Average or Central Tendency: Arithmetic Mean, Median, and Mode

Let's say I've got
a set of numbers.

2, say I've got three 3's, I've
got a couple of 4's, and

I've got a 10 there.

And what we want to do is find
the middle of these numbers.

We want to represent these
numbers with the center of the

numbers, or the middle of the
numbers, just so we have a

sense of where these numbers
roughly are.

And this central tendency that
we're going to try to get out

of these numbers, we're going
to call the average.

The average of this
set of numbers.

And you've, I'm sure, heard the
word average before, but

we're going to get a little
bit more detailed on the

different types of averages
in this video.

The one you're probably most
familiar with, although you

might have not seen it referred
to in this way, is

the arithmetic mean, which
literally says, look, I, the

arithmetic mean of this set of
numbers, is literally the sum

of all of these numbers divided
by the number of

numbers there are.

So the arithmetic mean for this
set right here is going

to be 2 plus 3 plus 3 plus 3
plus 4 plus 4 plus 10, all of

that over, how many
numbers do I have?

1, 2, 3, 4, 5, 6, 7.

All of that over 7.

And what is this equal to?

This is 2 plus 9, which is 11,
plus 8, which is 19, plus 10,

which is 29.

So this is going to be equal to
29/7, or you could say it's

equal to 4 and 1/7.

If I got my calculator out,
we could figure out

the decimal of this.

But this is a representation
of the central tendency, or

the middle of these numbers.

And it kind of makes sense.

4 and 1/7, it's a little
bit higher than 4.

We're kind of close to the
middle of our number range

right there.

And you might say, well, it's
a little skewed to the right

and what caused that?

And well, gee, 10 is a little
bit larger than all of the

other numbers.

It's kind of an outlier.

Maybe that skewed this average
up, the arithmetic mean.

So there are other types of
averages, although this is the

one that, if people just say,
hey, let's take the average of

these numbers, and they don't
really tell you more, they're

probably talking about
the arithmetic mean.

The other forms of average,
though, are the median, and

this literally is the
middle number.

If there are two middle numbers,
you actually take the

arithmetic mean of those
two middle numbers.

You actually find the number
halfway in between those two

middle numbers.

So the median of this set
right here-- let me just

rewrite them.

So I have a 2, a 3,
3, 3, 4, 4, 10.

So, let's see, we have seven
numbers right here.

The middle number, if
I go 1, 2, 3, to the

right, we're there.

If we go 1, 2, 3 to the
left, we're there.

The middle number is
that 3 right there.

I just listed them in order, and
I said, well, look, 3, you

could think of it as the fourth
number from the right,

and it's also the fourth
number from the left.

3 is the middle number.

And this case, it
is the median.

So in this case, 3, if you use
the median, is our average.

And that also makes sense.

I mean, it's literally the
middle number, and if you look

at this set of numbers, it kind
of does represent the

central tendency of this set.

Now just to be clear, it was
very clear what the middle

number was, because I had an
odd number of numbers.

I had three on each side of the
three, so it was very easy

to figure out the median,
the middle number.

But if I had a situation-- let's
say I have the situation

where I have 2, 3, 4, and 5.

Let's say that's my
set of numbers.

Well, here, there is no
one middle number.

The 3 is closer to the left
than it is to the right.

The 4 is closer to the right
than it is to the left.

There's actually two middle
numbers here.

The two middle numbers here
are the 3 and the 4.

And here, when you have two
middle numbers, which occurs

when you have an even number
in your data set, there the

median is halfway in between
these two numbers.

So in this situation, the median
is going to be 3 plus 4

over 2, which is equal to 3.5.

And if you look at this data
set, that's not what our

original problem was, but if
you look at this data set

right there, you're actually
going to find that the

arithmetic mean and the median
here is the exact same thing.

Let's calculate it.

What 's the arithmetic
mean over here?

It's going to be 2 plus 3 plus
4 plus 5, which is what?

5 plus 9, which is equal
to 14, over 4.

And what's this equal to?

14/4 is 3 and 2/4, or 3 and
1/2, the exact same thing.

So for this data set, they
were the same thing.

For this data set, our median
is a little bit lower.

It's 3, while our arithmetic
mean is 4 and 1/7.

And I really want you to think
about why that is.

And it has a lot to do with this
10 that sits out there.

All of these other numbers are
pretty close to whichever

average you want to pick,
whether it's the arithmetic

mean or it's the median.

But this 10 is kind of
an outlier, or it

skews the data set.

Maybe it's so much larger than
the other numbers, that it

makes the arithmetic mean seem
larger than maybe is

representative of
this data set.

And that's something important
to think about.

When you're finding the average
for something, most

people will immediately go
to the arithmetic mean.

But in a lot of cases, median
will make a lot more sense, if

you have these really large or
really small numbers that

could skew the data set.

I mean, you can imagine, if this
wasn't a 10-- or let's

imagine adding another
number here.

If I added the number 1 million,
if I added 1 million

to this data set, if that was
the eighth number, the

arithmetic mean is going
to be this huge number.

It's going to be much larger
than what is representative of

most of the numbers
in this data set.

But the median is still
going to work.

The median is still going to be
about 3 and a half, right?

If you had 1 million here,
it would be 1, 2, 3, 4.

The middle two numbers
would be that.

It would be 3 and 1/2.

So the median is less sensitive
to one or two

numbers at the extremes that
otherwise would skew the mean.

Now, the last form of average
I want to talk

about is the mode.

It has nothing to do
with ice cream.

The mode is literally the
most frequent number.

And in this data set, it's
pretty clear what the most

frequent number is.

I only have one 2, I have three
3's, I have two 4's, I

have one 10, and even if want to
include the million, I only

have one million there.

So here, the number
that occurs most

frequently is the 3.

So, once again, the mode seems
like a pretty good measure of

central tendency or a
pretty good average

for this data set.

Now the mode, it's a little
tricky to deal with, and you

won't see it used that often,
because it becomes a little

ambiguous when-- you know,
look at this data set:

2, 3, 4, and 5.

What is the mode there?

All of these numbers are
equally frequent.

So if you have a situation
like this, then you might

just-- the mode really
loses its meaning.

It might force you anyway to
take the median or the

mean in some form.

But if you really do have
numbers that one shows up a

lot more than the other, then
the mode starts to make sense.

So, hopefully, this has given
you a pretty good overview of

how to represent the central
tendency of a data set.

Very fancy word.

But it's just saying, look,
we're trying to represent with

one number all of this data.

And you might say, hey, why do
we even worry about that?

It only has seven numbers here
or eight numbers here.

But you can imagine if you had 7
million numbers or 7 billion

numbers, and you don't want to
show someone all of that data.

You just want to give someone a
sense of what those numbers

are on average.

And as we said, the arithmetic
mean is what I see being used

the most. But in situations
where you might have numbers

that would skew the arithmetic
mean, because they're so large

or they're so small,
the median might

make a lot of sense.