Statistics Regression

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Today we are going to be talking about regressions today.0002

Here is the big goal of this lesson.0007

Basically we want to set up a conceptual understanding of regressions before we actually learn to calculate them and find it.0010

Today we are going to do just a brief review of linear equations and talk about the regressions as the center of line.0018

Instead if a center point like the mean when you talk about a center line and then we are going to talk about prediction and error.0026

What is a linear equation?0037

Y = mx + b should be pretty familiar to a lot of you and whenever we think of y= mx + b.0040

You can think of y as the output or f(x), x will be the input or often whatever it is on this horizontal axis the x axis.0048

B is the y intercept.0064

Another way you could think of the y intercept is where x = 0 what is b?0073

x would be here and apparently anything where x is 0 that will mean that y will have to be somewhere on this y axis.0080

That is what we mean by y intercept.0093

M is this slope.0096

Slope of something pretty much numbers but just in case you do not, here is how we calculate slope.0103

Slope is the change of y over the change of x.0118

When we say change we think of delta, the change of y over change of x.0124

More commonly people refer to it as rise/run.0132

When you think of rise, you think of going up vertically or down vertically.0140

The entire rise and they are running in a sort of more horizontal.0146

What is the rise/run?0149

That is what we think of as a slope.0151

When we think about rise/run we mean in the direction of the positive direction is up and right.0154

The negative direction will be down and more to the left.0162

You could think of rise/run as an indication of rate of change.0170

How much x changes in relation to y or vice versa?0177

These are the components of our linear equation and every line it does not matter what the line looks like.0183

Every straight line has an equation and from that equation you can figure out any x what the y is at any y what the x is?0197

If you have xy in a slope you could figure out the intercept and if you have the intercept of y you could bring out the slope.0211

This is a useful equation for us.0224

We are going to be trying to find a line that is the mean.0227

That is the center at the data point.0231

In order to do that, we would have to find its equation because the equation is the mean of the line.0235

In statistics, we are going to use this equation but we are going to write it in a different way.0243

It is just writing conceptual change but we are going to change it around just very slightly and superficially.0250

The first thing we do is we talk about the y intercept first.0258

In statistics, that is y=b comes first and it is the first b so it is b sub 0 or b knot.0262

Instead of the y intercept being added second we start from the y intercept and then we add the slope × x.0276

Notice that the slope is not called n anymore, it is called b sub 1.0287

We have b sub 0 which is the y intercept and b sub 1 which is slope.0293

Same idea as before.0303

This how I will refer to things when we talk about the equation of a regression line.0306

What do we mean by the center line?0320

If you think about a scatter of data, if you have a whole bunch of data you want to think of a line that somehow cuts through the middle of all of these points.0325

Right now we could just roughly draw a line and try to make it cut through the center of all these points.0337

That is a very rough line.0353

In order to find what that equation of this line is, I can as long as I just have 2 of the points on this line.0356

For example, if I take this data point and this data point I could find the equation of that line.0365

It is because by having 2 axis and 2 y, a set of x and y and another set of x and y I will calculate rise/run.0374

From having slope and x and y, I could calculate the y intercept.0384

That is a rough line but because it just depends on which 2 points I take.0390

If I pick these 2 points I will get this line but let us say I pick this point and this point then I would get an entirely different of line.0396

Moreover if I pick this point and that point I will get an even more different line.0414

The question is which 2 points will you pick?0426

It might be not good enough for us to just eyeball things because we are not sure which 2 points to pick.0430

If we have 2 data points then life is easy like you could just use those 2 but usually we have more than 2 data points.0439

Just eyeballing a rough line may not be good enough for us.0448

If we could just show then we only have 2 data points we could manually find slope and intercept and find the equation of that line.0453

Let us talk about this regression as a center line instead of a center point.0465

Here are some reasons for summarizing with a regression line and notice that for all of these I’m talking about scatter plot.0470

Regression lines are used for scatter plots.0478

Here what we want to do is we want to have some variable.0482

Here is my first variable, variable 1.0487

Here is my other variable, variable 2.0491

We want to have a line that describes the center of all of these cases whatever the cases maybe.0496

Why do we have a line?0509

Why not just a mean?0512

Sometimes there is not enough info from just a point.0516

If you just have a point, for instance this point is the mean of my x and y.0522

That would be x bar and y bar.0533

Let us say that is my center point, that point might not give us enough information about this whole distribution.0538

We are going to be talking about how to summarize a distribution and what about trend we do not just want a point we like a trend.0548

The most information from that point it is useful to have center line.0557

We want to find the summary that describes the relationship between the 2 variables.0563

It is not enough just to have a point, the point would not describe the relationship between the 2 but the line does.0574

A line will tell you whether its slope is negative or positive.0580

The line will tell you what kind of information you would want from a trend.0587