In this video, we’re gonna talk about how to make box and whisker plots. Now what you need to be able to do? Is you need to be able to identify five key data points in your data set? The first two are very straightforward. It’s the minimum and the maximum now. The other three data points are the first quartile, the second quartile and the third quartile. Now, once you have that, you can plot those things on a number line and then draw the box and whisker plot or round out. So let me show you an example, so you can see how this is going to work, So let’s say we have the numbers 11 22 2014 29 and then 835 27 1349 1024 and 17 So first, let’s determine the three quart’s house. In addition to the minimum and the maximum, the first step is to arrange the numbers in ascending order, so the lowest number that we have here is 8 and then 10 and then 11 now lets cross out those numbers. The next number is 13 and then 14 and then 17 after 17 we have 20 22 24 and 27 after that is 29 which we can see it here 35 and then 49 so at this point, we have a total of 13 numbers in our list now. The first thing is to break the data into two equal parts, so let’s calculate q2 which is the median of the entire data set. So if we eliminate the first two numbers on both sides. And then if we keep doing that until we get the middle number, this will give us the median, which is 20 now what I’m gonna do is I’m gonna draw online and then put 20 on top. So this is the second quartile So now I have two equal parts of data that have six numbers on the left side and six numbers on the right side now. I need to find a median of the lower half of the data. So if I eliminate the first two and the last two, I mean the next two. I have two numbers in the middle. So the median is going to be the average of those two numbers. The average of 11 and 13 is 12 so this is q1 the first quartile. Now the median will be between these two numbers that is the median of the upper half of the data, so the average of 27 and 29 is 28 so that’s q3 the third quartile, so we have q1 q2 and q3 now. The next thing we need to do is identify the minimum value and the maximum value. The minimum value is the lowest value in the data set, which is 8 and the maximum value is the highest value in the data set, which is 49 now. We need to check to make sure that these two values are not outliers, because if they’re outliers, they’re not going to be part of the box and whisker plot. They will exist outside of that. Now what we need to do Is we need to determine a range of numbers in which the outliers can’t be, so it’s going to be q1 minus 15 times the IQR value to q3 plus 15 times the IQR value, so any number that is outside of this range? That is part of the data set is an outlier now the IQR value The interquartile range is basically the difference between q3 and q1 so q3 is 28 Q1 is 12 so the interquartile range is 16 so using this interval it’s going to be q1 which is 12 minus 15 times 16 and then q3 is 28 plus 15 times 16 Now let’s get a calculator and plug these numbers in so 12 minus 15 times 16 That’s gonna be negative 12 and then 28 plus 15 times 16 That’s 52 so negative. I mean, not negative 8 but 8 is in this range so 8 is not an outlier. It is the minimum of the data set. 49 is also between negative 12 and 52 so 49 is not an outlier. So now that we know, we don’t have any outliers. At this point, we can draw the box plot or the box and whisker plot. Let’s begin by creating a number line. The lowest value is 8 and the highest is 49 So let’s start from 0 and let’s go by tens until we get up to 50 so first, lets plot. Q 1 Q 1 is 12 which is approximately right there, and then Q 3 is 28 which is just under 30 and then draw a rectangle now. Q 2 is 20 So we’re going to put a line here so as you can see, this is. Q 1 Q 2 and Q 3 now. Our next step is to plot the minimum, which is around 8 so there it is, and then the maximum is at 49 so we’re gonna put a line just below 50 and so that is the Box in whisker plot that corresponds to the data set that we see here, so that’s how you can draw, but now what about if we had an outlier? How would that impact the box and whisker plot? So let’s consider an example in which that’s the case so first let’s write out a list of numbers. Let’s say we have the numbers 18 34 76 29 15 41 46 25 5438 twenty thirty to forty three and then 22 So if you think you know what to do, feel free to pause the video and try it, so let’s begin by put in two numbers in ascending order, so we have 15 18 20 and so here are those numbers and then, after 20 it’s 22 and then 25 and then 29 after the 29 it’s 32 34 38 and 41 and then after that it’s gonna be 43 46 54 and then the last one is 76 So we have a total of let’s see. This is 3 6 9 12 14 numbers, so we want to split it into two equal parts. Let’s put a line between the seventh and the eighth number, so we got seven numbers on the left side. Seven numbers on the right side. So the median is going to be the average of those two numbers. The average of 32 and 34 is 33 so this is the second quartile. Now we need to determine the median of these seven numbers, which is going to be the middle number 22 but let’s replace 22 with a line so we can split the left side into two equal parts of three numbers, But I’m gonna put 22 on top, so you can see that It represents q1 the median of the left side of the data, now 43 is the middle number of these seven numbers. So let’s do the same thing. Let’s replace 43 with a line and so this is going to be the third quartile and we’ll put 43 on top, so keep in mind the quartiles. They divide the data into four equal parts, so we have four equal parts of three numbers now. I’m gonna put a comma between 46 and 54 so it doesn’t look like 4650 for our next step is to calculate the interquartile range so its q3 minus q1 so it’s the difference between those two values, so it’s 43 minus 22 which is 21 So that’s our IQR value Im. Excuse me, our next step is to see if we have any outliers. So first, let’s calculate the range in which no outliers should exist. So q1 is 22 and IQR is 21 Q3 is 43 22 minus 15 times 21 that is equal to negative nine point five, and then 43 plus 15 times 21 is seventy four point five. Now, are there any numbers in our list of numbers and that is not in this range 15 is in this range, but 76 is not so therefore 76 is an outlier, so we can’t include that in the box and whisker plot 76 will be outside of it and so now we can plot the box and whisker plot, so let’s start with the number line, so let’s go up to 80 so this is going to be zero. We’re going to say 80 is over here, so this is 40 and this is 20 and 60 and in between are 10 30 50 and 70 so let’s start with Q 1 Q 1 is 22 which is just above 20 and Q 2 I mean Q. 3 is 43 so that’s gonna be to the right of 40 so this is just a rough estimate now. Q 2 is 33 which we’re going to put here. So this is Q 1 Q 2 Q 3 now! Our minimum value that is not an outlier is 15 the highest value that is not an outlier is 54 So we’re gonna plot. Those two numbers 15 is right between 10 and 20 and then 54 is almost in the middle between 50 and 60 but a little bit closer to 50 and so this is the minimum, and this is the maximum that is not an outlier now to show the outlier. All we need to do is basically put a point at 76 which should be somewhere around here and that’s basically it. So that’s how you can show the outlier that exists basically outside of the box and whisker plot. Thanks for watching.