Box-and-whisker Plot

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Box-and-whisker Plot

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In this video we're going to talk about box-and-whisker
diagrams. And these are actually quite common,
box-and-whisker diagrams, especially if you ever do any
stock trading.
If you go to most stock chart providers, a box-and-whisker
diagram is usually one of the options.
Let's understand how to generate one, and essentially
how to read one using some that data
that we have up here.
So the first thing we want to do is order it, because we're
going to be finding a lot of medians when we want to
generate our box-and-whisker diagram.
So let's just order the numbers up here.
So it looks like the lowest number here is 49.
So I have that 49, then I have another 49 right here.
Then I have another 49 right over there.
And then do I have any 50s?
Yes, I have two 50s.
One, two 50s.
I have two 50s.
One, two.
Do I have any 51s?
I have one 51 as far as I can tell.
So one 51.
Do I have a 52?
One 52.
I have a 52.
A 53?
I have one 53, two 53s.
53 and 53.
54, there's one.
54, I don't have any other ones.
55, I don't see one 55 here.
56, I see a 56 right there.
And then 57, I have three of them.
One, two, three.
One, two, three 57s, and I've got a 58, a 59.
Let me scroll over a little bit.
And then I have a 67, and I'm done.
I've ordered all of these numbers.
And now it's a lot easier to find the median.
Now the median, as you may or may not remember, is the
middle of these numbers.
So how many total numbers do we have?
We have one, two, three, four, five, six, seven, eight, nine,
ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen,
seventeen numbers.
So the middle number, if you have seventeen numbers--
there's two ways to think about it.
However many numbers you have, we have 17, you add 1 to 17
and you divide by 2, and you count that many spaces from
either the left- or the right-hand side, and you will
get the middle number.
So this is 18 divided by 2, which is equal to 9.
So one, two, three, four, five, six, seven,
eight, nine-- 53.
Should work from this side as well.
One, two, three, four, five, six, seven, eight, nine.
You got 53.
And if you ever have a situation where it's like,
let's say you only had 16 numbers-- it would be 16 plus
1 divided by 2, you get 8 1/2.
That means, count eight spaces and then go halfway between
that number and the number right after that, which we saw
that in a few diagrams before.
But anyway, 53 is the median.
53 right here is the median.
Now the next thing we want to do when we want to generate a
box-and-whisker diagram is figure out the median of the
two halves of the data.
Notice, if you pick 53 as kind of the dividing line, half the
data is above it, half the data is below it.
If these are people's ages and they say they're 53, they're
at the 50th percentile.
50% of the people are above their age, 50% of the people
are below it.
This is the middle number.
What we want to do now is find the middles of the middles.
Or find, essentially, what is the 25th and the 75th
percentile.
So what's the number where only 1/4 of the numbers are
less than that?
And there's actually a little ambiguity here.
So if I want to find the middle of this left-hand
portion right here, it's debatable whether I want to
include this median right here.
And actually, both of those are valid ways of generating
box-and-whisker diagrams. The typical algorithm-- an
algorithm is just a way, a process for doing things-- is
to include this median in the bottom list, and to include
this medium in the top list.
So let's find the median of this list right here.
So we have one, two, three, four, five, six, seven, eight,
nine numbers.
9 plus 1 over 2 is equal to 10 over 2, which is equal to 5.
So we count, one, two, three, four, five right there.
That is the median of this bottom list, and we actually
called this, right here, we call this the first quartile.
This number right here is the first quartile.
And what it does is, it separates-- there's a bunch of
ways you could view it.
You could say that 25% or 1/4-- that's where the word
quartile comes from-- you have 1/4 of the values are below
this number, or you could say 3/4 of the values are above
this number.
Now the other way you could do a box-and-whisker diagram is
you don't include this median, but I think you would have
gotten the same answer.
If you didn't include this median, you would have had
one, two, three, four, five, six, seven, eight values.
8 plus 1 divided by 2 is 4 1/2.
So you go one, two, three, four and a half.
So you'd go halfway between these two 50s, but you would
still get 50-- that's the number that's between a 50 and
a 50 is a 50.
So either way our first quartile
number wouldn't be different.
Now, the second quartile is the median.
This is the second quartile.
I always imagine you have 25, this is 25% of the numbers.
25% or 1/4 of the numbers is the first quartile, or it's
the first fourth of the numbers.
And it's upper-bound is the first quartile.
Then you have this 1/4 of the numbers, this is also 25% of
the total numbers.
But it's upper-bound is the second
quartile or it's the median.
This is from the 25th to the 50th percentile.
And then if we want to go from the 50th, so this is 25th
percentile, this is 50th percentile, which means that
50% of the values are below it, which means 25% of the
values are below it.
And then if we want the 75th percentile, we find the median
of these values over here.
And we'll include this median right there.
So once again we have one, two, three, eight, four, five,
six, seven, eight, nine values.
9 plus 1 over 2 is 5.
So you go one, two, three, four, five.
So the median of the higher half of our data set is 57.
So this right here, this number right there, the 57, we
call that our third quartile, and it represents the upper
bound on the section from our 50th to 75th percentile.
So that's 25% of our numbers there.
You could imagine, that's what's
called the third quartile.
One quartile, two quartiles, three quartiles.
And, of course, the number 57, you would say it's in the 75th
percentile, which means that 75% of numbers, or it is the
75th percentile, and 75% of numbers are below it.
And if you count these you would see that
that's indeed the case.
Anyway, so we've been able to classify some interesting
numbers here.
Let's actually plot it now using a
box-and-whisker diagram.
So what I'm going to do here, let me draw my
x-axis just like that.
It doesn't have to be horizontal like this, it could
be in any form.
So let's say that this right here is 0.
That's 0, that right there is 100.
Halfway between 0 and 100, I'm just making a scale right
here, is 50.
This right here would be 75.
Now what I'm doing has nothing-- well I want to
include these numbers, but I'm just trying to make a nice
scale to measure by.
This is 25.
And so let's draw this data using a
box-and-whisker diagram.
So the median is 53.
Maybe 53 is sitting right about there.
Right, that's right, about 53.
That's our median.
Our first quartile is 50.
So 50 is right there, actually.
Our third quartile is 57.
So 57 might sit right around here.
I might want to scale this a little bit differently.
And I'm going to draw a box here.
So this right here, that is 53, that right there is 57--
that's 57-- and that right there is 50.
So what I'm essentially doing is representing the middle 50%
of the data is in this box, and this line represents the
actual median.
Now it's called a box-and-whisker diagram.
I've only drawn a box.
Let me draw the whisker or the whiskers.
The whiskers essentially show us the entire range of data.
So the lowest data point is 49, which is right over here.
I should zoom in on this a little bit.
Highest data point is 67, which is
maybe right over there.
So we draw whiskers.
So what a box-and-whisker diagram is showing us, it says
OK, most of the data is sitting inside of the box.
50% of the data.
This 25% and this 25% is within this box.
But just to get an idea of the range of the data, it shows
whiskers to show the low point all the way to the high point.
Let me draw this one a little bit-- let me zoom in a little
bit because I realize that this is a little bit all
scrunched up.
So let me draw it a little bit nicer.
So let me change my scale a little bit.
Let's say that this right here is, let's say that this is 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68.
So this right here is 67.
Where was-- 46, 47, 48, 49, 50, that's 50 right there.
So if we use this as our scale, I think it'll be a
little bit clearer what's going on with the
box-and-whisker diagram, we have our
median, our second quartile.
Our second quartile is 53-- one, two, three, 53.
It's right here.
Our first quartile.
The number that is greater than 25% of the values is 50.
It's 50 right there.
That's that.
Let me color code it.
This number right there is this number right there.
This number, our third quartile, the number that is
larger than 75% of the values is 57.
One, two, three, four, five, six, seven-- it is this value
right here.
The median.
Just to be clear, it was 53.
That is this value right here.
So now we can draw our box.
The box is telling us is that 50% of the values are between
here and here, between 50 and 57.
And then we draw our whiskers to show the entire range of
the values.
So the highest value here is 67-- let me do that in another
color-- is 67.
So we draw a whisker that goes all the way to 67.
And then the lowest value here is 49.
So we draw a whisker that goes to 49.
So what you see when you look at a box-and-whisker, so let's
say you didn't even see this data.
If you just looked at this box-and-whisker diagram, you'd
immediately say, OK, the middle number looks like 53.
Most of the numbers are 50%, I should say, of the numbers are
between 50 and 57.
So most of them are scrunched up over here.
But the entire range goes all the way to 67, but it doesn't
go too much below the 50th-- the inner quartile range or
the range between the 25th and 75th percentile.
So hopefully that gives you a good overview, and now when
you see these in stock charts-- although on stock
charts you'll see these vertical as opposed to
horizontal like that-- you'll have a good understanding of
what they're telling you.