note the quartiles are the 25th 50th and 75th percentiles okay right because 25 below 50 below and 75 percent below and q2 is another way of defining the median median meaning the middle value and that's exactly what i just said here so the median is the middle value all right so here's more of a visual representation of what the quartiles look like so there's q1 q2 q3 and we have 25 percent of the data blocked into these quartiles and then of course there's the minimum and the maximum value as well these values bring us to what we call the five number summary okay so the five number summary includes the min the max and all three quartiles okay so this can give us very useful information when finding um when finding the quartiles when finding the data set because it can tell us how spread the data is it can tell us you know is a lot of the data really low really high is there a lot in the middle um so these are all things that we need to look at when dealing with the five number summary now when you're obtaining the five number summary okay they're giving a step by step here but basically you're just trying to find the middle of everything so first you find the smallest value that's the minimum okay of course the largest value is the maximum then you want to find the median or the second quartile okay so what you want to do is you want to just find the middle value so you go right into the middle of the data set that's your median if there is no single number to let you find the median then you need to take the average of the two middle numbers okay and we'll see an example of that in just a second and then you do that similarly for q1 and q3 by splitting those in half so let's look at an example so here we have the data of 16 randomly selected weights of models from a model agency find the five number summary and include the interquartile range so what we're gonna have we need the min we need q1 we need q 2. q3 and the maximum okay and then let me just put in the q1 q2 and q3 there we go okay so when we're looking at the data set uh so we can find the min and the max pretty quickly so the smallest value is 85 largest value is 138. so the min is 85 the max is 138. okay now to find q2 which is also the median we need to start working our way towards the middle so we have 16 data points here so that means i'm going to cut that in half so that's eight on each side so we're gonna go one two three four five six seven eight one two three four five six seven eight so this right here is where q2 is now it's not a single value right because there's not one value in the middle so what we have to do is take the average now lucky for us these two are the same number so it's not going to matter but we need we would just need to do 110 plus 110 over 2 which of course is just 110 but that's the process that we would use okay so the second quartile is 110 so that means half the data lies below that number then to find the first and third we need to split these in half okay so now i have up here i have a set of eight data points half of that is four so i'm going to go one two three four and one two three four so the middle here is between the 85 and the 98 okay and so now we do really need to do this calculation so this is 85 plus 98 over 2 which is 91.5 so that's our first quartile so that tells us that 25 of the data lies below that number and 75 percent lies above that number okay and then we do the same thing for the third quartile so we have eight data points so one two three four one two three four and we're right in the middle again [Music] that's q3 and it's between 115 and 115 so we know it's going to be 115. that is our five number summary and then it also asked us for the iqr that interquartile range so remember that's just q3 minus q1 so it's that middle 50 percent which in this case that's 115 minus 91.5 which is 23.5 so that means the middle 50 percent of the data falls within a range of 23.5 units now sometimes when we're dealing with data like this you can get data points that are what we call outliers so outliers are data points that are really far out there away from the data set that can start to skew our results so we always want to check to see if there are any outliers in data sets so that we know is there a data point that's really pulling my averages okay so first we need to determine q1 and q3 okay find the iqr that interquartile range and then determine what we call the fences so the lower fence and the upper fence are what we use to identify um where the outliers would be so we take the first quartile and subtract 1.5 times the iqr and that gives us our lower fence so that tells me right all my data is in here and my lower fence is right here by keeping all the data in from the bottom the upper fence you start at q3 and add 1.5 times the iqr and that bounds in my data above then the question is were there any data points out here right out beyond those fences those would be considered outliers so let's look at the example we just did are there any outliers in this data set well we need the lower fence and the upper fence so lower fence is quartile one which we got up there was 91.5 minus 1.5 times the interquartile range which is 23.5 run that calculation we get 56.25 and the upper fence we start with q3 which is 115 plus 1.5 times 23.5 which is 150.25 so if we go back and look at our data set so bear these numbers in mind 56.25 150.25 our lowest data point is 85. that's within that range our highest data point is 138 that's also within that range so there are no data points that fall outside of these fences so that tells us this one has no outliers which is good for us we definitely like not having outliers now when it comes to finding the five number summary you'll see at the top of the next page here they show you how to do it on the calculator i'm going to show you how to do it on desmos it can be very quick and simple so we're still using the weight data so that's my w up there at the top so let me pull this up here okay so all we need to do if you go into the functions on desmos notice there's a stats one right here there's a you can see a min button and a max button but i'm not even going to use those i'm just going to use the quartile button you can also just type in quartile and i'm going to do the quartile of w and the nice thing about what desmos does is if i try to close that off it says hey wait a second there's a problem quartile requires two arguments so it needs the data set and it needs you to tell it which quartile you're using this is very helpful that decimals can tell you that way you know kind of how to set it up so let's find the first quartile we found in the example the first quartile was 91.5 and there we have it 91.5 then to find the second quartile that's just 110 and to find the third quartile put in 3 and we get 115. now the nice thing about the quartile function we don't even need to use the max and min functions because if you think about quartiles one two and three just picture the minimum as quartile zero right there's zero percent of the data beneath it and that's 85. okay that was our min and think of the max as the fourth quartile right so it's up there 100 of the data is below it so we put 4 and 138 that is our maximum value okay so just using the quartile function on desmos can get us those values really quickly since we already have the data set plugged in here all right so next up interpreting percentiles quartiles and median so let's go back and look at this example uh looking at the cumulative relative frequency column determine the 78th and 34th percentile where does the median lie okay so remember when we're talking about relative frequency we're talking about what portion of the hole is in that section so for example we're looking at number of cars in a household eight percent of houses have zero cars 26 percent have one car 44 have two cars etc etc when we're looking at the cumulative relative frequency we're taking those values and adding them as we go so this says eight percent have zero cars 34 percent have either zero or one right because it's eight plus twenty six seventy eight percent have two one or zero ninety two percent have three two one or zero et cetera et cetera so now let's use the cumulative frequencies to find the 34th percentile and the 78th percentile and then we'll figure out where the median is okay so the 34th percentile that's going to be right there 34 percent of the data lies beneath this value okay 34 of the data lies beneath this value so that means the 34th percentile is going to be one car then the 78th percentile so 78 of the data lie beneath that value so that means that the 78th percentile is two cars okay so 78 percent of people have two cars or fewer and then the median so the median is where fifty percent of the data is above and fifty percent is below well i know that thirty four percent is below this seventy eight percent is below that and 50 is somewhere in between those two okay so the median has to fall between these two values now this 78 encompasses all of the twos the ones and the zeros the 34 only covers the ones and the zeros so this tells me that the median is actually going to be two cars okay and you just picture like if i actually listed these out i'd list four then 13 more than 22 more the median is going to be somewhere among those 22. okay somewhere among those 22 households is where the median would be right there in the middle right so we can read these off of frequency tables as well okay let's look at another example so on a timed math test the first quartile for the time it took to finish the exam was 35 minutes interpret the first quartile in the context of this situation so we have q1 the first quartile is 35 minutes so what does this tell us well this means that since it's the first quartile 25 of the data is less than this so 25 of students took less than 35 minutes to finish that's what it's telling us okay which means that 75 took more than 35 minutes okay and then for the next example taking the obtained information from 2.8 interpret the median so example 2.8 that was the weight example so go back up here that was this one okay so interpret the median the median we got was 110 so that means that half the models weigh more than 110 pounds half way less than 110 pounds that's what the median tells us all right so now heading into section 2.4 we're still talking about quartiles we're still talking about the median talking about this interquartile range but we're going to look at how we can visually represent these values and that is using a box plot so a box plot is a graphical representation of a five number summary it's sometimes called a box and whisker plot this is what a box plot looks like the box plot highlights using a box where the first and third quartiles are okay you draw a rectangle for those two you draw a line vertical line to represent the second quartile or the median and then from the edges of the box you draw lines these are called the whiskers that's why it's called a box and whisker plot to the minimum and to the maximum okay so that's how we draw a box plot okay and we can also represent outliers on this and we'll see an example of that soon okay so let's look at this example so every six months the united states federal reserve board conducts a survey of credit card plans in the u.s the following data are the interest rates charged by 10 credit card issuers randomly selected for the july 2005 survey determine the five number summary of the data from the calculator so so this one we'll just go through it quickly so we need the five number summary so the minimum first quartile second quartile third quartile and and the maximum so the min is the smallest value that's 6.5 the max is the largest value which if you go through that's going to be 14.5 okay the median is the middle value so what you would need to do is you need to write these out in order right because notice they're not they're not all in order right some of them are kind of out of place here once you list them all out in order do exactly like we did in the previous example find the middle in this case there are 10 data points you're going to have to find an average that'll give you 13.6 it's a good one to practice if you're wanting some extra practice with finding these quartiles and then do the same thing we did on the last one you get the first quartile is 12 third quartile is 14.4 now to construct the box plot you get the five number summary you draw the horizontal axis okay to give you scale find the fences locate the quartiles and draw the vertical lines okay draw the horizontal lines finish your box you can label the fences using brackets and place an asterisk for any outliers one important note it is important and required that you draw a horizontal axis with correlating tick marks otherwise the box plot is not to scale like it doesn't tell me any information unless i see that scale on the horizontal axis okay so let's do that so i'll draw my horizontal axis all of the data points in here go back and look are between like 6 and 15. so let me go ahead and put a scale on here so i'm going to start at 6. seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen skip to twenty okay so that gives us our very important scale and now we start drawing our boxes so we've already found the five number summary so let's go ahead and put those in so we've got the 6.5 for the min the 14.5 for the max and then we've got our q1q2q3 so 12 13.6 and 14.4 so we've got 12 13.6 is right around here and 14.4 is right in there there we go let's go ahead and finish off that box okay so again here's our q1 q2 q3 now before we draw our whiskers on this i also want to look at defenses so we need the iqr that's going to be 14.4 minus 12 which is 2.4 so that's going to give us our lower fence and our upper fence so lower fence is q1 which is 12 minus 1.5 times the interquartile range that is 8.4 and the upper fence is q3 plus 1.5 times the interquartile range so 2.4 and that is 18. now on the upper end of this so 18 is where our upper fence is so right right there that's our upper fence our maximum value which was 15 falls within that so i'm going to go ahead and draw a little tick mark for 15 and then connect connect that but on the lower fence notice the lower fence value is 8.4 but one of our data points the 6.5 the min is actually below that number so that is actually an outlier okay so what do we do when the min is an outlier well we go ahead and just make it an outlier so draw it as an asterisk and then the box plot so that whisker will only go to whatever the next lowest number is okay so let me draw the fence first so our fence is at 8.4 so it's right about there we're going to have our value of 6.5 is out here it's an outlier so that's going to be right here so we draw a little asterisk and then our new value our new value that's going to be within the fence will be nine point nine right i feel like that's the next lowest number nine point nine so right about ten here we're going to have just our little tick mark and then connect it like that all right and that is our box plot okay so we can draw a box plot um notice when you're doing the five number summary 6.5 is still the min okay so numerically that's still the minimum value but when drawing the box plot it is considered an outlier so we just stop the box plot at the next lowest value that's inside the fence now when you're doing these using like technology so on desmos let me show you so there is a box plot function on desmos if i can oh there it is so yo see box plot right here right across from histogram so you can draw a box plot of anything so let me just do one with my set w okay and there's a box plot does most will draw it for you um the one thing that desmos does not do it does not identify outliers for you so if you just look at the box plot on desmos that's not going to tell you if there are outliers so you still have to make that determination for yourself and that'll help you in completing these drawings of the box plots