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Video transcript

the owner of a restaurant wants to find out more about where his patrons are coming from one day he decided to gather data about the distance in miles that people commuted to get to his restaurant people reported the following distances traveled so here all the distances traveled he wants to create a graph that helps him understand the spread of distances is the key word the spread of distances and the median distance and the median distance that people traveled or that people travel what kind of graph should he create so the the answer of what kind of graph he should create that might be a little bit more straightforward than the actual creation of the graph which we will also do but he's trying to visualize the spread of information and at the same time he wants the median so what graph captures both of that information well a box and whisker plot so let's actually try to draw a box and whisker plot and to do that we need to come up with the median and we'll also see the median of the two halves of the data as well and whenever we're trying to take the median of something it's really helpful to order our data so let's start off by attempting to order our data so what is the smallest number here well let's see there's one two so let me mark it off and then we have another two another two so we got all the twos then we have this three then we have this three I think we got all the threes then we have that four then we have this four do we have any fives no do we have any sixes yet we have that six and that looks like the only six any sevens yep we have this 7 right over here and I just realized that I missed this one so let me put the one at the beginning of our set so I got that one right over there actually there's two ones I missed both of them so both of those ones are right over there so I have one two three four s no fives this is one six there was one seven there's one eight right over here one eight and then let's see any 9s no 9s any tens yep there's a 10 and the 11s we have an 11 right over there any 12s nope 13 14 then we have a 15 then we have a 20 and then a 22 so we've ordered all our data now it should be relatively straightforward to find the middle of our data the median so how many data points do we have 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 so the middle number is going to be a number that has eight numbers larger than it and eight numbers smaller than it so let's think about it 1 2 3 4 5 6 7 8 so the number 6 here is larger than 8 of the values and if I did the calculations right it should be smaller than 8 of the values 1 2 3 4 5 6 7 8 so it is indeed it is indeed the median now when we take a box-and-whisker when we were trying to construct a box and whisker plot the convention is ok we have our median and it's essentially dividing our data into two sets now let's take the median of each of those sets and the convention is to take our median out and have the sets that are left over sometimes people leave it in but the standard convention take this median out and now look separately at this set and look separately at this set so if we look at this first the bottom half of our numbers essentially what's the median of these numbers well we have one two three four five six seven eight data points so we're actually going to have two middle numbers so the two middle numbers are is this 2 and this 3 3 numbers less than these 2 3 numbers greater than it and so when we're looking for a median you have two middle numbers we take the mean of these two numbers so halfway in between two and three is 2.5 or you say 2 plus 3 is 5 divided by 2 is 2.5 so here here we have a median of this bottom half of 2.5 and then the middle of the top half once again we have 8 data points so our middle two numbers our middle two numbers are going to be this 11 and this 14 and so if we want to take the mean of these two numbers 11 plus 14 is 25 halfway in between the two is 12 point 5 so 12 point 5 is exactly halfway between 11 and 14 and now we've figured out all of the information we need to actually plot or actually create or actually draw our box and whisker plot so let me draw a number line let me draw a number line so my best attempt at a number line so that's my number line and let's say that this right over here is 0 I need to make sure I get all the way up to 22 or beyond 22 so let's say that's 0 let's say this is 5 this is 10 that could be 15 and that could be 20 this could be 25 it could keep going 30 maybe 35 so the first thing we might want to think about there's several ways to draw it we want to think about the box part of the box and whisker essentially represents the middle half of our data so it's essentially trying to represent trying to represent this data right over here so the data between the two between the medians of the two halves so this is a part that we would represent attempt to represent with the box so we would start right over here at this lower this this 2.5 this is essentially separating the first quartile from the second quartile the first quarter of our numbers from the second quarter of our numbers so let's put it right over here so this is 2.5 2.5 is halfway between 0 and 5 so that's 2.5 and then up here we have 12.5 and 12.5 is right over let's see this is 10 10 so this right over here would be this halfway between well halfway between 10 and 15 is 12.5 so let me do this so this is 12.5 right over here 12.5 so that separates the third quartile from the fourth quartile and then our boxes everything in between so this is literally the middle half of our numbers the middle half of our numbers and we'd want to show where the actual median is and that was actually one of the things we wanted to be able to think about in our original when when when the owner of the restaurant wanted to think about how far people are traveling from so the median is six so we can plot it right over here so if this this right over here looks this is about six so that is we doing that same pink color so this right over here is 6 and then the whiskers of the box and whisker plot essentially show us the range of our data and so let me do that I could do this in a different color that I haven't used yet I'll do this in orange so essentially if we want to see look the numbers go all the way up to 22 so they go all the way up to so let's say that this is 22 right over here our numbers go all the way up to 22 our numbers go all the way up to 22 and they go as low as 1 so they go 1 is right about here they go as low we label that so that's 1 and they go as low as 1 so there you have it we have our box and whisker plot and you can see if you have a plot like this just visually you can immediately see ok what is the median it's the middle of the box essentially it shows you the middle half so it shows you how far they're spread or kind of where the the meet of the spread is and then it shows well beyond that we have the range that goes well beyond that it goes or or how far the total spread of our data is so this gives a pretty good sense of both the median and the spread of our data