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Lecture Comments (5)

1 answer

Last reply by: Winfred Chiles
Fri Sep 26, 2014 1:13 PM

Post by Manoj Joseph on May 18, 2013

I thought buffering problem has something to do with my monthly subscription and I renewed to six month. Now, it is taking too much time in buffering. can some one tell me how to have an uninterrupted lecture session

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Post by David Nilsen on April 13, 2013

If your ever in Rockford Illinois let me know, I will introduced you to my teacher and you can teach him how to teach. I have learned more in 30 minutes than listening to him for hoooooooours.

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Last reply by: Professor Son
Wed Aug 15, 2012 2:12 PM

Post by Ryan Mulligan on January 26, 2012

Very clear explanation....cheers!

Normal Distribution

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Roadmap 0:13
    • Roadmap
  • What is a Normal Distribution 0:44
    • The Normal Distribution As a Theoretical Model
  • Possible Range of Probabilities 3:05
    • Possible Range of Probabilities
  • What is a Normal Distribution 5:07
    • Can Be Described By
    • Properties
  • 'Same' Shape: Illusion of Different Shape! 7:35
    • 'Same' Shape: Illusion of Different Shape!
  • Types of Problems 13:45
    • Example: Distribution of SAT Scores
  • Shape Analogy 19:48
    • Shape Analogy
  • Example 1: The Standard Normal Distribution and Z-Scores 22:34
  • Example 2: The Standard Normal Distribution and Z-Scores 25:54
  • Example 3: Sketching and Normal Distribution 28:55
  • Example 4: Sketching and Normal Distribution 32:32

Transcription: Normal Distribution

Hi and welcome back to

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

Does it look like more than 34%?0546

That means that this is not a normal distribution.0548

Even though it looks like one superficially.0552

On the other hand, let us say we have something that looks like too skinny to be a normal distribution0556

but if in this case going 1 stdev out, looks like that.0566

But this is sort of 34% of the total curve, right?0578

Because of that, even though it looks skinny it is a normal distribution.0583

In this one, it looks like a perfect in a very typical normal distribution but it is not because the area does not fit.0589

That is one thing you need to know, that 34%.0598

If you flip around that side because it is a symmetrical distribution, if you flip it around the area between 0 and stdev 1, this is also 34%.0601

When you add those together, it is about 68%.0619

Just by going 1 stdev out on each side from the line of symmetry this should get 68% of the area covered more than halves.0623

If you go out another stdev and you look at the area between 2 stdev out and 1 stdev out, that area is equal to about 14% or 13 ½.0636

Let us write this t our new notation where we use P and I will just use this as my values for now.0656

Where x is in between 0 and 1 = .3413 and the area in between 1 and 2 = .1359, this is how we would write this in algebraic notation.0672

Finally let us cover the small area right here where now we are going 3 stdev out.0707

I’m just going to write my notation down here.0714

Here I’m going to write P where in between 2 and 3, that area is about 2%.0716

This is what we call the empirical rule.0730

I advice that you memorize these 3 numbers, 34, 14, and 2.0734

Note that by knowing this you could know at least approximately how much of the area is covered by going 1 stdev out and 2 stdev out, 3 stdev out.0740

Because we are rounding if you add all of these numbers out 34, 34, 14, 14, 2, 2 = 100% but in a normal distribution it is asymptotic.0754

It is continuous.0769

It is a little bit less than 100%, it is 99. Something percent.0770

There is a tiny little bit area that goes out and out forever.0777

It is handy to know that if you 1 stdev out then you will cover 68% of that total area.0785

If you go 2 stdev out, that is about 95% or 95 ½ %.0799

When you go out all the way to 3 stdev you will get .9974%.0809

It is really close to 100% but not 100%.0821

That is important for a normal distribution.0827

Let us talk about the different kind of problems that you might be able to solve just by knowing the empirical rule.0831

Usually in a normal distribution problem, you have to first look for rather the distribution that we give you is normally distributed.0833

If it do not say that it is normal, then you cannot use the empirical rule.0844

Let us read the prompt and let us get with the different kinds of problems there might be.0850

The distribution of SAT scores for incoming students in a university is approximately normal with a mean of 550 and a stdev of 100.0855

They told us it was approximately normal so we could use our empirical rule.0866

Usually in a normal distribution problems where you need to use the empirical rule they will probably give you the mean and stdev.0871

In other problems with a normal distribution that we will look at later, we would not give you the mean and you might be able to figure it out.0880

They might ask you what percentage of scores with 450 or below?0888

Here they give you the score and you are supposed to find the probability of getting that score or below.0893

Where x is less than 450.0907

Note here that what is missing is probability.0911

Another question that they could ask you is that the same prompt above, they could ask you what math scores separates the lowest 2% from the rest?0916

Here they give you the probability.0925

What they actually want you to find out is, what are the scores?0929

What is this blank right there, right?0933

Here we are missing the score.0940

Here I will write missing probability.0944

Here we are missing the score or value.0952

Now let us try to solve these 2 problems by using the empirical rule.0961

What percentages of scores will 450 and below?0965

It has to stretch out in a normal distribution first.0970

Here is 550, that is the mean, point of symmetry.0978

450 is just 1 stdev in a way.0984

The distance is 100.0993

You could think of stdev as new units of measurements.0996

We knew about inches and meters but now we are interested in how far way things out in terms of stdev rather than inches or feet.1002

Here the stdev is 100.1012

We know that even though these are the scores or what we call the raw scores, we know that these in terms of stdev,1015

this means that this is 1 stdev away on the negative side.1025

We could just draw a little border here and sketch the part that we need to find and that is the area that we need to find.1031

We know that this part from here to here is 34%.1041

We know that this part is 14%.1048

We know that this part is about 2%.1053

We could just add 14 + 2 and get the probability where x is less than 450 = about 16%.1057

That is the percentage of scores.1071

What about when we are missing a score that we have the probability?1075

Let us sketch this one as well.1080

Here we have this 2% and we want to know what is this value right here.1085

We know that although we do not know the raw score right now, we do know in terms of stdev1094

that at about in between 2 or 3 stdev away, that is about the 2% mark.1104

Just to show you I will draw the other one.1112

In terms of stdev, here is 0, -1, -2, -3.1117

This seems to correspond with about being 2 stdev away on this side.1127

We know that this middle is 550, if we take stdev 100 jump then this would be 450 and this value will be 350.1133

That is a little small but I will write it up here.1151

We know that what we are looking for is 2% that is equal the probability where x is less than 350.1153

So far it has been pretty easy just adding and subtracting, and memorizing the empirical rule so let us go ahead and do some more problems.1178

Before some more problems we are going to just look at this in terms of a shape analogy.1190

Just to sum it in, when we think about shapes like rectangles, we know that rectangles are defined by their length and width.1195

If you know their length and width, you could draw that rectangle.1205

In the same way for the normal distribution, all normal distributions follow the empirical rule.1209

All you need to know is the mean and the stdev.1214

You could draw that particular normal distribution and we know that it is unimodal, symmetric, asymptotic, and continuous.1221

Rectangles can all look a little bit different like sometimes their length is longer than the width.1235

Sometimes the width is longer than the length.1242

Sometimes the length is equal the width in the special case of square.1244

They could all look different but they are all rectangles.1248

In the same way normal distributions can look slightly different from each other.1251

Because the x axis can be stretched out or can be shrunk.1256

But as long as that normal distributions follows the empirical rule where about 1 stdev out1261

on either side is about 68% then you know that it is still a normal distribution.1269

Even though it looks a little too skinny or too fat.1275

What can you find out when you know rectangles?1278

If you have the area and width, let us say you have this and this, you could find out the length because they have this relationship with each other.1282

Same thing with perimeter, if you know the perimeter and the length then you could figure out the width.1293

These are what we call constraints because they constrain the system.1301

They are like these little boundaries and you can balance within the boundaries and figure out the things that are missing.1306

In the same way, when you have normal distributions you could balance from probability to raw score because they have this relationship.1313

And that relationship that you have for now is called the empirical rule.1321

We have only covered knowing that relationship between probability and score and only wonder even intervals away.1327

That 1 stdev, 2 stdev, 3 stdev.1340

But we really do not know how to get the probabilities when it is like 1.5 stdev away.1344

That is what we will cover in the next lesson.1350

Let us go into some problems.1354

Example 1.1356

Using my empirical rule, what percentages of values in a standard normal distribution is used to solve below the SAT score of -1.1358

What falls above of these scores of -1?1368

Although we have not yet covered the standard normal distribution yet,1373

let us assume that it is the standard normal distribution we have been talking about.1377

We will define it in the next lesson.1381

It is actually when we do not know any of the values and we just know the c scores or what we call the standard normal deviations.1385

That is the c scores.1395

We could easily do this problem just by knowing the empirical rule.1399

In a standard normal distribution, we pretend that we do know the actual values, but we really do not know.1407

We just know the standard deviation or what we know the c scores.1419

What they want to know is what percentage of values fall below a c score of -1?1432

Here we know that this area is about 14% and this area is about 2%, this area is pretty negligible.1441

If we add these up this would be .16.1454

To write it in an algebraic expression it is p where z is less than -1 = .6.1462

Once you know this, it is now asking about what about above the z square of -1.1476

Now it is talking about this area.1482

There are two ways you could do this to figure out p where z is greater than -1, we know that the entire area is 1.1485

We could just subtract out this red part, .16.1499

We could just do that and we know that it will end up like this area which will then be .84.1508

There is another way that we could this.1515

What we could do is out at this part, this part over here which we know it is exactly half of the normal distribution .5.1517

And then add 2 to it to this little part right here which we know is the 34%.1528

When we add those together, we could do 34 and that also gives us 84% of the curve.1537

Those are two different ways of doing it either way, whatever your preference.1548

Here is another example.1556

Using the empirical rule, what percentage of values in a standard normal distribution fall below the z squares or stdev of -1 and 2.1560

It is nice to sketch it so that you will know where you are headed.1574

Here is 0 and what we need is between the stdev of 1 and -2.1581

Once again there are multiple ways that we could find this out.1593

Probably a very simple and straightforward way is knowing p and we are trying to find p where x or z lies in between -1 and 2.1596

What we could do is add up all this separate little probabilities.1612

The probability between -1 and 0 , and add that with the probability between 0 and 1.1616

Add that with the probability between 1 and 2.1630

You could just add all those up and that would be 34, 34, 14.1637

That would give us 68, 70, 82% of the curve.1658

That is one way of doing it.1668

Another way you could possibly do this is I just know that this is 2%, I could deduce that from this whole thing being 50%.1670

I could just subtract that 2 and get 48.1686

I could actually just do 34 + 48 that would give us the same answer of 82%.1690

Just different ways of summing this up figuring out the distributions.1701

Some of you may memorize the middle part between -1 and 1 is 68 then it would be like 68 + 13.1706

Either way you want to do it but the point is I want to show you there are lots of different ways you could cut this out.1717

You want to think of this little distributions like a chocolate piece or something that you could1723

just break this off in lots of different ways and add them together again in lots of different ways.1730

Example 3.1737

What is this problem missing, sketch and find what is missing.1741

We know that there are only two things that could be missing in the problems that we have introduced so far.1744

One is probability that could be missing or the value, the boundary.1748

It says, given a normal distribution with a mean of -25 and a stdev of 10, find out where the middle 95% of values would lie.1756

Right now we have the probability here.1766

What we do not have is the actual values.1778

This is a values missing problem.1784

That is what we are going to need to do.1786

It helps to sketch out what we have done, my x axis is a little bit off.1789

We know that the mean is -25, here my raw scores and here I’m going to write my z scores or standard deviations.1805

Here is 0, -1, -2, -3, 1, 2, 3.1816

We are trying to find the middle 95%.1828

If you remember the empirical rule, we know that around here and here this is approximately 95%.1832

If you want to check that you could add 34, 34, 13.5, 13.5.1843

That is about 95% of the curve.1850

Let us try to find what these values are right here.1855

We know that the stdev is 10, each of these little jumps are worth 10.1861

Let us go out 10 jumps from -25 and that would be -35, -45, in this side.1868

If we go on the positive direction it would be -15, -5.1877

To solve it we would say this probability .95 is the probability between -45 and -5.1887

Many way of writing it without these values is to write it in terms of the standard deviation but it asks about values.1903

I just wanted to show you this other way.1912

The other way we could write it is also like this.1915

What z scores does the middle 95% cover and that would between 2 and -2.1924

Note that the 45 corresponds to -2 and the -5 corresponds to 2, just like here.1932

That is the nice thing.1945

There are these relationships between the raw scores and the z scores.1946

We are going to get more into that in the next lesson.1950

Here is example 4.1954

What is this problem missing?1956

Sketch and find what is missing.1958

Given the normal distribution of the mean of 46.4 and a stdev of 6.1, find the score that comes at the largest 16% of values.1960

This one gives us again the probability and we need to find the score or the values.1981

We know that the missing thing is the missing score.1987

Using our empirical rule, we could find that 16% is.1991

Here are my raw scores and here are my z scores or standard deviations.1996

Since I am looking at the top 16% that need to go in this side, 0, 1, 2, 3.2003

We know that this about 14% and this is about 2%.2014

Here is my elusive 16% and we need to find this cut off score.2020

That cut off square is 1 stdev away and here my stdev is 6.1 so my little jump is 6.1.2030

What I need to do is add 1 jump to 46.4 and that would be 52.5 or you could write is the probability where x is greater than 52.5 or 16%.2040

That is it for using the empirical rule to find the answers for normal distributions problems.2065

Thanks for using