Interquartile range (IQR)

let's get some practice calculating interquartile ranges and I've taken some exercises from the Khan Academy exercises here and I'm going to solve it on my scratch pad the following data points represent the number of animal crackers in each kid's lunch box sort the data from least to greatest and then find the interquartile range of the data set and I encourage you to do this before I take a shot at it all right so let's first sort it if we were actually doing this on the Khan Academy exercise you could just drag these you could just click and drag these numbers around to sort them but I'll just do it by hand so let's see the lowest number here looks like it's a 4 so I've had that 4 and then I have another 4 and then I have another 4 and let's see are there any fives no fives but there is a 6 so then there's a 6 and then there's a 7 there doesn't seem to be an 8 or a 9 but then we get to a 10 and then we get to 11 12 no 13 but then we get 14 and then finally we have a 15 so the first thing we want to do is figure out the median here so the median is the middle number I have 1 2 3 4 5 6 7 8 9 numbers so there's going to be just one middle number I have an odd number of numbers here it's going to be the number that has 4 to the left and 4 to the right and that middle number the median is going to be 10 notice I have 4 to the left and 4 to the right and the interquartile range is all about figuring out the difference between the middle of the first half and the middle of the second half it's a measure of spread how far apart all of these data points are and so let's figure out the middle of the first half so we're going to ignore the median here and just look at these these first four numbers and so out of these first four numbers I have since I have only have an even number of numbers I'm going to calculate the median using the middle two numbers so I'm going to look at the middle two numbers here I'm going to take their average so the average of 4 and 6 halfway between 4 & 6 is 5 where you can say 4 plus 6 is 4 plus 6 is equal to 10 but then I want to divide that by 2 so this is going to be equal to 5 so the middle of the first half is 5 you can imagine it right over there and in the middle of the second half I would have to do the same thing I have four numbers I'm going to look at the middle two numbers the middle two numbers are 12 and 14 the average of 12 and 14 is going to be 13 is going to be 13 if you took 12 plus 14 over 2 that's going to be 26 over 2 which is equal to 13 but an easier way for numbers like this you say Li 13 is right exactly halfway between 12 and 14 so there you have it I have the middle of the first half this 5 I have the middle of the second half 13 to calculate the interquartile range I just have to find the difference between these two things so the interquartile range for this first example is going to be 13 minus 5 the middle of the second half minus the middle of the first half which is going to be equal to 8 let's do some more of these this is strangely fun find the interquartile range of the data and the dot plot below songs on each album in Shane's collection and so let's see what's going on here and then like always encourage you to take a shot at it so this is just representing the data in a different way but we could write this again as an ordered list so let's do that we have one we have one song or we have we have one album with 7 songs I guess you could say so we have a 7 we have two albums with 9 9 songs so we have two nines let me write those we have two nines then we have three tens cross those out so 10 10 10 then we have an 11 we have an 11 we have 2 12 to 12 and then finally we have so use those already and then we have an album with 14 songs 14 so all I did here is I wrote this data like this so we could see okay this album has 7 songs I sell them as 9 this album has 9 and the way I wrote it it's already in order so I can immediately get get I can immediately start calculating the median let's see I have one two three four five six seven eight nine ten numbers I have an even number of number so to calculate the median I would have to look at the middle two numbers so the middle two numbers look like it's these two tens here because I have four to the left of them and then four to the right of them and so since I'm calculating the median using two numbers it's going to be halfway between them it's going to be the average of these two numbers well the average of 10 and n is just going to be 10 so the median is going to be 10 median is going to be 10 and in the case like this where I calculated the median using the middle two numbers I can now include this left hand in the first half and I can include this right 10 in the second half so let's do that so the first half is going to be those five numbers and then the second half is going to be these five numbers and it makes sense because I'm literally looking at first half it's going to be five number second half is going to be five numbers if I had a true middle number like the previous example then we ignore that when we look at the first and second half or at least that's the the way that that we're doing it in these examples but what's the mid what's the median of this first half if we look at these five numbers well if you have five numbers you have an odd number of numbers you're going to have one middle number and it's going to be the one that has two on either sides this has two to the left and it has two to the right so the median of the first half the middle of the first half is 9 right over here and the middle of the second half I have one two three four five numbers and this 12 is right in the middle you have two to the left and two to the right so the median of the second half is 12 interquartile range is just going to be the median of the second half 12 minus the median of the first half nine which is going to be equal to three so if I was doing this on the actual exercise I would fill out a 3 right over there