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Box and whisker plots, which we also sometimes call box plots or box charts are a great way to display

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data when we want to show both the median and spread of the data at the same time.

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So in general, a box and whisker plot might look something like this.

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We always have the plot itself, which is this blue part above, and then we have it set next to a number

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line.

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This number line think of as a reference so we know where the box plot is actually positioned.

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For instance, we know that the left edge of this box and whisker plot is sitting at the value two and

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the right edge of the box and whisker plot is sitting at 14.

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So this number line is for reference.

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It tells us where this box plot is positioned.

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Now, when we talk about these, we need to know that this rectangular box in the center, this is the

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box part.

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And then these little extensions on the left and right hand side are called the whiskers.

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So this extension here out to the dot on the left and then this little linear extension here, out to

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the dot on the right.

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These are the whiskers attached to the box in the middle.

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Keep in mind that sometimes we'll see box and whisker plots shown vertically, but the concept is exactly

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the same.

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We have the box in the center and then the whiskers extend up at the top and down at the bottom and

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there's a number line for reference.

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So we have the box and we have the whiskers.

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And the great thing about a box and whisker plot, again, remember, all of these plots were just looking

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for the best way to communicate the data, to communicate the point we're trying to make that we see

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in the data.

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And the great thing about the box and whisker plot is that it does show us a lot of data.

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At the same time, this picture looks simple, but it's communicating a lot of different things all

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at once.

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So this far left edge of the whisker represents the minimum value of the data.

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So this is the minimum in the data.

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So we know that whatever data set we use to create this box and whisker plot, the smallest value in

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the data set is two.

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And then, as you may expect, this far right edge of the right whisker represents the maximum point.

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Of the data set.

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So we know that the maximum value in this data set is 14.

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So that's already two pieces of information.

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And then if you remember, we talked about this before, we know that the range of a data set is always

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equal to the maximum minus the minimum.

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So because we know that the maximum here is 14 and the minimum is two, we can say that the range for

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this data set is 14 minus two or 12.

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And very quickly, we can know from the box plot that the range of the data is 12.

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So now we have three pieces of information.

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We also know that this line in the middle of the box, there will always be one line here in the middle

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of the box, not necessarily in the center of the box.

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As you can see, this one here is not centered within the box.

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It's just contained somewhere inside the box.

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But this is always representative of the median.

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So in our case, the median is at 11.

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And then finally, the box plot also gives us the quartiles of the data.

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If you remember earlier, we talked about the quartiles of a data set and the interquartile range,

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well, the left edge of the box.

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So this point right here always represents the first quartile which we call Q one.

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As you know, the median is by definition Q two, and then the right edge of the box.

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So in our case, this point right here represents.

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Q three or the third quartile.

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So for this particular data set, we know the first quartile is five.

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The third quartile is 13.

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And the second quartile, which is equivalent to the median, is of course 11.

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But then if you remember the interquartile range or IQ, R is always the third quartile minus the first

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quartile or Q three minus Q one.

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And so in our case we can quickly see that's 13 minus five, so 13 minus five or eight.

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We know that our interquartile range is eight based on the fact that the quartiles are represented by

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the left edge, this median line and the right edge of the box.

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Remember that the quartiles divide the data set into quarters.

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So we know that 25% of the data has to be contained between two and five, that another 25% of the data

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has to be between five and 11, that another 25% has to be between 11 and 13, and that the last 25%

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of the data has to be between 13 and 14, which means then that we can also say that the middle 50%

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of the data set.

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In other words, if we wrote out all the values in the data set, maybe it was something like two,

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three, three, four, et cetera.

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All the way up to the last value of 14.

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So we took all the values in our data set.

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We line them up in ascending order.

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Maybe there are 100 data points.

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Well, the middle 50 data points that middle 50% of the data are data points that span from a value

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of five to a value of 13, since those are the values that define the edges of the box.

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So you can see that with a quick box and whisker plot, we have the minimum maximum and range.

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That's three pieces of information.

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The three quartiles and IQ R, that's four more pieces of information or seven total plus the median.

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Of course, that's the same as Q two.

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But if we think about it as a separate thing now we have eight pieces of information, plus we can think

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about each quarter of the data, one being represented by the left whisker, one being represented inside

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the box to the left of this median line, one being represented by the inside of the box to the right

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of the median line and the last quarter being represented by this right whisker.

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But we have those 4/4 of the data.

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And so then we can quickly identify the middle 50% of the data or the upper 75% of the data or lower

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75% of the data.

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So we have tons of information just from this box and whisker plot available to us at a glance quickly.

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So if this is the kind of information that we're trying to communicate based on the data set and the

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goals that we have, this box and whisker plot idea might be a really good choice.

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And then the only other thing we want to say about box and whisker plots is that we often represent

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a plot like this with what's called a five number summary, because we can essentially pull five critical

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numbers from this plot and present them in a table instead.

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So those five numbers are these values here.

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One, two, three, four and five.

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In other words, minimum value, left edge of the box, median right edge of the box, maximum.

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If we collect those values in a five number summary, it looks like this.

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We show it as the minimum.

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Here.

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Our minimum value was two, so we're showing a minimum value of 2q1 or the first quartile, the boundary

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between the lower 25% of the data and the second 25% or second quarter of the data that's right there

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at five.

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So we have the first quartile, we have the median, or we could also call that Q two, we have the

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third quartile or we could also call that Q two or the second quartile we have.

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Q three, the third quartile and then the maximum.

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This five number summary is essentially displaying all of the same information as the box and whisker

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plot.

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It's a little less visual, but the numbers are all right there.

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So this is kind of two different ways of representing the same thing.