And then we talked about a confidence interval, that was the last step,

this is now where we take that standard error and

we convert it into an uncertainty statement, a confidence interval,

an interval that tells us what's going on across all possible samples.

And we had a result from our sample size 20 that we did in one of our previous

lectures, where we had an interval from 66-98, so

that's kind of wide, we're not very certain about it, it's not our best,

we prefer it to be narrower, how might we improve that?

Well, we could do that by increasing the sample size, because we know now

something about that standard error, and we haven't looked at it, but

it turns out that it is inversely related to sample size, and remember,

that's the result of algebra, some of you just didn't think about it, and

make up that formula, there was an algebraic derivation from that conceptual

framework that we're dealing with.

So, how can we think about this now, this process, in terms of evaluating quality?

And here is a display where we're going to look at two measures of sample quality,

and this is a fairly,

classic representation of what happens with quality in research investigations,

and you'll see this in other kinds of descriptions of research investigations.

And we have a five targets that could be used for some kind of sport,

and we have, in each one of them, representations of multiple shots at it.

Think of our samples now as those multiple tries, so, we're going to get six shots

at it, we're going to draw six of those possible samples,

there could be a lot more, but what kinds of things could happen when we do that?

In the one on the far left, we have our six shots and

they are all very close to the middle, very tightly grouped, very close,

not only are they tightly grouped, but they're right at the middle.

In the second representation, the next one to the right,

they're still grouped very close together.

As a matter of fact, the same pattern, no, not quite the same pattern, but very

similar pattern, grouped tightly together, but they're away from the middle.

I don't know if you ever watch people playing darts, and the really good ones,

they're very precise, they can group their shots, their darts, in the same

location very consistently, and they can move around what they're aiming at.

They could aim for the middle or they could aim at a target off to the side,

there are point totals for doing this, of course.

Now, that kind of thing is similar to what we're seeing here,

they're grouped tightly together, they're very precise, but

they can be thought of as being accurate or inaccurate.

Low accuracy in the second one, as opposed to the other two representations here,

the third one is where we have a spread, a wider spread of the results,

this would be equivalent to our samples giving us means that are more widespread.

There are more differences among those means than in the tightly grouped kind,

less precise but, in this case, on average, we've got it right,

they're targeting the middle of that pretty closely.

And then finally, on the far right, they're spread out and

their average is away from the targeted middle, they are low precision and

low accuracy, that allows us to think about two kinds of measures

here that are typically used to summarize this kind of result.

And now, we're thinking about this now in terms of our samples producing

estimates and how close they come to the true population value.

So we have one measure that is the distance from what we're getting on

average to the thing that we're aiming for, and so in the second display, we've

put in a double-headed arrow, a nice black double-headed arrow there to represent

the difference between what we're getting on average and the true value.

Now, in our sampling realm, we're not aiming to be away from the middle,

not like playing darts, here, we really want to get towards the middle and

there's a discrepancy, we're going to call that bias, that's the term that's

typically used, the difference between what it is we're trying to estimate, and

what we're getting on average.

But then we also can talk about the spread, and

we've highlighted here that the spread is larger, particularly

larger than the other displays, and so now we can talk about variance.

Variance, that standard error that we're talking about,

is a measure of the size that, if you will, the diameter of that circle,

and that larger standard error means that circle is wider in diameter and

were less precise in our results, our survey, there's something about our

sampling processes producing results that are more widespread.

Okay, so we have bias, well, how do we determine that?

In practice, this is very difficult, theoretically,

with the sampling processes that we're dealing with, it turns out that if we

go back to that conceptual framework, that sampling distribution,

we can determine whether or not there is a bias in our process.

On average, if we're going to get the right thing, and so,

we can measure the size of that bias, or actually determine that there is no bias,

so simple random sampling, that we're going to talk about next,

turns out to be unbiased, on average, we're getting them grouped together.

Now, it may be that they're spread out because we have a small sample size, or

narrowly concentrated because they have a large sample size, but, either way,

they're centered on the true population value.

The variance, there we've got to measure a standard error that we can compute from

the data, we do not need to rely on some theoretical derivation that says

what's going to happen overall, we can take a particular sample and

calculate the spread, the diameter of that circle.

And so, in the second diagram,

where they're fairly narrowly concentrated together, small diameter,

small standard error, precise, wider, larger diameter circle,

less precise, and that means that we can compare those diameters and

compare the quality of what we're getting, that's the quality measure, something

that we can use to compare different ways of doing the sampling operation.

So, [COUGH], it all depends on the random process used to select the sample as to

what these outcomes are.

If we have a process that is biased but precise,

we could also have a process that is unbiased but not very precise and so on,

across these, these are all possible ways of evaluating the sample.

We are going to use those two dimensions, biased and variance,

in order to assess what's going on with the quality of our results,

and we're going to calculate one of those from a given sample.

If we've drawn a probability sample, we can use the standard error calculation

to get that diameter calculated for all possible samples from just one.

I was trained as a statistician, you always need at least two things to measure

variance, well, that's natural, right?

We can't talk about variance unless we have two experiences with something,

at least two.

Here's a case where we can have an experience with one of them, but

if we've selected it, our process has been done in a certain way.

I can tell you what's going on across all of them from just one, so

this is really pushing our thinking here, that it is not just a simple process,

but one that has some properties that can be extremely valuable.

So to return to something that we said at the end of the last lecture,

there are two measures here, now we've explained them, the bias and

the variance, they both depend on understanding this process in which we've

taken random digits and applied them to a frame, and gotten a sample,

one sample, of many possible ones in theory, and we can measure

the variability across all of them by looking at just one possible sample.