Measures of Variability (Range, Standard Deviation, Variance)

PROFESSOR: In this
video, we're going

to learn about measures
of variability,

another form of
descriptive statistics

that people often want to
know in addition to measures

of central tendency.

But before we get to any of
the nitty gritty details,

I want to motivate why we
need measures of variability

with two examples.

So here's two different
data sets, one on the top

and one on the bottom.

I'll just go ahead and tell you
that the mean for both data sets

is 87.

Now if I were to just tell
you the mean of these data,

I would be misleading
you a little bit,

because in reality, the
situation in each data set

is quite different.

If I were to plot
it out, for example,

you would see this
difference clearly.

In the top data
set, all the scores

are very clustered together.

Everything is close.

But in the bottom data set,
scores are very spread out.

So again, I need some way to
quantify these differences.

And a measure of central
tendency, like the mean,

simply can't capture that alone.

Here's another example.

Let's say you're working for
a pharmaceutical company,

something like that.

And you need to decide between
two different medications

for depression.

We'll call them medication
A and medication B. So

let's say you did a
study where you measured

how much improvement
happened when people took one

over the other, and
this is what you got.

So let's say over here
that higher scores mean

more improvement
and lower scores

mean little to no improvement.

Well, let's compare.

The means in this
case are the same.

In both cases, people improved
by about 10-ish points or so.

But the variability
is very different.

On the left, some people
benefited very greatly,

whereas others really
didn't benefit at all.

But on the right, everyone
benefits a good amount.

In this case, I would
personally pick medication B

because it's more consistent.

And so this is an example of
why knowing the variability

might help us to make
some real-life decisions.

So in general in statistics,
measures of variability

are ways to describe these
differences statistically.

They describe how scores
in a given data set

differ from one another.

And they capture things
like how spread out

or how clustered
together, the points

are things we've been
looking at already.

So there are three that
we're going to talk about.

We have the range, standard
deviation, and variance.

Let's start with the range.

The range is nice because
it's a really simple measure

of variability, of dispersion,
of how spread out points are.

It can often be calculated
in 5 or 10 seconds.

Here's the formula.

So we have the range, r.

Don't get confused
later on when we

learn about correlations, which
are often also described by r.

We'll use some
different subscripts

to make that difference
clear when the time comes.

But for now, range is r.

And then we have r
equals h minus l.

h means the highest
score in the data set,

l means the lowest
score in the data set.

So you can see that this is
a very simple calculation.

And if we go back
to the example we

were working with
a minute ago, we

can calculate the
range very quickly.

So for the first data set,
we have 95 negative 80.

So the range is 15.

And in the second data set,
we have 150 negative 25,

giving us a much
larger range of 125.

So in this case, I would do
well to report both to you.

I'll tell you the mean and
this measure of variability,

because that gives you a more
full picture of what's going on.

So a mean of 87
and a range of 15

describes a very
different situation

compared to a mean of
87 and a range of 125.

So again, it's a great
idea for me to report both.

And this is what's often done.

A big limitation of
the range, though,

is that by using it,
even though it's simple

and it's pretty
effective, you might

miss a little bit of the data,
a little bit of the information

in your data set.

Let me show you an
example to illustrate.

Here's a data set here.

Although these bars
are quite high,

there's really just one
sort of value in each bar.

So we have one person who
scored a 30, one person who

scored a 40, and so on.

Now the range here is 120.

It's 150 minus 30.

But let's look at
a second data set.

In this case, the
range is still 120

because our highest and
lowest values are the same.

But everybody is
kind of over here,

and there's just a couple
outliers beyond that.

So again, if I were to
just tell you the range,

I might be misleading you a
little bit because you're not

sure if it looks
like this on the left

or if the data looks
like this on the right.

And this is where standard
deviation and variance

come into play.

Standard deviation, just
like the name suggests,

describes the standard
or typical amount

that scores deviate from the
mean, hence standard deviation.

Now we'll get into exactly
what this looks like

once we learn to calculate
standard deviation.

But I just want to show
you some symbols for now.

So like with means, we
have different symbols

to describe population standard
deviation versus sample standard

deviation.

Population standard deviation
is described by sigma.

It's this sort of O with a Elvis
hair, I like to think of it as,

not to be confused
with this sigma, which

is a capital S.
Unfortunately, they're

named the same thing, which
means take the sum of.

We learned about
that previously.

This is sigma with a little s.

So for a sample,
standard deviation

is simply described by S.

So I want to take a
step back and talk

about why standard
deviations are really useful.

Whenever you have a normal
curve, a normally distributed

set of data, which is very
common in the world, things

like height, weight, and so on
are all normally distributed,

standard deviations have this
really interesting property

of telling you a lot of
information about what's common

and what's uncommon.

So if we have 0, this is right
at the mean of whatever we're

talking about.

This is the mean.

0 standard deviations away
from the mean is right here.

You're right at the mean.

We can look at one standard
deviation above the mean and one

standard deviation below,
and we automatically just

because of how standard
deviations work,

that 68% of people will
fall within this range.

We can go beyond that.

We know that between
two standard deviations

in either direction of the mean,
95% of people will be contained.

And 3, you're getting really
extreme, really far out, really

rare, 99.7% of the
data will be contained

within three standard
deviations in either direction

from the mean.

To illustrate this a little bit
more, let's talk some specifics.

So let's say I'm
looking at IQ scores.

We know a lot about IQ scores.

We know for example, the
population mean of IQ is 100.

And we know that the population
standard deviation, sigma,

is 15.

So let's go ahead and draw
that same sort of normal curve.

We know that intelligence
is normally distributed.

And let's take a look
at what information

we have just by knowing
standard deviation.

So average IQ is
right here at 100.

One standard deviation
above the mean would be 115.

Two standard deviations
above the mean would be 130.

And three standard
deviations would be 145.

And we could do the same
in the opposite direction.

One standard deviation below
the mean of intelligence is 85.

Two standard
deviations below is 70.

And three standard deviations
below the mean of intelligence

is 55.

So again, I automatically
know 68% of people

will fall between
an IQ of 85 and 115.

I also know that 95%
of people will fall

between an IQ of 70 and 130.

And finally, that
99.7 or so will fall

between an IQ of 55 and 145.

So this is great to know
because if you tell me

you have an IQ of 146,
I'm really impressed.

This is rare.

This is very extreme.

But if you tell me you have
an IQ say, 106, something

like that, that's fine.

Good for you.

Not very impressed.

So knowing standard
deviations helps

you to get this extra
information about a data set.

So finally, we have variance.

Variance is very simple.

It's just the square
of standard deviation.

So it's the average squared
deviation from the mean.

Unfortunately, for variance,
it doesn't get its own symbols.

We just take the
symbols we already

have for standard deviation.

And we put a squared
because it's just

squared standard deviation.

So here, for a population,
we would call the variance

in a population sigma squared.

And for a sample, we would call
the sample variance, s squared.

So in the next
video, we'll learn

how to calculate
some of these things.

But I want to at least highlight
some of the formulas you're

going to see.

So we have four
different formulas

because we have standard
deviation and variance.

And we have the population
versions and the statistic

for sample versions.

So for standard deviation
in the population.

This is our formula.

Notice we have
sigma on the left.

And we have all this mess,
which I'll get into next time.

One thing I'll mention is that
for all of these formulas,

the numerator is called
the sums of squares, ss.

And we're going to learn
about what the sums of squares

really means in the next video.

But for now, just
keep that in mind.

So for our sample
statistic, we have this.

You're going to see
an s on the left here.

And it's going to have
some similarities,

but you're going to notice a
difference or two that we'll

talk about in the next video.

For variance, we
have sigma squared.

And for sample statistic version
of variance, we have s squared.