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Average or Central Tendency: Arithmetic Mean, Median, and Mode - YouTube
Channel: Khan Academy
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Let's say I've got
a set of numbers.
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2, say I've got three 3's, I've
got a couple of 4's, and
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I've got a 10 there.
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And what we want to do is find
the middle of these numbers.
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We want to represent these
numbers with the center of the
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numbers, or the middle of the
numbers, just so we have a
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sense of where these numbers
roughly are.
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And this central tendency that
we're going to try to get out
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of these numbers, we're going
to call the average.
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The average of this
set of numbers.
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And you've, I'm sure, heard the
word average before, but
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we're going to get a little
bit more detailed on the
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different types of averages
in this video.
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The one you're probably most
familiar with, although you
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might have not seen it referred
to in this way, is
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the arithmetic mean, which
literally says, look, I, the
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arithmetic mean of this set of
numbers, is literally the sum
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of all of these numbers divided
by the number of
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numbers there are.
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So the arithmetic mean for this
set right here is going
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to be 2 plus 3 plus 3 plus 3
plus 4 plus 4 plus 10, all of
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that over, how many
numbers do I have?
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1, 2, 3, 4, 5, 6, 7.
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All of that over 7.
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And what is this equal to?
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This is 2 plus 9, which is 11,
plus 8, which is 19, plus 10,
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which is 29.
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So this is going to be equal to
29/7, or you could say it's
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equal to 4 and 1/7.
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If I got my calculator out,
we could figure out
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the decimal of this.
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But this is a representation
of the central tendency, or
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the middle of these numbers.
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And it kind of makes sense.
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4 and 1/7, it's a little
bit higher than 4.
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We're kind of close to the
middle of our number range
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right there.
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And you might say, well, it's
a little skewed to the right
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and what caused that?
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And well, gee, 10 is a little
bit larger than all of the
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other numbers.
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It's kind of an outlier.
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Maybe that skewed this average
up, the arithmetic mean.
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So there are other types of
averages, although this is the
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one that, if people just say,
hey, let's take the average of
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these numbers, and they don't
really tell you more, they're
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probably talking about
the arithmetic mean.
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The other forms of average,
though, are the median, and
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this literally is the
middle number.
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If there are two middle numbers,
you actually take the
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arithmetic mean of those
two middle numbers.
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You actually find the number
halfway in between those two
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middle numbers.
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So the median of this set
right here-- let me just
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rewrite them.
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So I have a 2, a 3,
3, 3, 4, 4, 10.
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So, let's see, we have seven
numbers right here.
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The middle number, if
I go 1, 2, 3, to the
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right, we're there.
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If we go 1, 2, 3 to the
left, we're there.
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The middle number is
that 3 right there.
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I just listed them in order, and
I said, well, look, 3, you
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could think of it as the fourth
number from the right,
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and it's also the fourth
number from the left.
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3 is the middle number.
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And this case, it
is the median.
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So in this case, 3, if you use
the median, is our average.
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And that also makes sense.
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I mean, it's literally the
middle number, and if you look
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at this set of numbers, it kind
of does represent the
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central tendency of this set.
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Now just to be clear, it was
very clear what the middle
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number was, because I had an
odd number of numbers.
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I had three on each side of the
three, so it was very easy
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to figure out the median,
the middle number.
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But if I had a situation-- let's
say I have the situation
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where I have 2, 3, 4, and 5.
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Let's say that's my
set of numbers.
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Well, here, there is no
one middle number.
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The 3 is closer to the left
than it is to the right.
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The 4 is closer to the right
than it is to the left.
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There's actually two middle
numbers here.
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The two middle numbers here
are the 3 and the 4.
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And here, when you have two
middle numbers, which occurs
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when you have an even number
in your data set, there the
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median is halfway in between
these two numbers.
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So in this situation, the median
is going to be 3 plus 4
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over 2, which is equal to 3.5.
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And if you look at this data
set, that's not what our
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original problem was, but if
you look at this data set
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right there, you're actually
going to find that the
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arithmetic mean and the median
here is the exact same thing.
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Let's calculate it.
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What 's the arithmetic
mean over here?
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It's going to be 2 plus 3 plus
4 plus 5, which is what?
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5 plus 9, which is equal
to 14, over 4.
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And what's this equal to?
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14/4 is 3 and 2/4, or 3 and
1/2, the exact same thing.
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So for this data set, they
were the same thing.
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For this data set, our median
is a little bit lower.
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It's 3, while our arithmetic
mean is 4 and 1/7.
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And I really want you to think
about why that is.
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And it has a lot to do with this
10 that sits out there.
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All of these other numbers are
pretty close to whichever
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average you want to pick,
whether it's the arithmetic
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mean or it's the median.
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But this 10 is kind of
an outlier, or it
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skews the data set.
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Maybe it's so much larger than
the other numbers, that it
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makes the arithmetic mean seem
larger than maybe is
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representative of
this data set.
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And that's something important
to think about.
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When you're finding the average
for something, most
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people will immediately go
to the arithmetic mean.
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But in a lot of cases, median
will make a lot more sense, if
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you have these really large or
really small numbers that
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could skew the data set.
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I mean, you can imagine, if this
wasn't a 10-- or let's
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imagine adding another
number here.
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If I added the number 1 million,
if I added 1 million
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to this data set, if that was
the eighth number, the
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arithmetic mean is going
to be this huge number.
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It's going to be much larger
than what is representative of
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most of the numbers
in this data set.
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But the median is still
going to work.
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The median is still going to be
about 3 and a half, right?
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If you had 1 million here,
it would be 1, 2, 3, 4.
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The middle two numbers
would be that.
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It would be 3 and 1/2.
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So the median is less sensitive
to one or two
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numbers at the extremes that
otherwise would skew the mean.
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Now, the last form of average
I want to talk
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about is the mode.
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It has nothing to do
with ice cream.
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The mode is literally the
most frequent number.
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And in this data set, it's
pretty clear what the most
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frequent number is.
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I only have one 2, I have three
3's, I have two 4's, I
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have one 10, and even if want to
include the million, I only
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have one million there.
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So here, the number
that occurs most
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frequently is the 3.
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So, once again, the mode seems
like a pretty good measure of
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central tendency or a
pretty good average
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for this data set.
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Now the mode, it's a little
tricky to deal with, and you
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won't see it used that often,
because it becomes a little
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ambiguous when-- you know,
look at this data set:
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2, 3, 4, and 5.
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What is the mode there?
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All of these numbers are
equally frequent.
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So if you have a situation
like this, then you might
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just-- the mode really
loses its meaning.
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It might force you anyway to
take the median or the
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mean in some form.
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But if you really do have
numbers that one shows up a
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lot more than the other, then
the mode starts to make sense.
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So, hopefully, this has given
you a pretty good overview of
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how to represent the central
tendency of a data set.
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Very fancy word.
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But it's just saying, look,
we're trying to represent with
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one number all of this data.
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And you might say, hey, why do
we even worry about that?
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It only has seven numbers here
or eight numbers here.
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But you can imagine if you had 7
million numbers or 7 billion
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numbers, and you don't want to
show someone all of that data.
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You just want to give someone a
sense of what those numbers
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are on average.
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And as we said, the arithmetic
mean is what I see being used
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the most. But in situations
where you might have numbers
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that would skew the arithmetic
mean, because they're so large
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or they're so small,
the median might
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make a lot of sense.
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