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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.

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