Box and Whisker Plots: A Beginner’s Guide | Math with Mr. J

Welcome to Math with Mr. J. In this video, I'm going to cover box and whisker plots, also called box plots. We will first take a look at what each part of a box and whisker plot represents, so a basic overview. We will go through two examples. And then after that, we will go through two examples of how to make a box and whisker plot. Now at first, box and whisker plots may look complex and not make any sense, but it's just a matter of understanding what you're looking at and what box and whisker plots represent. Now simply put, box and whisker plots are a way to display data and the spread of that data. They give us a visual. Let's jump into our example and see exactly how to read and interpret a box and whisker plot. For our example, we're going to be taking a look at years of teaching experience. So 10 teachers were surveyed and here are the results. Again, this is years of teaching experience. This data is in order from least to greatest and the box and whisker plot has been created below. Now when it comes to box and whisker plots, there are five key parts. They show us a five number summary of the data set. Box and whisker plots display the minimum, the first quartile, the median, the third quartile and the maximum. Let's start by taking a look at the minimum and the maximum. So the minimum number of years of teaching experience is three. The maximum is 18. So the minimum is three years and the maximum is 18 years. So the minimum and maximum are just the smallest and largest numbers in value within the data set. So as far as the box and whisker plot, we have the minimum right here and the maximum right here. So let's label this minimum and maximum. So the whiskers that extend out from the rectangle extend to the minimum and to the maximum. Now that we covered the minimum and maximum, we're actually going to move to the middle and we're going to take a look at this line right here inside the rectangle. That line represents the median. The median is the middle point of the data, the halfway point or the 50th percentile. The median is also referred to as the second quartile. Now since we have 10 numbers within the data set, the halfway point is going to be in between the 8 and the 10. So this is the median. It splits the data set in half. We have five numbers to the left and five numbers to the right. Now we are directly in between 8 and 10. So we take the average of those two numbers to find the median. So we do 8 plus 10 and then divide by 2. 8 plus 10 is 18. Divided by 2 is 9. So the median is 9. Now for this one, we can just think that 9 is directly in between 8 and 10. But keep in mind, whenever you have two numbers and you need to find the median in between, you can take the average in order to do so. So again, the line inside the rectangle is the median. So that's the halfway point within our data. Now we need to take a look at the first quartile, also known as the lower quartile, and then the third quartile, also known as the upper quartile. Let's take a look at the first quartile first. So we need to take a look at the bottom half of our data. So right here. Now the first quartile is going to be the median, or halfway point, within our bottom half of the data. So we have five numbers within that bottom half. So this is going to be the median of the bottom half of the data. That means that's going to be the first quartile. That's the one-fourth mark within our data, so the 25% mark, or 25th percentile. So the first quartile is 7. That's represented by this part of the rectangle. So this is the first quartile, or lower quartile. Again, it's 7. Now we need to find the third quartile, so the upper quartile. Let's take a look at the upper half of our data. We need to find the median, so the midpoint of that upper half. That's going to be right here. So 12. 12 is our upper quartile, the third quartile. That's the three-fourths mark within our data, so the 75% mark, or 75th percentile. So the third quartile is 12. And that is going to be this part of the rectangle, so third quartile. Those are the five parts of a box-and-whisker plot. We have the minimum, which was three years of teaching experience. So minimum was right here. Then we have the first quartile, which was 7, so right here. Then we have the median, which was 9, so right here. Then we have the third quartile, which was 12, so right here. And then we have the maximum, which was 18 years of teaching experience, which was right here. Box-and-whisker plots use quartiles, so the data is split into four parts, so fourths. We have about 25% of the data right here. And then we have about 25% of the data right here. Now the box within the box-and-whisker plot represents the interquartile range, so the middle 50% of the data. And then a whisker extends to the minimum. That's going to be the bottom 25% of the data. And then the other whisker extends to the maximum. That's going to be the upper 25% of the data. Now for one final recap, I'm going to erase all of that writing on the box-and-whisker plot. That way it's a little more clear as to what we're looking at. Now without all of that writing, we can focus more on the box-and-whisker plot. So one final recap here. We have the minimum, which is 3, represented right here. Then we have the first quartile, which is represented right here, and that is 7. Then we have the median, which is represented right here and is 9. Then the third quartile is represented right here, and that is 12. And then lastly, we have the maximum, which is 18 and represented right here. So there's a basic overview of box-and-whisker plots. Let's take a look at a second example in order to get this down even more. Here is our second example. And for our second example, the box-and-whisker plot is displaying data about a specific student's scores on math tests throughout the year. And this is based on the percent earned on those tests. Feel free to pause the video here before I get started and see if you can identify the different parts of the box-and-whisker plot. Or feel free to work along with me. Let's start by finding the minimum and maximum. So we're going to start on the outside and work our way in. Now, the minimum and maximum are just the smallest and largest numbers in value within the data set. This part of the box-and-whisker plot is going to be the minimum. And then this part is going to be the maximum. The minimum in this example is 78. So 78% was the lowest score that that student had throughout the year. The maximum in this example is 94. So 94% was the highest score that that student had throughout the year. Now let's work our way into the outside of the box, the outside of the rectangle. This part of the box represents the first quartile, also called the lower quartile. This is the 25% mark, or one-fourth mark within the data. The 25th percentile. And then this part of the box is the third quartile, also called the upper quartile. So this is the 75% mark, or three-fourths mark within the data. The 75th percentile. So the first quartile, or lower quartile in this example, is 82. The third quartile, or upper quartile in this example, is 88. And then lastly, let's look inside the box, inside the rectangle. This line right here represents the median. Now the median is also called the second quartile. This is the 50% or halfway mark within the data. The 50th percentile. The median in this example is 86. So there you have it. There are the five parts of a box and whisker plot. We have the minimum and maximum. So the whiskers extend from the box to the minimum and maximum. And then we have the lower quartile and upper quartile. So the first quartile and third quartile. And then we have the median. So there is our second example. Let's move on to how to make a box and whisker plot. Now let's go through the steps of how to make a box and whisker plot. Let's jump into our example where we will make a box and whisker plot to represent this data right here. Let's say that 10 content creators were surveyed about how many videos they released last month. Here are the results. Now the first thing that we need to do is order the data from least to greatest. So we will start with three. So we have three. And then another three. Now I'm going to cross these out as we go through the numbers. That way we stay organized here. Then we have four, five, another five, nine, ten, 12, 15, and 18. Now we should have 10 numbers there. One, two, three, four, five, six, seven, eight, nine, ten. Now that we have the data in order from least to greatest, we need to find all of the parts that we will need for the box and whisker plot. The minimum, the first quartile, the median, the third quartile, and the maximum. Let's start with the minimum and maximum, which are just the smallest and largest numbers in value. The minimum is three, and the maximum is 18. So minimum is three, and maximum is 18. Now let's find the median, so the middle point of the data, the 50th percentile. Since we have 10 numbers in the data set, we will have five numbers on each side. Five numbers to the left and five numbers to the right. The median is right here in between the five and the nine. Since the median is between two numbers, we need to find the average or the middle point of those two numbers. That will be the median. To find the average of those two numbers, we need to add them and then divide by two. So let's come to the side here. Five plus nine, and then divide by two. Five plus nine is 14, and then 14 divided by two gives us seven. The average is seven, so that means the median is seven. The middle point between five and nine is seven. For this one, we may have been able to mentally just think of that median, that midpoint between five and nine. But in case you come across one that you can't do mentally, you can always find the average between the two numbers in order to find that median. Now that we have the median, we can find the first quartile and third quartile. Let's start with the first quartile, also called the lower quartile. This is the 25% or one-fourth point within the data, the 25th percentile. The first quartile is going to be the median or halfway point of the lower half of the data. So let's look at the lower half here. There are five numbers in the lower half, so the first quartile is going to be right in the middle. Two numbers on each side. The first quartile is four. Now let's find the third quartile, also called the upper quartile. This is the 75% or three-fourths point within the data, the 75th percentile. The third quartile is going to be the median or halfway point of the upper half of the data. So let's take a look here at the upper half. There are five numbers here as well, so two numbers on each side. The third quartile is 12. Now we have all of the information we need to create the box and whisker plot. We need to start by creating a number line. Our minimum is 3 and maximum is 18. So let's start at 0 and count to 20. That will include everything we will need. Now depending on the data you are working with, you can adjust the number line. But for this example, again, 0 to 20. So let's start with 0 here and I'm going to count off by fives here. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. Now let's create the box of the box and whisker plot. We will need the first quartile and third quartile. The first quartile is 4. The third quartile is 12. So let's create this box or rectangle here. This is the box of the box and whisker plot. It goes from the first quartile to the third quartile. Then inside the box, we have the median or second quartile. So the median is 7. So we draw a line here for the median. And then lastly, we have the whiskers, which extend to the minimum and maximum. Let's do the minimum first. The minimum, again, is 3. So let's draw a mark here and extend that whisker to the minimum. Then we have the maximum, which is 18. So let's make a mark at 18 and then extend the whisker to that maximum of 18. And that's our final box and whisker plot. We have the minimum at 3, the first quartile at 4, the median at 7, the third quartile at 12, and then the maximum at 18. So that was our first example. Let's move on to our second example. Here is our second example for how to make a box and whisker plot. Let's say that we have 15 people in a class and they had a test that had 50 questions. These are the results as far as answers correct out of the 50 questions. Now, the first thing that we need to do is put the data in order from least to greatest. So let's start with the smallest number in value, which is 21 here. So let's start with 21. And then go from there. Now, I'm going to cross off numbers as we go along in order to stay organized. Next, we have 28, then 33, another 33, 39, another 39, 40, another 40, and then another 40, so three 40s, 42, 45, 46, another 46, 47, and then 48. Now, there should be 15 numbers here. So let's double check and make sure. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Now that we have the data in order from least to greatest, we need to find the minimum, the first quartile, the median, the third quartile, and the maximum. Let's start with the minimum and maximum, which are just the smallest and largest numbers in value. The minimum is 21, and the maximum is 48. So the minimum, 21, and the maximum, 48. Next, let's find the median, so the middle point of the data, the 50th percentile. Since we have 15 numbers in the data set, we will have seven numbers on each side, and the number in the middle will be the median. So with seven numbers on each side, this is our median right here. So 40 is right in the middle. The median, 40. Now that we have the median, we can find the first quartile and the third quartile. Let's start with the first quartile, also called the lower quartile. This is the 25% or one-fourth point within the data, the 25th percentile. The first quartile is going to be the median or halfway point of the lower half of the data. So let's look at the lower half here. Now we have seven numbers within that lower half. So there are going to be three numbers on each side, and then the lower quartile, that first quartile, is going to be the number in the middle, which is going to be 33. So the first quartile, 33. Now let's find the third quartile, also called the upper quartile. This is the 75% or three-fourths point within the data, the 75th percentile. The third quartile is going to be the median or halfway point of the upper half of the data. So let's look at the upper half here. Now there are seven numbers in the upper half as well, just like the lower half. So we're going to have three numbers on each side, and then that number in the middle is going to be the third quartile, and it's going to be 46. The third quartile, again, 46. Now we have all of the information we need to create the box and whisker plot. We need to start by creating a number line though. So let's take a look at our minimum and maximum, because we need to make sure we include all of our data. The minimum is 21 and the maximum is 48. So let's start our number line at 20 and end it at 50. That way, again, we include everything. Now depending on the data you are working with, you can adjust the number line. So something to keep in mind. But again, for this example, we are going to start at 20 and end at 50. So let's start with 20 here and count by 5. So 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50. Now let's move on to the box of the box and whisker plot. So that's going to be the first quartile and the third quartile. The first quartile is 33. So let's make a mark at 33 here. And then the third quartile is at 46. So let's make a mark here. And those are the sides that the box or rectangle will extend to. So let's create that rectangle. And that's the box of our box and whisker plot. Again, it goes from the first quartile or lower quartile to the third quartile or upper quartile. Then inside the box, we have the median or second quartile. So that's going to be 40. So let's draw a line here at 40 to represent the median. And then lastly, we have the whiskers, which extend to the minimum and to the maximum. Let's do the minimum first, which is 21. So let's make a mark at 21 and then extend the whisker out to 21. So again, that's the minimum. Now let's extend a whisker to the maximum, which is 48. So let's make a mark at 48 and then extend the whisker to that maximum of 48. And that's our final box and whisker plot. We have the minimum, the first quartile, the median, the third quartile, and the maximum represented. Now, one thing you can do if you are drawing out a box and whisker plot by hand is use a ruler in order to make the lines a little straighter than what I did here in this example. So something to keep in mind. So there you have it. There's a basic general overview of box and whisker plots and how to make box and whisker plots. I hope that helped. Thanks so much for watching. Until next time, peace.