Statistics Normal Distribution

Hi and welcome back to www.educator.com.0000

Today we are going to be starting to talk about normal distributions.0002

When will talk about normal distributions in the next couple of lessons and it will come up again and again in the future.0006

It is a pretty important one.0012

First let us talk about what a normal distribution actually is.0016

I want to distinguish it to what we have been talking about before.0019

When a distribution looks normally shaped, a normal distribution is an entire different thing.0022

We are also going to talk about a lot of normal distribution yet they all have the same shape and what that means.0030

Then we are going to talk about some normal distribution problems but only problems that use the what we call empirical rules.0037

What is a normal distribution?0046

It is also called a normal curve.0048

A normal distribution is a theoretical model.0050

It is not based on data necessarily but it is a projection.0054

So far we have looked at frequency distributions actual data points that have a normal shape.0059

By normal shape what we really meant was symmetrical, unimodal, point of inflection, those kind of features.0066

They do not just have a normal shape, they are not lead normal distribution.0084

The normal distribution, the actual model is actually a probability density function.0089

When I draw a normal distribution, you might say that it just looks like those normally distributed shapes that we have looked at before.0096

The difference is that her, notice this curve is actually represented by a function that we are going to talk about in a later lesson.0105

What is underneath the curve, the area underneath the curve is the probability with which some value will occur.0115

On the x axis we have some values, presumably values of some variable.0124

For instances, height.0131

The values might be like 60 inches, 61 inches.0132

Here what we see is the mean, mode, median, but here in this area that is the probability of0139

which a value that is clocked out of the road with far in between x1 and x2.0154

We are interested in the relationship between the area and values.0167

That is why a normal distribution is really important because there is this nice, regular, relationship between probabilities and values.0174

Let us talk about the range of the probabilities that are possible underneath the normal distribution or normal curve.0188

The area underneath the curve represents probability.0196

The entire area underneath the curve, if I colored everything in, that probability is equal to 1.0200

All individual probabilities, this one and this one.0209

All of these add up to, from negative infinity to infinity, all the way equals 1.0215

A probability of 0 means that , that value never occurs.0231

By never, I mean never.0236

What we are going to introduce is this new notation.0239

It might not be new to you if you have probability before but this just means that this is a probability function.0247

Remember that we plot a probability – density function, it is a probability function we are given some value x0254

you will get as the output of the probability.0260

P of an exact square, a very particular square, the probability of height being 60m of dot is 0 in a normal distribution.0264

Because it is a density distribution, it is not about frequencies.0276

It is about probability density.0281

What we are interested in is the probability of a square following a certain range.0283

That probability is not 0 and it actually does not matter how far you will get, that probability is not actually 0.0289

It might be like .000000000001 but is not 0.0300

The normal distribution, we could summarize it a couple of ways.0311

We could summarize it using the mean and the standard deviation.0315

Here we see that the mean is represented by the mu sign and the standard deviation is represented by the sigma (∑).0320

If you recall back to population versus samples, this is not much more like a population than it is a sample.0327

That is because the normal distribution is a theoretical model, it is not actual data.0334

It is a theoretical model, a projection very similar to a population.0342

It is where we think the samples are coming from.0347

Some properties are just like the normally distributed shapes that we have looked at before, symmetrical, unimodal, asymptotic.0351

This is just talking about this fact but it actually approaches but never reaches 0 probability.0363

It asymptotes with the x axis and it comes close but does not actually touched it.0373

The final things is that it is continuous.0381

The normal distribution actually goes from negative infinity to infinity.0385

Most values and most measurable things in the world do not actually have values for negative infinity to infinity.0389

For instance, height.0398

We do not really have anybody who is infinite height or negative height for that matter.0401

But the normal distribution stretches on forever and ever.0410

There are going to be what seems like a lot of different normal distributions.0415

For instance, you might see normal distributions that look like this.0420

You might see normal distributions that look like that.0424

You might see normal distributions that look like that.0427

You might see them as having slightly different shapes.0429

Like this one is sort of fat, this one is skinny, this one is even skinnier.0434

But you do see that each of that are symmetrical, unimodal, asymptotic, and continuous.0439

Actually there is more to the normal distributions than that.0445

It is not that those for features.0448

There are other properties that has an empirical rule.0450

We talked about how can they be fat or skinny, but not all shapes that look like this is a normal distribution.0458

Let me show you why.0468

In a normal distribution, it follows some strict set of relationships between the probabilities and values down here.0470

That is going to be important.0481

Here is what I mean.0482

When you go 1 stdev out form the mean or the line of symmetry, when you go 1 stdev out, that area right here should be about 34% of the curve.0484

That is always the case in a normal distribution.0505

Let us say we have this shape and it looks like a normal distribution, but we go 1 stdev out.0508

Let us say we go 1 stdev out from the normal distribution, does this looks like just 34%?0524