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## Topic B: Summarizing a distribution that is approximately symmetric using the mean and mean absolute deviation

## Video transcript

We will now begin our journey
into the world of statistics, which is really a way
to understand or get our head around data. So statistics is all about data. And as we begin our journey
into the world of statistics, we will be doing
a lot of what we can call descriptive statistics. So if we have a bunch
of data, and if we want to tell something
about all of that data without giving them
all of the data, can we somehow describe it
with a smaller set of numbers? So that's what we're
going to focus on. And then once we
build our toolkit on the descriptive
statistics, then we can start to make
inferences about that data, start to make conclusions,
start to make judgments. And we'll start to do a lot
of inferential statistics, make inferences. So with that out of
the way, let's think about how we can describe data. So let's say we have
a set of numbers. We can consider this to be data. Maybe we're measuring
the heights of our plants in our garden. And let's say we
have six plants. And the heights are 4 inches,
3 inches, 1 inch, 6 inches, and another one's 1 inch,
and another one is 7 inches. And let's say someone just
said-- in another room, not looking at your
plants, just said, well, you know, how
tall are your plants? And they only want
to hear one number. They want to somehow
have one number that represents all of these
different heights of plants. How would you do that? Well, you'd say, well,
how can I find something that-- maybe I want
a typical number. Maybe I want some number that
somehow represents the middle. Maybe I want the
most frequent number. Maybe I want the number
that somehow represents the center of all
of these numbers. And if you said any
of those things, you would actually have
done the same things that the people who first came
up with descriptive statistics said. They said, well,
how can we do it? And we'll start by thinking
of the idea of average. And in every day
terminology, average has a very particular
meaning, as we'll see. When many people
talk about average, they're talking
about the arithmetic mean, which we'll see shortly. But in statistics, average
means something more general. It really means
give me a typical, or give me a middle number,
or-- and these are or's. And really it's
an attempt to find a measure of central tendency. So once again, you have
a bunch of numbers. You're somehow trying
to represent these with one number we'll call
the average, that's somehow typical, or middle,
or the center somehow of these numbers. And as we'll see, there's
many types of averages. The first is the one that you're
probably most familiar with. It's the one-- and
people talk about hey, the average on this exam
or the average height. And that's the arithmetic mean. Just let me write it in. I'll write in yellow,
arithmetic mean. When arithmetic is a noun,
we call it arithmetic. When it's an adjective like
this, we call it arithmetic, arithmetic mean. And this is really just the
sum of all the numbers divided by-- this is a human-constructed
definition that we've found useful-- the sum of
all these numbers divided by the number of
numbers we have. So given that, what
is the arithmetic mean of this data set? Well, let's just compute it. It's going to be 4 plus
3 plus 1 plus 6 plus 1 plus 7 over the number
of data points we have. So we have six data points. So we're going to divide by 6. And we get 4 plus 3 is 7,
plus 1 is 8, plus 6 is 14, plus 1 is 15, plus 7. 15 plus 7 is 22. Let me do that one more time. You have 7, 8, 14, 15,
22, all of that over 6. And we could write
this as a mixed number. 6 goes into 22 three times
with a remainder of 4. So it's 3 and 4/6, which is
the same thing as 3 and 2/3. We could write this as a
decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any
one of those ways. But this is kind of a
representative number. This is trying to get
at a central tendency. Once again, these are
human-constructed. No one ever-- it's
not like someone just found some religious
document that said, this is the way that
the arithmetic mean must be defined. It's not as pure
of a computation as, say, finding the
circumference of the circle, which there really is--
that was kind of-- we studied the universe. And that just fell out of
our study of the universe. It's a human-constructed
definition that we found useful. Now there are other ways
to measure the average or find a typical
or middle value. The other very typical
way is the median. And I will write median. I'm running out of colors. I will write median in pink. So there is the median. And the median is literally
looking for the middle number. So if you were to order
all the numbers in your set and find the middle one,
then that is your median. So given that, what's the
median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have 1. Then we have another 1. Then we have a 3. Then we have a 4, a 6, and a 7. So all I did is
I reordered this. And so what's the middle number? Well, you look here. Since we have an even number of
numbers, we have six numbers, there's not one middle number. You actually have two
middle numbers here. You have two middle
numbers right over here. You have the 3 and the 4. And in this case, when you
have two middle numbers, you actually go halfway
between these two numbers. You're essentially taking the
arithmetic mean of these two numbers to find the median. So the median is going
to be halfway in-between 3 and 4, which is
going to be 3.5. So the median in
this case is 3.5. So if you have an even
number of numbers, the median or the middle two, the--
essentially the arithmetic mean of the middle two, or
halfway between the middle two. If you have an odd
number of numbers, it's a little bit
easier to compute. And just so that
we see that, let me give you another data set. Let's say our data
set-- and I'll order it for us--
let's say our data set was 0, 7, 50, I don't know,
10,000, and 1 million. Let's say that is our data set. Kind of a crazy data set. But in this situation,
what is our median? Well, here we have five numbers. We have an odd
number of numbers. So it's easier to
pick out a middle. The middle is the number that is
greater than two of the numbers and is less than
two of the numbers. It's exactly in the middle. So in this case,
our median is 50. Now, the third measure
of central tendency, and this is the
one that's probably used least often in
life, is the mode. And people often
forget about it. It sounds like
something very complex. But what we'll see
is it's actually a very straightforward idea. And in some ways, it
is the most basic idea. So the mode is actually the most
common number in a data set, if there is a most
common number. If all of the numbers
are represented equally, if there's no one single
most common number, then you have no mode. But given that
definition of the mode, what is the single most common
number in our original data set, in this data
set right over here? Well, we only have one 4. We only have one 3. But we have two 1's. We have one 6 and one 7. So the number that shows up
the most number of times here is our 1. So the mode, the most typical
number, the most common number here is a 1. So, you see, these
are all different ways of trying to get at a typical,
or middle, or central tendency. But they do it in very,
very different ways. And as we study more
and more statistics, we'll see that they're
good for different things. This is used very frequently. The median is really good if you
have some kind of crazy number out here that could
have otherwise skewed the arithmetic mean. The mode could also be useful
in situations like that, especially if you do
have one number that's showing up a lot
more frequently. Anyway, I'll leave you there. And we'll-- the next few videos,
we will explore statistics even deeper.