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

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

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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.