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We will now begin our journey
into the world of statistics,

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which is really a way
to understand or get

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our head around data.

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So statistics is all about data.

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And as we begin our journey
into the world of statistics,

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we will be doing
a lot of what we

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can call descriptive statistics.

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So if we have a bunch
of data, and if we

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want to tell something
about all of that data

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without giving them
all of the data,

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can we somehow describe it
with a smaller set of numbers?

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So that's what we're
going to focus on.

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And then once we
build our toolkit

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on the descriptive
statistics, then we

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can start to make
inferences about that data,

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start to make conclusions,
start to make judgments.

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And we'll start to do a lot
of inferential statistics,

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make inferences.

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So with that out of
the way, let's think

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about how we can describe data.

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So let's say we have
a set of numbers.

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We can consider this to be data.

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Maybe we're measuring
the heights of our plants

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in our garden.

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And let's say we
have six plants.

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And the heights are 4 inches,
3 inches, 1 inch, 6 inches,

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and another one's 1 inch,
and another one is 7 inches.

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And let's say someone just
said-- in another room, not

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looking at your
plants, just said,

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well, you know, how
tall are your plants?

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And they only want
to hear one number.

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They want to somehow
have one number that

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represents all of these
different heights of plants.

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How would you do that?

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Well, you'd say, well,
how can I find something

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that-- maybe I want
a typical number.

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Maybe I want some number that
somehow represents the middle.

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Maybe I want the
most frequent number.

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Maybe I want the number
that somehow represents

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the center of all
of these numbers.

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And if you said any
of those things,

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you would actually have
done the same things

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that the people who first came
up with descriptive statistics

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said.

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They said, well,
how can we do it?

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And we'll start by thinking
of the idea of average.

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And in every day
terminology, average

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has a very particular
meaning, as we'll see.

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When many people
talk about average,

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they're talking
about the arithmetic

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mean, which we'll see shortly.

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But in statistics, average
means something more general.

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It really means
give me a typical,

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or give me a middle number,
or-- and these are or's.

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And really it's
an attempt to find

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a measure of central tendency.

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So once again, you have
a bunch of numbers.

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You're somehow trying
to represent these

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with one number we'll call
the average, that's somehow

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typical, or middle,
or the center somehow

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of these numbers.

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And as we'll see, there's
many types of averages.

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The first is the one that you're
probably most familiar with.

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It's the one-- and
people talk about hey,

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the average on this exam
or the average height.

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And that's the arithmetic mean.

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Just let me write it in.

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I'll write in yellow,
arithmetic mean.

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When arithmetic is a noun,
we call it arithmetic.

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When it's an adjective like
this, we call it arithmetic,

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arithmetic mean.

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And this is really just the
sum of all the numbers divided

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by-- this is a human-constructed
definition that we've

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found useful-- the sum of
all these numbers divided

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by the number of
numbers we have.

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So given that, what
is the arithmetic mean

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of this data set?

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Well, let's just compute it.

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It's going to be 4 plus
3 plus 1 plus 6 plus 1

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plus 7 over the number
of data points we have.

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So we have six data points.

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So we're going to divide by 6.

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And we get 4 plus 3 is 7,
plus 1 is 8, plus 6 is 14,

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plus 1 is 15, plus 7.

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15 plus 7 is 22.

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Let me do that one more time.

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You have 7, 8, 14, 15,
22, all of that over 6.

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And we could write
this as a mixed number.

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6 goes into 22 three times
with a remainder of 4.

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So it's 3 and 4/6, which is
the same thing as 3 and 2/3.

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We could write this as a
decimal with 3.6 repeating.

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So this is also 3.6 repeating.

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We could write it any
one of those ways.

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But this is kind of a
representative number.

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This is trying to get
at a central tendency.

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Once again, these are
human-constructed.

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No one ever-- it's
not like someone just

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found some religious
document that said,

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this is the way that
the arithmetic mean

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must be defined.

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It's not as pure
of a computation

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as, say, finding the
circumference of the circle,

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which there really is--
that was kind of-- we

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studied the universe.

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And that just fell out of
our study of the universe.

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It's a human-constructed
definition

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that we found useful.

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Now there are other ways
to measure the average

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or find a typical
or middle value.

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The other very typical
way is the median.

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And I will write median.

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I'm running out of colors.

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I will write median in pink.

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So there is the median.

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And the median is literally
looking for the middle number.

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So if you were to order
all the numbers in your set

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and find the middle one,
then that is your median.

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So given that, what's the
median of this set of numbers

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going to be?

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Let's try to figure it out.

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Let's try to order it.

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So we have 1.

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Then we have another 1.

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Then we have a 3.

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Then we have a 4, a 6, and a 7.

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So all I did is
I reordered this.

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And so what's the middle number?

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Well, you look here.

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Since we have an even number of
numbers, we have six numbers,

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there's not one middle number.

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You actually have two
middle numbers here.

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You have two middle
numbers right over here.

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You have the 3 and the 4.

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And in this case, when you
have two middle numbers,

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you actually go halfway
between these two numbers.

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You're essentially taking the
arithmetic mean of these two

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numbers to find the median.

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So the median is going
to be halfway in-between

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3 and 4, which is
going to be 3.5.

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So the median in
this case is 3.5.

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So if you have an even
number of numbers, the median

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or the middle two, the--
essentially the arithmetic

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mean of the middle two, or
halfway between the middle two.

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If you have an odd
number of numbers,

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it's a little bit
easier to compute.

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And just so that
we see that, let

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me give you another data set.

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Let's say our data
set-- and I'll

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order it for us--
let's say our data set

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was 0, 7, 50, I don't know,
10,000, and 1 million.

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Let's say that is our data set.

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Kind of a crazy data set.

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But in this situation,
what is our median?

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Well, here we have five numbers.

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We have an odd
number of numbers.

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So it's easier to
pick out a middle.

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The middle is the number that is
greater than two of the numbers

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and is less than
two of the numbers.

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It's exactly in the middle.

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So in this case,
our median is 50.

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Now, the third measure
of central tendency,

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and this is the
one that's probably

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used least often in
life, is the mode.

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And people often
forget about it.

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It sounds like
something very complex.

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But what we'll see
is it's actually

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a very straightforward idea.

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And in some ways, it
is the most basic idea.

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So the mode is actually the most
common number in a data set,

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if there is a most
common number.

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If all of the numbers
are represented equally,

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if there's no one single
most common number,

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then you have no mode.

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But given that
definition of the mode,

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what is the single most common
number in our original data

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set, in this data
set right over here?

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Well, we only have one 4.

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We only have one 3.

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But we have two 1's.

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We have one 6 and one 7.

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So the number that shows up
the most number of times here

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is our 1.

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So the mode, the most typical
number, the most common number

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here is a 1.

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So, you see, these
are all different ways

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of trying to get at a typical,
or middle, or central tendency.

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But they do it in very,
very different ways.

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And as we study more
and more statistics,

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we'll see that they're
good for different things.

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This is used very frequently.

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The median is really good if you
have some kind of crazy number

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out here that could
have otherwise

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skewed the arithmetic mean.

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The mode could also be useful
in situations like that,

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especially if you do
have one number that's

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showing up a lot
more frequently.

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Anyway, I'll leave you there.

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And we'll-- the next few videos,
we will explore statistics even

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deeper.