Statistics Normal Distribution: PDF vs. CDF

Hi and welcome to www.educator.com.0000

Today we are going to be talking about normal distributions again but this time breaking it down into the PDF0002

or probability density function and CDF or the cumulative distribution function.0008

Here is what we have for today.0014

We are going to be talking about frequency charts which we have been doing before0018

and then contrast it to this new idea but still fairly elementary, cumulative frequency of charts.0022

We are going to do a very brief review of calculus.0031

It is going to be a very elementary review, no actual calculations just conceptual.0034

Then we are going to talk more deeply about the probability density function and the cumulative distribution function.0042

You have to reprogram yourself to see PDF and not think of it as a portable document or format.0048

Let us talk about frequency versus cumulative frequency.0056

So far what we have been doing is we have some sort of variables such as score on SAT verbal.0061

We have all these values that value can potentially hold.0067

We have been talking about what percentage of our sample or population have achieved that score.0072

1% score is 800, 3 score is 750.0081

That is what we have been talking about so far.0086

Here when I write percentage I am talking about relative frequency but it is largely the same thing as frequency.0089

Not a big deal.0100

When we talk about cumulative percentile, what we are really talking about is not just the people0102

who has achieved that value but an accumulation of everybody who has come before it.0112

Let us start off at the bottom.0118

Here only 1% of the population has achieved 250 points.0120

It think that is one of the minimum or something.0126

But 3% have achieved 300 or below.0129

If you are in the third percentile, you have out performed 3% of every else that has taken the test.0136

Not something to write home about yet.0144

This 8% actually accounts for this 3% as well plus a little bit extra.0147

The 16% encapsulates everybody who has come before it.0163

Cumulative percentile is helpful is you want to know your ranking in a performance.0169

For instance you want to know what percentile of the population you are in.0175

You just do not want to know what percent of everybody who is around you has also achieved that score.0179

You want to know how many people you have out performed.0185

Cumulative percentile continuously adding all the people that have come before you.0188

It gives you that ranking.0195

If you are in the 95 percentile, you know out performed or equally 95% of that sample.0198

That is cumulative frequency is really helpful to us.0210

One of the things that you want to notice is that cumulative frequency, when you just look at the number right away,0215

you do not know how common that score is.0222

When you look at 98% you do not know how common 750 as a score is but you could easily find that out0226

just by looking at the difference between 98 and whatever cumulative frequency came before it.0235

That difference is 3, so 3% of people have been in that bracket.0240

Another thing about cumulative frequency I want you to notice is that it is a monotonic increase.0247

It means that there is no going up and then going back down.0254

There are no changes in direction.0260

It is continuously going up, up, and up.0262

That makes sense because you have to add up everybody who has come before you.0265

That is cumulative frequency.0269

When you look at it on a visualization, you could see what I mean by monotonic increase.0272

Here we have an example of monotonic increase.0280

This curve goes up, and up, and up because it is adding up everybody who has come before you.0286

You have to be at least the score before you or higher.0293

Every score is improving on the previous score.0298

Whereas in frequency we can have non monotonic curves because here you could go up, down, we could have the uniform distribution.0304

We could have all kinds of things but in cumulative frequency distribution you cannot have a uniform distribution.0315

If everybody has only see the bottom score then you would have a uniform distribution because you will be adding 0 everytime.0321

Otherwise the most frequent shape that you will see is a monotonic increase that looks like this.0329

Here we see just like this normal looking distribution and what ends up happening when you have this normal-ish distribution0336

you have the s shaped curve when you transform it into the cumulative frequency.0345

This part in the middle, that part corresponds from about 400 to 600.0352

That part corresponds to the biggest jumps and as you see there are big jumps here too but the jumps are just going in one direction.0362

That is how it looks.0377

Let us put that on hold for a second and let us talk about calculus in brief.0382

A lot of people when they think about calculus, they think immediately derivatives and intervals like integrating stuff and getting derivative of equations.0397

It is what they think about.0398

Let us start packing and think about it conceptually.0399

Calculus in some way you could think of an equation as being on a continuum.0403

Let us just say that there are some equation that we are thinking of Y=x² or something like that.0409

We all know what that function looks like.0418

It looks something like this a parabola and when we think about this we are saying this is just plotting exactly what y is given from x.0421

That is all that graph is.0434

When x is 0 y is 0.0435

When x is 1 y is 1.0437

When x is 2 y is 4.0439

We are just plotting those precise points.0442

When you go and take the integral, I’m going to put the integral on this side versus you go and take the derivative,0447

you are describing 2 different aspects of the same graph.0459

When you talk about the integral, what you are doing is you are no longer plotting these particular point but now you are plotting these areas.0467

When you think about integral, think about area.0480

Whenever you have some curve or shape or line, when you take the integral you are pointing towards the area.0485

You could get these areas of this weird bizarre shapes and at the same if you go on the other side of the continuum0495

and you go towards the derivative which you will end up getting is not the area but instead you are just getting the slope.0504