College Math Lesson 4 3 Five Number Summary

lesson 4.3 five-number summary our objective for this lesson is to be able to give the five-number summary and draw box plot for data set in this lesson and the next lesson we're going to be talking about ways to measure the variation in distribution and the first way is going to be with the range the range is simply the difference between the highest and lowest values in data set so we find the highest or maximum value in the data set and subtract from that the lowest or minimum value in the data set and that difference we call the range next we need to define quartiles so quartiles are values that divide the data distribution into quarters the lower quartile will divide the the lower fourth of the distribution from the rest of the distribution and the upper quartile will divide the upper fourth of the distribution from the rest of the distribution and then the median is the middle quartile so the way to find these quartiles is to first find the median and and then find the median of the lower half and then to find that and that will be the lower quartile then you find the median of the upper half and that's the upper quartile all right so the five number summary is description of distribution consisting of the lowest value the lower quartile the median the upper quartile and the highest value so the lowest value and the highest value will be obvious and we've already talked about how to find the median the median will just be the middle number in the distribution or if the if there's an even number of data points in the data set then then we're going to take the middle two numbers and and halfway between those will be the median then to find the lower quartile we take the lower half of the data set or in if if there's an odd number we'll just ignore the median and take everything below the median and then we find the mean the median of that half and that's the lower quartile and then we do the same with the upper half and that will be the upper quartile and those five numbers together make our five number summary all right so to draw the to represent the five number summary visually we're going to draw box plot or it's sometimes called box and whiskers plot and the steps for drawing plot box plot are first to draw number line that spans all the values in the data set so we just need to look at the minimum and maximum values of the data set and make sure that our number line is is big enough to cover all of that that entire range step two is to enclose the values from the lower to the upper quartile in box so we'll put line where the lower quartile will be and then we'll put line representing the upper quartile and then we'll connect those lines together in box the thickness of the box doesn't matter it's just the length of the box between the lower quartile and the upper quartile step three then is to draw vertical line through the box at the median so what we'll have at this point is just box with line at some somewhere in the middle of that box and those three lines then the the left side of the box will represent the lower quartile the line in the middle will represent the median and the right side of the box will represent the upper quartile and then step four is to add whiskers or just straight lines extending from the box to the lowest and highest values so the ends of those whiskers will represent the range the lowest and highest values and then the box in the middle will represent the middle half of the data set from the lower quartile to the upper quartile our example gives us two different data sets and i've written them on the board here one in black and one in red and we're told that these data sets represent 100 meter dash times and there's 20 in each set and we're asked to compare the two data sets using five number summary and box plot so let's start by writing by giving five number summary for each of the data sets so first for the for the black data set we're going to write the lowest value the lower quartile the median the upper quartile and the highest value and that's going to be our our five number summary so the lowest value is just going to be notice that these are in order by the way they they go from lowest through the first row and then down to the second row up to the highest so the lowest value is just going to be this value here 9.92 and the highest value is going to be right out here at 10.38 all right now to find the median since these are all in order the median is going to be relatively easy we just need to find the the value that's halfway between this value the tenth value and this 11th value so halfway between 10.13 and 10.14 is going to be 10.135 now to find the lower quartile we're going to find the median of the first half of these values so that's going to be halfway between this value the fifth value and this value the sixth value so 10.06 and 10.07 halfway between those is going to be 10.065 and then for the upper quartile we're looking for the median of this second row here the upper the the upper half of our data set so halfway between 10.18 and 10.1 or 10.21 is going to be 10.195 all right so that's the that's the five number summary for that black data set for the red data set our lowest value is going to be here 9.89 our highest value is out here 12 flat our median is going to be halfway between this value 10.98 and 11.05 so that's going to be 11. and then our lower quartile is going to be the median of the top row which will be halfway between 10.35 and 10.41 which is 10.38 and then the upper quartile will be the median of the second row which will be halfway between 11.55 and 11.76 which would be eleven point all right so now let's draw box plot so first step is going to be to draw number line so we'll make our number line with values that are sufficient to to cover the entire range so our lowest value in any of these data sets is 9.89 and our highest value is 12. so we want to go maybe we'll start here at 9.5 and we'll go 10.0 then 10.5 then 11.0 then 11.5 and 12.0 so there's our number line and then above this number line we're going to draw our box plots so let's start by drawing the red one so we're going to start by drawing box with the left half of the box is going to be at the lower quartile the right half of the box will be at the upper quartile so we start at 10.38 and we'll draw line right about there for 10.38 and then 11.655 is going to be about here and we'll connect those two lines and make box like that then we're going to draw line in this box where the median will be so the median is going to be right right about at 11. so right about there will be our median and then we're going to draw whiskers so the little just line out from the edge of the box to the lowest value which is at 9.89 so about here and then another whisker out to the highest value which is at 12. so that'll bring us out to there so here's our box plot for this red data set now let's draw the box plot for the black data set here our lower quartile is 10.065 so that's going to be 10.065 is right about there and then 10.195 is the upper quartile so this is just this is going to be much smaller box here and then our median is at 10.135 so maybe right about there for our median and then we'll draw the whiskers out to 9.92 which is just going to be that little bit of whisker right there and then out to the highest value at three 10.38 and so that's gonna be let's see think think this lower quartile no that's that's right this this should go out to 10.38 which is where this lower quartile was there let me make sure that the upper quartile is one that yeah that's that's right so you can see that this is much smaller box plot than this one meaning that the the variation in this black data set is much smaller than the variation in this red data set you can also see that the center of the data set represented by the median is quite bit lower in the black in the black box so we have lower center and lower variation in the black set which would tell me that maybe the red set represents times from some regional qualifier where the the black set would be the championship the olympic games these are very fast times for 100 meter dash and so once you get to the to the highest level you're going to see lower variation as well as lower times