Box and whisker plots, which we also sometimes call box plots or box charts are a great way to display
data when we want to show both the median and spread of the data at the same time.
So in general, a box and whisker plot might look something like this.
We always have the plot itself, which is this blue part above, and then we have it set next to a number
line.
This number line think of as a reference so we know where the box plot is actually positioned.
For instance, we know that the left edge of this box and whisker plot is sitting at the value two and
the right edge of the box and whisker plot is sitting at 14.
So this number line is for reference.
It tells us where this box plot is positioned.
Now, when we talk about these, we need to know that this rectangular box in the center, this is the
box part.
And then these little extensions on the left and right hand side are called the whiskers.
So this extension here out to the dot on the left and then this little linear extension here, out to
the dot on the right.
These are the whiskers attached to the box in the middle.
Keep in mind that sometimes we'll see box and whisker plots shown vertically, but the concept is exactly
the same.
We have the box in the center and then the whiskers extend up at the top and down at the bottom and
there's a number line for reference.
So we have the box and we have the whiskers.
And the great thing about a box and whisker plot, again, remember, all of these plots were just looking
for the best way to communicate the data, to communicate the point we're trying to make that we see
in the data.
And the great thing about the box and whisker plot is that it does show us a lot of data.
At the same time, this picture looks simple, but it's communicating a lot of different things all
at once.
So this far left edge of the whisker represents the minimum value of the data.
So this is the minimum in the data.
So we know that whatever data set we use to create this box and whisker plot, the smallest value in
the data set is two.
And then, as you may expect, this far right edge of the right whisker represents the maximum point.
Of the data set.
So we know that the maximum value in this data set is 14.
So that's already two pieces of information.
And then if you remember, we talked about this before, we know that the range of a data set is always
equal to the maximum minus the minimum.
So because we know that the maximum here is 14 and the minimum is two, we can say that the range for
this data set is 14 minus two or 12.
And very quickly, we can know from the box plot that the range of the data is 12.
So now we have three pieces of information.
We also know that this line in the middle of the box, there will always be one line here in the middle
of the box, not necessarily in the center of the box.
As you can see, this one here is not centered within the box.
It's just contained somewhere inside the box.
But this is always representative of the median.
So in our case, the median is at 11.
And then finally, the box plot also gives us the quartiles of the data.
If you remember earlier, we talked about the quartiles of a data set and the interquartile range,
well, the left edge of the box.
So this point right here always represents the first quartile which we call Q one.
As you know, the median is by definition Q two, and then the right edge of the box.
So in our case, this point right here represents.
Q three or the third quartile.
So for this particular data set, we know the first quartile is five.
The third quartile is 13.
And the second quartile, which is equivalent to the median, is of course 11.
But then if you remember the interquartile range or IQ, R is always the third quartile minus the first
quartile or Q three minus Q one.
And so in our case we can quickly see that's 13 minus five, so 13 minus five or eight.
We know that our interquartile range is eight based on the fact that the quartiles are represented by
the left edge, this median line and the right edge of the box.
Remember that the quartiles divide the data set into quarters.
So we know that 25% of the data has to be contained between two and five, that another 25% of the data
has to be between five and 11, that another 25% has to be between 11 and 13, and that the last 25%
of the data has to be between 13 and 14, which means then that we can also say that the middle 50%
of the data set.
In other words, if we wrote out all the values in the data set, maybe it was something like two,
three, three, four, et cetera.
All the way up to the last value of 14.
So we took all the values in our data set.
We line them up in ascending order.
Maybe there are 100 data points.
Well, the middle 50 data points that middle 50% of the data are data points that span from a value
of five to a value of 13, since those are the values that define the edges of the box.
So you can see that with a quick box and whisker plot, we have the minimum maximum and range.
That's three pieces of information.
The three quartiles and IQ R, that's four more pieces of information or seven total plus the median.
Of course, that's the same as Q two.
But if we think about it as a separate thing now we have eight pieces of information, plus we can think
about each quarter of the data, one being represented by the left whisker, one being represented inside
the box to the left of this median line, one being represented by the inside of the box to the right
of the median line and the last quarter being represented by this right whisker.
But we have those 4/4 of the data.
And so then we can quickly identify the middle 50% of the data or the upper 75% of the data or lower
75% of the data.
So we have tons of information just from this box and whisker plot available to us at a glance quickly.
So if this is the kind of information that we're trying to communicate based on the data set and the
goals that we have, this box and whisker plot idea might be a really good choice.
And then the only other thing we want to say about box and whisker plots is that we often represent
a plot like this with what's called a five number summary, because we can essentially pull five critical
numbers from this plot and present them in a table instead.
So those five numbers are these values here.
One, two, three, four and five.
In other words, minimum value, left edge of the box, median right edge of the box, maximum.
If we collect those values in a five number summary, it looks like this.
We show it as the minimum.
Here.
Our minimum value was two, so we're showing a minimum value of 2q1 or the first quartile, the boundary
between the lower 25% of the data and the second 25% or second quarter of the data that's right there
at five.
So we have the first quartile, we have the median, or we could also call that Q two, we have the
third quartile or we could also call that Q two or the second quartile we have.
Q three, the third quartile and then the maximum.
This five number summary is essentially displaying all of the same information as the box and whisker
plot.
It's a little less visual, but the numbers are all right there.
So this is kind of two different ways of representing the same thing.