Statistics F-distributions

Hi, welcome to educator.com.0000

We are going to talk about F distributions today.0002

So first we are going to review other distributions recovered besides F, namely the NT.0004

Then we are going to introduce the F statistic also called the variance ratio.0011

Then we are going to talk about the distribution of all these S, distribution of all these ratios and finally 0024.5 what Alpha and P value mean in an F distribution.0017

Because eventually were in a deep hypothesis testing with the F statistic.0029

Okay , first, these other distribution so we know how to calculate the Z statistic and we also know how to0031

find the probability of such V value in a normal distribution.0044

But what is EZ distribution?0050

Well, imagine this.0053

Take a data set, let us just call it a population.0056

We take a data set, I will just draw a circle and we take some sort of sample from it, of size.0059

And we actually calculate the Z statistic for this sample so we calculate the goals, get the mean of this little0068

sample minus the mu divided by the standard error.0087

So you do that and then you plot the Z.0093

So imagine you replace all those that sample again to with replacement and you draw another sample and0098

you do this again and then you plot that guy and you dump it back in, you draw another sample, you calculate Z.0115

So you do that over and over again many times which end up getting is a normal distribution overtime.0123

So many times if you plot Z you get a normal distribution and because of that we also call this a Z0136

distribution because the distribution made up of a whole bunch of Z and it has the shape of a normal0150

distribution so that is what we call a Z distribution.0159

Now, if you take that same idea and you do it, you get a sample, and instead of calculating Z for that simply0162

you calculate T, if you do this then and then you plot that and you do that over and over and over and over again you get a T distribution.0175

And this resulting t-distribution follows the rules of the t-distribution where it depends on the degrees of0195

freedom, how wide it is, at the lower your degrees of freedom assertive variable but the higher the bigger0209

your degrees of freedom assertive, less variable and more normal it looks.0217

And so that is what we call the t-distribution.0222

So that is how Z statistic and the Z distribution sort of go together.0225

And this is how the T statistic and the t-distribution sort of go together.0232

And they just have to imagine taking a whole bunch of this sample, calculating whatever statistic and0237

implying that statistic and then looking at the shape of those statistic.0245

So really what this is a sampling distribution of Z.0250

And this is a sampling distribution of T, instead of using means or Z squares to plot your plane instead use the T statistic.0260

And you could do that for anything you could do that for standard deviation and you can do for inter0281

quartile, you can make the sampling distribution of anything you want.0286

That is important to keep in mind as we go into F distribution.0290

Okay so first thing is what is the F statistic?0293

We know how to calculate the T statistic and the V statistic, what is the F statistic?0301

Well, later on in these lessons were going to come across what we call the ANOVA, the analysis of variance.0307

Analyze means to break down and variance is well, you know what variances is, the spread of usually0316

around the mean of your data set and so when we analyze variance, we are going to be breaking down0325

variances into its multiple component and the F ratio happens to be ratio of those component variances.0332

And so I just want you to get sort of the big idea behind the F ratio not exactly how to calculate it, well get0343

into the details of that later on but the general concept.0352

So the S statistic usually is this idea that we have let us say two samples, x1, x2, x3, y1, y2, y3.0356

Now there is always some variation within the sample within exit there is some variation.0370

And within the Y there are some variation.0379

So there is definitely some variation but there is another variation here that we are really interested in.0385

We are really interested in the difference between these two things.0392

Between these two samples, so the F statistic really is taking those ideas and turning it into a ratio and here is what a ratio looks like.0397

It is really the between sample variance all over the within sample variance.0408

I remember variance is always squared, the average squared distance away from the mean and so because0425

of that this is a squared number, this is the square number, they are both positive so this number is always going to be greater than zero.0433

There is no way that this number could be less than zero so the statistic is always going to be greater than zero.0442

Now another way to think about between sample variance and within sample variance is this.0449

Whenever we do these kind of test, we are really interested in the differences between the samples like that is really important to us.0454

But sometimes their difference is also like a part of that difference is going to be just inherent variation.0464

So sometimes there might be a difference between let us say,men and women, or people who got a0478

tutorial versus people who did not, right?0486

People who study for the test versus people who did not, people went to private school versus people with public school.0488

There might be some difference between them?0495

But that difference is also going to have variation.0497

So this between sample variance often has inherent variation just variance you cannot do anything about0500

inherent variation plus real difference the effect size between samples.0508

And noticed that we keep using this word between and that is to indicate that part, so between, that is the part that we are really interested in.0520

Over within sample variance and so here there is inherent variation between X and between the Y and that0534

is not something we are interested in but it is good to know how variable are in the our little samples are.0557

Everyone very similar to each other, is very different, we need to compare the difference between the sample to the difference within the samples.0565

So this the inherent sample of the within sample variation is just inherent variation.0574

So these are all different ways of seeing the same thing and the reason why I want to say I also like this is0583

because later on we are not just going to be talking about between sample and within simple differences, we are going to add onto those ideas.0593

The final way I want you sort of think about the F statistic is basically this.0601

Ultimately in hypothesis testing, where going to want to know about differences between sample, that is the thing that were really interested in.0608

So it is going to be the variation that we want to explain because that is the reason that we did our research in the first place.0616

All versus the variation we cannot explain, not with this design at least.0631

So in our experimental design we will have these two groups and hopefully these groups will be similar to0646

each other but different, similar within the group but different between the groups.0653

And that is why in a S statistic we want this variation that we want to explain to be quite large and this0660

variation that we cannot explain or do anything about to come along for the ride where we want that to be relatively small.0667

Okay so let us do a limited thinking about the F ratio.0676

Now if we had a very big difference between the groups what kind of F ratio would we have?0679

When it is greater than one, less than one?0688

Well if our variation between the groups is bigger than the variation within the group then we should have0690

a very large F so that should be F that is greater than one right so at least greater than one but maybe a lot 1144.0 greater than 1, it could be 2 over one or 2 over .5.0697

Any of those values which show between sample variances are a lot larger than within sample variance.0708

And so if there is a lot of within sample variance then that competes with the between sample variance so0715

let us say there is a vague between sample difference but there is also a lot of differences within the0728

samples themselves and sort of evens out and you might see an F that is smaller or even less than one right if this one is bigger than this one.0734

So that is how you could sort of think about the S statistic.0745

Now imagine getting that F statistic over and over and over again from the population and plotting a sampling distribution of S statistics.0748

What would you get?0761

Well, remember that F cannot go below zero because both numbers are going to be positive so the F really stops at zero.0763

But this is what the S statistic ends up looking like.0774

This is a skewed distribution and it has a positive tail.0778

That means it goes for a really long time on the positive side.0786

Its one-sided so it is not is not symmetrical, it is actually asymmetrical there is only a positive side and it is0792

because of the proportion of variances and variances are positive.0803

And like T is a family of distribution and you are going to be able to find the particular F distribution you are0810

working with by looking at the degrees of freedom in the numerator, the one about between sample0819

differences and by looking at the denominator the sort of leftover or within sample differences variation.0829

So you are going to need both of those numbers in order to find out which S statistic you are working with0847

and in Excel, it will actually ask you for the degrees of freedom for the numerator and denominator.0854

Now let us talk a little bit about what Alpha means here.0861