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Five Number Summary and Boxplots

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Source: Image created by the author

In this tutorial, you're going to learn about the five number summary of a data set. Now the five number summary takes larger data sets and makes a more manageable and easier to understand. By breaking them down from lots of numbers to just five, they can help to summarise the center and variability. Two of the numbers in the five number summary are the smallest and largest, the minimum and the maximum.

So we'll talk about those first real quick here. So suppose you have the example of the Chicago Bulls basketball team. It's easy to see, just by inspection of the list, that the shortest person on the team is here at 71 inches tall. And the tallest person on the team is 84 inches tall. Those are two of the numbers in the five number summary. The three remaining numbers will be based on the median.

So just a little bit of review. The median measures of center of a data set. And it's the middle of an ordered set of data. Currently, this is alphabetical by last name. So we should rearrange it so that it's in height order.

Then we just work our way in until we find the middle number of the ordered data set. That number in the middle is the median, 79. What that leaves us with is two groups-- a low group and a high group.

What we can do is take the median of each of those data sets. So now we have 74 down here, 81 up here, and 79 in the middle. Those three numbers are called quartiles. And they divide the data set into four equal parts.

This is called the first quartile. The median is the second quartile. And then you have the third quartile up here. What you'll notice is that 25% of the data falls at or below the first quartile. Half the data set falls at or below the median. And 75% of the data falls at or below the third quartile.

So the five number summary consists of the minimum, the first quartile, the median, the third quartile, and the maximum. The benefits of this particular summary is that you'll notice is that about 25% of the data falls within each of these bands here. So what you can understand about the data set is where lots of data values lie.

For instance, there are more data values in a narrower range. There's the same amount of data values here between 79 and 81 as there are between 74 and 79. It's the same number of data values, but they fall in a more narrow range. So you can tell the data are more clustered together in this area than they were in this area.

And again, this makes it fairly obvious that 25% of the data falls below the first quartile-- at or below the first quartile. 50% falls at or below the median. 75% falls at or below the third quartile. And obviously, all the data falls at or below the maximum.

So to recap, the five number summary is a nice way to summarize a larger data set. Now it will work for pretty much any size data set so long as it has five numbers, obviously. Then you can make those your five numbers for your five number summary. But it's ideally for larger data sets to summarise more values.

It consists of the minimum, first quartile, median, third quartile, and the max. And it allows us to understand where clusters of data points might be and where the data might be more spread out.

So we talked about the five number summary, which consists of the min, the max, and the quartiles. And so we had the first quartile, which was called the lower quartile also. The second quartile is the median. Typically we don't use the term second quartile. We use the term median. And then we have the third or upper quartile.

Good luck. And we'll see you next time.

Source: GRAPHS AND TABLES CREATED BY THE AUTHOR

In this tutorial, you're going to learn about a graphical display called a boxplot. They're also sometimes called box and whisker plots. But the real name actually is boxplots. So you might wonder what this thing actually is and what it looks like. A boxplot is a way to graphically display the five-number summary for a data set.

So suppose I have the heights of the Chicago Bulls basketball team. The five-number summary consists of the minimum value, which was the shortest individual on the team at 71 inches, the maximum, which is the highest value in the data set. This is the tallest individual on the team. And the three quartiles-- first quartile, the median, and the third quartile.

This is the value at which 25% of the data falls at or below it. Half the data falls at or below 79. And 3/4 of the data falls at or below 81. And obviously, all the data falls at or below 84.

So let's get to actually making this thing. First, we're going to draw an axis. It can be horizontal or vertical. I've made mine horizontal. You should also scale it with equal increments. So I've gone from the lowest number, 71 and a little bit lower, to the tallest number, 84.

First, make some kind of mark at the five numbers in the five-number summary, so 71, 74, 79, 81, and 84 for our example. I've chosen to make vertical lines.

Second, draw a box from the first quartile to the third quartile. This box is going to show you where 50% of the data lie, the middle 50%. And about 25% of the data falls in this whisker out in the left side. And about 25% of the data falls in this whisker out to the right-hand side. This is why it's sometimes called a box and whisker plot.

Different statistical packages might show this a little bit differently. You'll notice, the boxplots from this statistical package don't have the vertical lines out here at the edges, and that's fine. We can use boxplots to compare two distributions. For instance, if we're talking about the height of girls versus boys, we might be able to compare them by saying the spread or the variation with the girls is much less than the variation with boys.

We can see that not only in the width of the boxes but also in the total width from the minimum to the maximum in each of these two data sets. So we can use boxplots to use as sort of a summary distribution for the boys and for the girls.

So to recap, boxplots will allow us to display, visually, the five-number summary. We can interpret it to see where the data points are close together-- that's where the vertical lines are close together-- and where the data points are further apart. We can analyze skewness, and we can look for it for symmetry as well. And we can use multiple boxplots on the same set of axes that will help us to compare two or more distributions. Good luck, and we'll see you next time.