MATH 110 Sec 13.2 (F2019): Measures of Dispersion

MATH 110 Sec 13.2 (F2019): Measures of Dispersion

Today, we'll cover section
13.2 on measures of dispersion.

In the previous section, we
studied the mean, median,

and mode.

Those are measures
of central tendency--

in other words, locations--

around which the data
tends to cluster.

For example, suppose we
plot a small data set--

say it looks like that--

and then a second one.

It looks the same, but it's
in a different location--

and yet a third one.

If you plot them
all together, you

can see that the data sets
basically look the same.

They've just been shifted.

In other words, they have
different means, medians,

and modes.

But each is equally
spread out or dispersed.

That's what we
mean by dispersion.

Each of those three data
set is equally dispersed.

That's what leads us to
the study of measures

of variation or dispersion
of data as a contrast

to just a measure of
location, which we studied

in the previous section.

These measures will
tell us, in some sense,

just how much the data either
spread out or packed in

together.

The first and simplest measure
of variation's called a range.

The range of a
data set is simply

the difference between the
largest and smallest data

points.

So the range is just a maximum
value minus the minimum value.

It's a relatively crude
measure of the spread

of a data set because it doesn't
say anything about the values

in between the minimum
and maximum values.

Here's an example
that's sort of aged.

It's been in my slides
for quite a long time,

so the information
may be out of date,

but the point is the same.

This is a list of
the heights in inches

of four different people--

at least at the time,
the tallest person

in the world, the shortest
person, and two other people,

including LeBron James
and LaDainian Tomlinson.

In any case, the
range of that data set

would simply be the largest
number, the maximum,

minus the smallest
number, the minimum.

In this case, that
would be 102 minus 26,

which happens to be 76 inches.

And in case you're interested,
that's about 6 feet, 4 inches.

Another measure of
variation of a data set

is the standard deviation.

Now, we'll say here, I'm going
to flash up that formula.

And in fact, I'm even
going to show you

how to calculate it
a step at a time.

However, just so it
doesn't scare you too much,

we will find that there is a
keystroke on the calculator

that will do this for us.

I basically want to show you
how much value that keystroke is

going to have for
us by showing you

what you would do if you
did not have that keystroke,

so bear with me.

Let's take a data set--

16, 14, 12, 21, 22--

and calculate the
standard deviation

without the assumption
that there's

a button on the calculator
that can do it for us.

If you're still
having difficulty

calculating the mean, I'll
refer you back to the video

from the previous
section on the mean.

But at this point, I'll assume
that you can calculate the mean

without me mentioning it again.

It turns out to be
17 for that data set.

But to calculate the
standard deviation

from this sample of
1, 2, 3, 4, 5 numbers,

here's what you would do by--
what I would say, by hand.

You're actually using a
calculator in any case,

probably, but you
could do this by hand.

The first thing you do
is you make a column

for the deviations
from the mean.

If you look at that
formula, there's

one point where you
have x minus x bar.

So just calculate the
difference of each data

point from the mean.

So you take 17 away
from 16, get minus 1.

You take 17 away from
14, you get minus 3.

From 12, you get minus 5.

You take 17 away
from 21 and get 4.

You take 17 away
from 22 and get 5.

So there is your
deviations from the mean.

Then, if you look
back at the formula,

you'll notice they get squared.

So you need another column
for that squared deviation.

So you square minus 1.

You square minus 3. you square
minus 5, and you square 4,

and you square 5.

And then you see, in that
formula, there's that sigma.

Sigma means sum, so
you need to add up

all those squared deviations.

And if you add them up, it
turns out you'll get 76.

So the numerator inside that
square root symbol is 76.

The denominator
is n minus 1 where

n is how many values there are.

Well, we said there were
five, so n minus 1 would be 4.

So you'd get 76 divided by
4 under the square root.

That's the square root of 19.

And if you take that calculator,
and do the square root,

and round it to two decimal
places, you get about 4.36.

This is how you would
do the calculation

if we didn't have a
Standard Deviation

button on the calculator.

Luckily for us, we do.

So these calculations
are strictly educational,

just to show you
what you would do

if you didn't have the button.

Lucky for us, we
do have a button,

and we will not have
to do it this way.

Let's show the same
process given the knowledge

of a set of keystrokes that will
give us the standard deviation

from this sample.

And all it is a
procedure exactly

like calculating
the mean, except you

do Shift-9 instead of Shift-7.

Otherwise, it's identical.

So if I want to
calculate the mean,

I don't need any of
those other columns.

I just need the data
values themselves.

I enter those data values
exactly like I enter them

if I were calculating the mean.

So you press On to
clear out any old data.

You do Mode-period, which
puts it in stats mode.

And then you key in each number
and press M-plus afterwards.

This is exactly
what you did when

you were calculating the mean.

This is nothing new.

But when you're doing a
standard deviation of a sample,

after all the data
is in, you press

Shift-9 instead of Shift-7.

That's the only difference.

I'm going to throw up
the other useful tips

just so that list is complete.

But calculating the standard
deviation of a sample--

it's exactly the same as
calculating the mean except you

do Shift-9 instead of Shift-7.

Just remember that.

In this particular
case, the keystrokes

will be On to clear the old data
and to wake the calculator up,

Mode-period to put
it in stats mode,

and then you enter each data
point followed by M-plus.

Don't forget M-plus
after the 22.

And then do Shift-9.

And if you look in your
calculator display,

you'll see 4.358898944.

And if you round that to two
decimal places, you get 4.36.

This also might be a
good time to introduce

a new term called the variance.

The variance is nothing but
the standard deviation squared.

So once you get the
standard deviation,

if you want the
variance, you just

square the standard deviation.

For example, in this problem,
the standard deviation

was 4.358898944.

If you square that,
you get the variance.

And if you are going to round
it to two decimal places,

you'd get 19.00.

But do not use the
rounded standard deviation

to calculate the variance.

Take the unrounded value, square
it, and then round it back.

How about this example?

A company hired six interns.

After four months,
their work records

show the number of work days
missed for each worker--

0, 2, 1, 4, 2, 3.

Find the mean, sample standard
deviation, and sample variance

of this data set.

And then round your final
answer to two decimal places.

First, calculate the mean.

This is the material from
the previous section--

should be straightforward.

Press On.

Press Mode-dot, Mode-period,
whatever you call that.

Then enter the data
followed by M-pluses.

Then hit Shift-7.

That's the mean.

It turns out to be 2.

The data's already
been entered now.

So when you're doing something
beyond just the mean,

you do not have to
re-enter the data.

You could clear
everything and start over.

But if the data has
not been cleared,

you do not have to put it
back in in order to calculate

the standard deviation.

So all you need to do
now is press Shift-9.

And when you do, you'll see that
the sample standard deviation

is 1.414213562.

So you do not have to clear
the data and start over.

If the data is already
in there, and you

want to use that same data set
to calculate something else,

you do not have to start over.

And if you round that
to two decimal places,

you'd get a standard
deviation of 1.41.

If I ask for the
variance, you simply

square the standard deviation.

But remember, you're going to
square the number before you

round it, and then round.

So you're going to square
1.414213562, and then round it.

And it turns out to be
2.00 to two decimal places.

So far, we've been doing our
calculations with samples

without really explaining why.

And without going
into a lot of detail,

when we work with samples,
we're making the assumption

that the sample data is drawn
from some larger population.

For example, we might take a
sample of 10 student grades

from a class of 96 students.

This set of 10
grades is the sample,

while the 96 student
grades are the population.

It turns out that, when we
calculate the sample mean,

it's the same as
a population mean.

It doesn't matter whether
it's from a sample

or from the population.

Turns out, though, that there is
a slight difference when you're

doing standard deviation.

There's a slight change
in the standard deviation

formula when you're
calculating the population

of standard deviation.

So whereas it never mattered
with the distinction

between a sample and a
population with means,

it does matter if we're talking
about the standard deviation.

Just to illustrate,
suppose we take the data

from the previous example--

the 0, 2, 1, 4, 2, 3.

We found that sample
standard deviation

to be about 1.414213562.

That was from a sample.

If I told you that was
the entire population,

you have to do
something differently,

but it amounts to
just one keystroke.

If you look at it,
every single keystroke

is the same until you
get to the very end,

and it amounts to making
one press differently.

If you're doing a sample
standard deviation,

you do Shift-9.

If you're doing a population
standard deviation,

you do Shift-8.

So it's strictly
one press different.

But you do have to
be watchful to see

if you're asking for a
population standard deviation

or a sample standard deviation.

I do want to say,
when you're looking

at this on your calculator, if
you'll notice, above the nine

there is a sigma n minus 1.

That's what we're calling small
s, which is the sample standard

deviation.

If you look above the eight
on your calculator key,

you see sigma sub n.

And that stands for the
population standard deviation.

So you do need to notice
that on the calculator.

What we're calling s and
sigma, your calculator

calls sigma sub n minus
1 and sigma sub n.

I don't really want to
get into why that is.

There is a reason.

If we were studying this as
a whole course in statistics,

I would talk about
it more in depth.

But I do want to point it
out that the sample standard

deviation that's associated
with the nine keystroke

is going to be labeled
sigma sub n minus 1.

The population
standard deviation,

which is associated with
the keystroke eight,

is going to be sigma sub n.

So just look at that and
get it in your head--

sigma sub n for population
standard deviation, sigma sub

n minus 1 for sample
standard deviation.

Just take a minute,
and look at that.

And once you've got
it, you've got it.

Let's look at this example.

Consider the following--
12, 21, 13, 20, 27.

Compute the population standard
deviation of the numbers.

Round your answer to
one decimal place.

Remember, this is a
population standard deviation.

So when you put the numbers
in, it's exactly the same

as if you're doing the sample
standard deviation except,

at the end, you press
Shift-8 instead of Shift-9.

And you get
approximately 5.535341.

If you round it to one decimal
place, that's about 5.5.

Here's an additional question.

Add a non-zero constant c to
each of your original numbers

and compute the
standard deviation

of this new population.

Again, it's a population.

And again, we want to round our
answer to one decimal place.

So they're asking us to add
a constant c that's not zero.

So they're saying, pick
any number you like,

and add it to each of those data
points, and then recalculate.

So I'll pick two.

You can pick five.

It doesn't matter.

Pick some number that's not
zero and add it to each of them.

So if I add 2 to 12, 2 to 21,
2 to 13, 2 to 20, 2 to 27,

I get 14, 23, 15, 22, 29.

Now calculate the population
standard deviation again.

If you do that, you'll
find out it doesn't change.

If you round it, you still
get 5.5, approximately,

to one decimal place.

The B part asks you to look
at those two calculations

and make an inference.

It says, use the results from
part A in inductive reasoning

to state what happens to
the standard deviation

of a population when a non-zero
constant is added to each data

item.

Now, an inference is
not necessarily a proof.

But it looks like
it doesn't change.

And that's the
choice I would choose

based on what I just did.

I did the calculation both ways.

The answers came out the same.

So the inference would
be the standard deviation

remains the same.

It turns out that
is, in fact, true--

that when you add a single
number to each data point,

the calculation for
the standard deviation

does not change at all.

Now, this has nothing
to do with your ability

to do these problems,
but I do feel

that I should try to explain
a little bit about what

the standard
deviation really is.

And I'm going to give it a shot.

But understanding what I'm going
to say from here out really

doesn't affect your
ability to do the problems.

I'm just trying to add a little
bit more to your understanding.

So let's give it a shot.

Suppose the length that I
put up here on the screen

represents the difference
between two values.

And what if each of the line
segments I've displayed down

here in the lower left quadrant
is one standard deviation

in length?

If I take one of those
things, and then another,

and then another,
and then another,

and stretch them along the
length of that wider segment,

it turns out it
takes four of them

to equal the length
of that one wider bar.

So what I could say is, that
blue bar-- that wider bar--

is four standard
deviations long.

But what if the bars for
the standard deviations

were really longer?

What if, on the
right quadrant-- what

if those bars represented one
standard deviation of length?

What if I stretched
those bars out?

Well, it takes only
three of those.

So the wider bar is
three standard deviations

long if the standard
deviation is the length of one

of the bars on the right.

But the length of the wider
bar is four standard deviations

long if the standard
deviation is

the length of the bars on
the lower left quadrant.

So in effect, when we calculate
the standard deviation,

we're computing the length
of our measuring stick.

If the standard deviation
is really small,

it will take several
standard deviations

to span a certain distance.

If the standard deviation is
larger, it takes fewer of them.

Now, I don't know if
that helps or not.

It has no bearing on whether
you do the calculations or not.

But I do hope that it
gives you some little bit

of more understanding in
what a standard deviation is.

It's sort of the length
of our measuring stick.

One more reminder--
there is a summary sheet

that you can download if you
bring this presentation up

in PowerPoint.

The link will bring
you to a summary sheet

that you can actually
print out and keep.