Math 1080 Lecture 15 Normal Distribution 3 Sampling Distributions x bar and s squared

Okay, let's go back to basics.

Let's see sampling distributions.

We had something
called "population."

Like, let's take some
huge population,

say, all high school students
in the United States.

That's a huge population.

I don't know
how many there are,

and we want to study
something like weight,

obesity of students, and
we need the weight of them.

That's huge data.

What we usually do
when we have a huge data--

or it might be the case that we
do not have access to data--

[unclear] for example, you
want to study something about...

...some rabbits, for example,
in state of Minnesota.

Well, you cannot just gather all the rabbits
and maybe weigh them, for example.

So, what you do is you sample.
This one makes more sense

to just sample some rabbits at random from maybe
different places in the state and weigh them.

So, that's when you really don't
have access to the whole population.

The other case you might have
access to the whole population

of all high school students in the
country, but it's not feasible.

So, in that case, too, we choose
some samples from every state--

there are different
ways to sample them--

you can say based on population of each
state we sample according to that population.

So, a more populous state like
California you sample more students

from California than, say,
I don't know...Minnesota.

So, we have a population and we're
studying something about this population.

Usually we are talking about
the normal distribution,

although this applies
to any distribution.

So, this population
has some distribution.

It has some parameters.

Now, parameters for the
distribution of the population are--

usually what we study: mu is mean,
and sigma squared which is variance,

and sigma which is
standard deviation.

Well, there might be other things that
we study, depending on the situation,

if you're working for an insurance
company you might even consider other--

these are some kind of "moments,"
we call them "moments"--

so you can choose other moments.
You might need more than just these two.

So...

For the moment we
just need these two,

in fact this one [sigma/standard deviation]
and this one [mu/mean].

Now, that's for the population.

But, I said it's not feasible or it's not
possible to look at the whole population.

So, what we do is we sample.

And that's where statistics really
comes into the picture.

Statistics basically starts here.

Otherwise, if we know the population,
we know everybody and the weight

of every student in this country,
then that's it, we have data. So what?

Statistics comes in when it is not possible (for
whatever reason) to study the whole population.

In fact, when we do data science,
that's another story.

In data science, say, we
usually have the population.

Well, sometimes we would sample,
but we usually have the population.

And most often what
we want to do is predict.

So, we have to--

you want to predict something about
the population in the future.

But, anyways, this is not data science,
this is statistics, although they are related.

We have a sample here and this sample also has
some kind of mean and some kind of variance.

This mean and variance we denote
them differently. Remember? Sample--

This is population, and for sample
we have x̅ [x-bar] for mean,

and remember we have s
for standard deviation.

Okay, s for standard deviation
and this is for the sample.

Oh, it's getting so dark,
let me turn a light on.

Now...

First of all, you might say--
you might intuitively [unclear]--

You might feel like having more
sample, so you might say,

"A larger sample size will give me
a better idea of the population."

Where does that feeling
or intuition come from?

In fact, it's true. It's true that a larger
sample will give a much better idea.

But, there's something here, this
mean and this standard deviation,

so the larger sample will give me a mean and
a standard deviation. But, what do we want?

We want this [sample] mean and
this [sample] standard deviation

be close to this [population] mean and
this [population] standard deviation.

In other words, you want these two [from the
sample] to estimate [the population] for me.

Right? So, you want these
two [from the sample]

to estimate this [population] mean
and this standard deviation.

And that's where the
word "estimators" comes in.

So, this x-bar, which is
the mean of the sample,

and s, which is the standard
deviation of the sample,

these two are estimators for
mu and sigma. In fact, in--

Well, the best-- I mean, x-bar is
an unbiased estimator for mu,

and s squared is an unbiased
estimator for sigma squared.

So, this estimates this,
and this estimates this.

Not sigma, not sigma and s,
that's why we always write variance.

This guy x̅ estimates this μ, and
this guy s^2 estimates this σ^2.

And they're unbiased. Now, what "unbiased" is,
I'm not going to discuss in more detail, but it's...

Well...at some point I will
touch that, but anyways.

But, the word "unbiased" itself, should
give you some idea of what they really are.

So...

Okay, now...

This...

So, really there's something
that's coming into the picture.

We have the sample, and this
sample has some distribution.

So, the sample has
some distribution.

Now, what do we assume
for that distribution?

Well, most often what we do
is we graph the histogram

and by looking at the histogram
we guess, and we say,

"Okay, the histogram is telling me that
this distribution looks like normal."

Or, "This distribution looks like a
exponential distribution," so on and so forth.

The first thing we usually do is we graph
the histogram and look at the picture

and try to just guess
what the distribution is.

Well, anyway, the sample
comes with some numbers,

so if it is weight of students or age of students,
something [like that], you have some numbers.

Those numbers give you x-bar and s,
so you have at least 2 numbers here.

And these two numbers
definitely help you to--

not the distribution, but most often
you guess the distribution, but anyway--

to write down the distribution more clearly
by including these two in the distribution.

Okay, so...

First thing we do is histogram.

So, we have a sample
and we draw a histogram.

So, given a sample

graph the histogram first.

So, the histogram should give you an idea
of what the distribution should look like.

When you graph the histogram this means that
you are using some computer program,

whatever program you
are using, you'll find:

x-bar and s.
What is x-bar and what is s?

X-bar is [uppercase] sigma
[Σ meaning sum] of all x's

divided by n. And s squared
is [uppercase] sigma Σ,

x minus x-bar squared
over n minus 1.

That is s squared.
Now, what is n?

N is the sample size. Let me
write here: "n = sample size."

And these two formulas, in fact--
n is sample size-- these two formulas--

First of all, it means that your
sample must be more than 1.

[He points to denominator where 1 minus 1
would equal 0 and be undefined.]

Definitely, can you say anything just
by having 1 [item in a] sample?

So, this one and this one give me two numbers,
and these two numbers theoretically--

by theory we know-- that these are unbiased
estimators for mu and [lowercase] sigma squared.

Now, as I said, and I'll say it again, this chapter,
in fact, is about normal distribution.

So, what we usually do is we just say, "Okay, well
let's assume that the population is normal."

Although there are some theorems which have
nothing to do with normal distributions,

but they work for every distribution.
But, in order to work with a sample

and make some predictions
and other things,

we usually assume that the distribution of
the population is normal, and also the sample.

So, although there are some problems
with this assumption, but...

Well, we have to assume sometimes.

Unless we have a large sample,
if we have a large enough sample,

then by looking at the histogram we might say,
"I think this is not a normal distribution,

this is like an exponential distribution,
or like 'blah-blah' distribution."

That idea needs more experience and
knowledge of statistics and probability.

So, these two guys
are good estimators.

Now...

Let me erase...
what should I erase?

[unclear]

So, it's saying something like this.
When we study population--

So, this is the population...

You have to listen carefully
to what I'm saying.

So, we want to
study a population.

Say I'm a stats professor and I choose
some students from my class,

n students, say 10 students,

and send them to different places in the country
to measure weight of some students.

So, to weigh some students,
high school students.

Today I sent them and they go around
the country in the morning and weigh,

and bring me 10 numbers.
So, today, day 1...

they bring me ten numbers.
[counting]

So, they give me ten numbers.

These ten numbers, you
can find using that formula,

you can find for these ten numbers
both x-bar and s squared for day 1.

So for day 1 we have
x-bar and s squared.

And, again, day 2,

I send students again to give me
another x-bar and another s squared.

Day 3, another one.

Day-- well, how many days do
you think? Let's say 100 days.

Day 100, so I have x-bar [subscript] 100,
and s squared [subscript] 100.

Say I send them 100 days, and I have
no money to do that. [chuckles]

That's why I'm saying
"if it's feasible."

Anyway, so I have some x-bars
here and some s squares.

And samples themselves are kind
of random, the whole set of ten--

How many students did I send? Ten.

So, this sample has ten students from
high schools around the country.

This sample [day 2] has ten students,
but they do not have to be the same,

they can be at random, so it's unlikely
to have two samples exactly the same.

Right? So, just imagine. You choose some
students from the high schools in the country.

The next day you choose other students,
but since you are doing it at random,

some of these students might be the same.
But, just imagine out of-- I don't know,

like out of a hundred million students--
I don't know, let's say ten million--

out of ten million students you are choosing
ten in 2 days and they are the same?

It's really unlikely to have the same.

In fact, we can find the probability of being
the same, but that's not the point here.

And then day 3, another ten students,
up to day 100, ten students.

The point is, this sample
itself is like random.

Although each one of them is random,
the sample in whole is also like random.

Which makes these x-bars
and s squared's random.

Right? This makes them random.

So, being random, it means that we can
talk about some kind of random variable.

X-bar, for example, or s squared.

And this x-bar has values: x-bar [subscript] 1,
x-bar [subscript] 2, up to x-bar [subscript] 100.

This one [s squared] takes values from

s^2 [subscript] 1, s^2 [subscript] 2,
up to s^2 [subscript] 100.

Since they are random variables,

with these as values taken by these two
random variables, they have a distribution.

So, the distribution of these two, that's what
this thing is about: sampling distribution.

And that's what we usually graph.
That's what we graph. And...

So, we have this x-bar
and s squared for ten--

Well, ten's not enough, but say
I send 200 students, okay?

And then we graph it-- a histogram for
x-bar, and a histogram for s squared--

those two histograms will give us
an idea of what the distributions are.

Now, next time I will discuss
the distributions of these...

...random variables and also some amazing
theorems about those distributions.

So, see you next time.