Statistics Correlation

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Today we are going to talk about correlation.0002

First let us go back in and just briefly review summarizing scatter plots quantitatively0007

and talk about all the other things we have talked about scatter plots.0013

Then we will talk about eyeballing the correlation coefficient or what we call r, persons r.0018

Actually if you have a set of data that looks a particular way often you could sort of ballpark where the correlation coefficient falls.0026

We already talked about precisely calculating it.0036

Then we are going to go back and talk about the relationship between r and b1 or slope of our regression line.0039

First let us talk about summarizing a scatter plot quantitatively.0049

We did not deal with shape.0056

We just looked at it and maybe that is pretty good.0058

We will talk about shape in the next couple of lessons, but for now we are going to leave shape alone in terms of quantitatively calculating it.0062

We did look at how to precisely calculate the trend or by looking at the regression line,0072

that middle line in between all those lines that summarizes the middle of all those points.0080

And that middle line really gives us the relationship between X and Y, because it is the function that gives us if we have x, we get y and if we get y we get x.0087

We can get the relationship between those two variables.0101

Finally today we are going to talk about how calculating strength and not just looking at it as pretty strong right,0106

but instead we are going to actually calculate the correlation coefficients r.0115

That idea is simply, how pact around the regression line is our data points.0120

Are they tightly packed?0130

Is it a strong correlation like it strongly packed around that regression line?0132

Or is it very loose?0138

Is it dispersed?0139

It is not really sticking close to that line, then we would have low strength or low correlation.0141

For instance I am just eyeballing it and there are a lot of data.0152

You might have no relationship between two variables, and in that case, the spread looks something like this where there is no real line in there.0157

It is just sort of this cloud of dots.0168

Remember each of these points is a case.0172

Each of those cases has two variables.0178

One variable x is on it the x-axis and the other variable we will call it y is represented on the y-axis.0183

That point represents x here and y here.0192

In this case there is no relationship between X and Y just because you know what x is.0199

Let us say we know x is here.0204

Do you have any certainty as to where y might be?0209

There is some y down here and some y up here.0213

Even more so what about if we got X was here do we have any reason to say y is in a particular place?0218

No not really.0226

Because of that a line would not help us here.0228

A regression line does not actually summarize this very well, and it is because the correlation coefficient is very low.0233

There is very low strength.0240

There is very low adherence to the line.0241

Moving out a little bit further, you see that this one is starting to have more of a elongated shape.0245

This is still a fairly low correlation, but you can see that they are starting to be a linear relationship between X and Y, namely as x goes up, y also goes up.0255

This is what we call a positive correlation.0270

There is a relationship between x and y that is linear and positive.0273

Notice that y on the other side of this is the exact opposite where it is the same shape,0280

but it is been almost like flipped around like we put a mirror here and looked at the mirror reflection.0286

In this case it is the same shape cloud but now as x goes down y goes up.0294

Here we see the opposite relationship between X and Y and we call it a negative correlation.0305

Because of that the signs act accordingly.0312

Here the sign for this slope that is negative is -.4.0316

Here for slope that is positive as x goes up y goes up, as x goes down y goes down, that is a positive number.0325

We could easily just by looking at the correlation coefficient immediately know what kind of roughly what kind of relationship x and y have.0336

Notice that as we go out even further, not only do the numbers get bigger and bigger out from 0 but the numbers correspond to how whiny the data are.0346

How much they correspond to a line.0361

It is not really about having more dots but it is about how much do all those dots fit to a line.0364

That is what we often call collect fitting the data to a regression line.0373

We want to see is it a good fit? Is it a bad fit?0378

Correlation coefficient gives us the strength of that it is really strong fit or is it very week and loose.0381

The actual maximum for a correlation coefficient is 1 and the minimum -1.0390

That is as far as it will go.0398

You cannot have a correlation of like 1.1.0400

We will talk a little bit about y.0407

Here what you see is that it might have the same number of points of all of these, but it is just that there is very, very little variation from the line.0409

There is not a lot of variation out from the line, whereas coming like .8 you could see it is better than .4 but it is not quite as whiny as 1.0.0423

That is one way to just very quickly eyeball correlation coefficient.0435

You can just look at a data and it is elongated a little bit.0440

Maybe it is .4 but if it looks like tighter eclipse we will use .8.0445

If it looks very close to a line perhaps it is close to 1.0453

I want you to notice something.0466

Correlation coefficient, other than caring about positive and negative slope, it does not otherwise care very much about slope.0467

For instance, look at all of the situations, these are all lines.0477

They are all very lined, they are maximum lined.0482

Notice that these lines have positive slope but they all have the coefficient correlation of 1.0486

It does not matter whether x changes y changes very quickly or as x changes 1 y changes very slowly.0497

A slope does not matter, except for just the positive or negative part.0507

The same thing with the negative slopes, even though the slopes are all different they all have a correlation coefficient of 1.0514

There is an exception to this rule and it is this line right here, the perfect horizontal line.0527

Let us think about what the equation for the regular horizontal line is.0533

In a regular horizontal line it does not matter what x is y is always the same.0539

Let us say here like y=3 that would be like a horizontal line or y=-2 or y=.1.0547

Those are all example of horizontal lines.0556

Let us think about in the case of a horizontal line.0561

It is a perfect prediction because we know where the x is.0564

You could tell me whatever x you want.0570

I can exactly tell you the y because y is always negative.0572

y in this case is 3.0577

It is perfect prediction and it is perfect lining this but the correlation coefficient is 0.0578

We will try to figure out y as the horizontal line as we figure out the formula for correlation coefficient0589

and that hopefully that will become more clear.0599

There are many, many ways in which data can have seemingly no linear pattern0606

or very weak linear pattern because that is what the correlation coefficients tell us.0613

If you see our data have a 0 as its correlation coefficient do we know that it looks just like a cloud?0618

No, in fact it can look like anyone of these crazy shapes down here.0627

All of these distributions, all of the scatter plots have a very, very weak correlation because remember correlation just means how whiny is it.0634