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Lecture Comments (11)

0 answers

Post by Ryan Reddell on April 30, 2014

All these questions with no answers.  Isn't asking questions answered part of the monthly fee?  

0 answers

Post by Marc Patrone on February 18, 2014

In my exam I won't be allowed to use excel or any other program so I would appreciate doing this
the long way. Thanks.

0 answers

Post by Michelle Greene on October 7, 2013

Example 2 HELP!  When calculating manually the z-score I do get .7037, however I am not getting .24 from the table.  What am I doing wrong?  We cannot use excel so calculator or manual way would be be great. thanks.

0 answers

Post by Michelle Greene on October 7, 2013

yes,Please tell us how to solve on the TI 83.  On the exams we only have access to our TI83 calculator and not excel.

0 answers

Post by Abdulaziz Baathman on April 2, 2013

4. A normal random variable x has mean µ = 1.20 and standard deviation σ = 0.15. Find the probabilities these x-values:
a) 1.00 < x < 1.10
b) 1.35 < x < 1.50
c) x > 1.38

5. A normal random variable x has mean 35 and standard deviation 10. Find a value of x that has area 0.01 to its right.

pls help me

0 answers

Post by Matthew Compton on February 28, 2013

Is there any helpful links to use for TI calcuators.Since the lectures ignore the needs of your customers that do not use excel.

0 answers

Post by Kristen Gravlee on September 30, 2012

Where is normdist on the calc?

0 answers

Post by Mariya Kossidi on September 26, 2012

teach us on normal calculator, because on the exams students are allowed to have only calculators

0 answers

Post by Kathryn Connor on March 13, 2012

Please show how to do the calculations on a TI-84 calc

1 answer

Last reply by: Kristen Gravlee
Sun Sep 30, 2012 9:36 PM

Post by Kamal Almarzooq on January 22, 2012

you depend on excel too much :(
what about the students who don't have it
teach us how to find it in the normal calculations

Standard Normal Distributions & Z-Scores

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 0:06
    • Roadmap
  • A Family of Distributions 0:28
    • Infinite Set of Distributions
    • Transforming Normal Distributions to 'Standard' Normal Distribution
  • Normal Distribution vs. Standard Normal Distribution 2:58
    • Normal Distribution vs. Standard Normal Distribution
  • Z-Score, Raw Score, Mean, & SD 4:08
    • Z-Score, Raw Score, Mean, & SD
  • Weird Z-Scores 9:40
    • Weird Z-Scores
  • Excel 16:45
    • For Normal Distributions
    • For Standard Normal Distributions
    • Excel Example
  • Types of Problems 25:18
    • Percentage Problem: P(x)
    • Raw Score and Z-Score Problems
    • Standard Deviation Problems
  • Shape Analogy 27:44
    • Shape Analogy
  • Example 1: Deaths Due to Heart Disease vs. Deaths Due to Cancer 28:24
  • Example 2: Heights of Male College Students 33:15
  • Example 3: Mean and Standard Deviation 37:14
  • Example 4: Finding Percentage of Values in a Standard Normal Distribution 37:49

Transcription: Standard Normal Distributions & Z-Scores

Welcome to

Now let us talk about standard normal distributions and z scores.0003

First we are going to contrast the normal distribution against standard normal distribution.0008

It is pretty because just by knowing the normal distribution you already really know the standard normal distribution.0015

Then we are going to talk about some normal distribution problems and contrast that with standard normal distribution problems.0021

Before we talked about how the normal distribution really is a family of problems, it is not just one shape, 0031

but it could be stretched or shrunk on that x axis.0037

And because of that it is actually an infinite set of distribution with all the different means and different standard deviations.0042

You could have means of 10 and deviation of 10.0051

That is one way of distribution.0053

Another one could have a mean of 1 and a stdev of 2.0055

There is like an infinite number of these.0059

They all fit the empirical rule.0062

Because it is problematic to work with like everything will one of these different normal distribution 0066

and they have thought of this transformative system where you transform all normal distribution into what is called the standard normal distribution.0072

And there are what we are doing is where making that mean 0 and the stdev become 1.0081

This way we do not have to worry about the actual values.0089

We do not have to worry about the mean of actually 50. 0090

We just have to know the mean is in the middle.0095

And the standard deviation by being 1 makes it really easy for us to label this stuff.0102

In a standard normal distribution, what we really done is we have transformed the values into z scores or what we call standard deviation.0111

We have normalized everything we do not care that the standard deviation is actually 10.0133

We just care how many is the stdev the way they are.0142

If you are 1 standard deviation away, 2 standard deviation away, and because of that we have changed this x axis 0146

instead of actually putting the values we are now putting the z scores.0152

That is the only difference between the normal distribution and the standard normal distribution.0156

In the standard normal distribution we basically ignore the values and we only use the z scores.0160

We have everything in terms of standard deviations.0168

Are you 1 standard deviation away, 1/2 standard deviation away?0171

That is what we care about.0175

In a normal distribution, which I will draw in blue, in a regular normal distribution, we usually have is probability, which is the area.0180

This is represented by the area.0193

The raw score, with the mean and standard deviations and actual values.0197

Things like 150, 450.0207

We have the z score, 0, -1, 1.0212

That is what we think about a normal distribution, a regular normal distribution.0220

What we do in a standard normal deviation is we ignore that part.0224

It is the same thing.0230

You still have probability and we still have the z scores, right?0231

Now all the scores are z scores because we do not have any other scores other than that.0235

That is the only thing that is different about a standard normal distribution.0240

Now let us talk about the z score, raw score, and the mean, and the stdev, and the relationship that we have to each other.0250

Before we really did it in an intuitive way. 0257

For instance, if our mean was 10, the mean is 10 and our standard deviation is 5.0262

How do we know where to draw these notches?0276

Well if it is a normal distribution we know that this is approximately this area should be About 68% more than half.0280

What you could do is look for the point of inflection and often in my drawings the point of inflections is hard to see 0289

but in a nicer drawing you could see the point of inflection.0296

What we could see is 1 standard deviation away means that it is a distance of 5.0300

What would this value be right here?0309

Since it is 5 on the negative side, all we have to do is subtract 5 and it is pretty intuitive.0312

If we would want to go another five steps to the left and that would be 0.0318

If we wanted to go yet another 5 steps to the left that would be -5.0323

If we wanted to go yet another 5 steps to the left that would be -10 and so on and so forth.0328

Same thing on the positive side, if we wanted to go 5 steps to the right and that would be 15, 20, 25, 30 and so forth. 0335

On to infinity and negative infinity. Now before we just did it in a very intuitive way and these notches just corresponds with z scores. 0348

This is 1 standard deviation out, 2 standard deviations out three standard deviations out, 0363

but there is systematic relationship that we could exploit here in turn with the formula between the z scores, raw score, and mean, and standard deviation.0373

If we wanted to go 2 standard deviations out and find what this value was, in order to find the raw score what we do is take the mean 0384

and then add the z score and multiply by the stdev.0402

This actually works with the negative side too because when we have a z score of -1 then you would subtract the stdev instead of adding to the mean.0408

This formula ends up being good for us.0422

Instead of raw score, we usually call the raw score the axis.0425

Instead of writing out mean we would write mule.0429

Then we could keep z and write sigma as our stdev.0436

Now we see our formula here that allows us to go from the mean, z score and standard deviation into a raw score.0442

We could use this formula to find anything, any of those four things.0452

If we wanted to find the z score for instance, and we did some algebraic to this 0457

and we would just see that is actually the distance between the raw score and the mean divided by the standard deviation.0465

Let us think for a second about what that means.0477

That means the z score is take this distance and cut it open to these chunks.0481

How many of those chunks do you have right?0487

the z score really 0489

is telling you how many standard deviations are you away from the mean?0490

It is totally relative to the mean and standard deviation. 0495

Also you can use this very same formula to solve for the mean and standard deviation.0501

We could do that over here right.0509

Let us say we wanted to solve for the mean, what would that look like?0513

All we have to do is move this over so that would be x - z score multiplied by the standard deviation.0519

Let us say we wanted to find standard deviation, what would we do then?0528

That is pretty easy as well.0535

All we have to do is you could just take this formula and swap these guys and that would be stdev = x to find out what is z.0538

Here we see that using this very simple formula and you could just remember one of these 0552

because having one of them, you can derive the other ones from it.0559

Just by having one of these formulas if you had any 3 of these you could solve for the fourth one.0563

You do not even have to do the algebraic transformations.0571

You could just plug them in and find out what is missing. 0574

So far we have only talked about the z scores for its nice and even like 1, -1, 2, -2.0583

We have not talked about weird z scores.0592

For instance, what about a z score of something like .5?0594

What would be the area over here?0608

Some of you maybe tempted to take .34 and divide it in half, but as you will see from this picture that would not work.0615

You are not really dividing this area in half, this area still ends up being slightly bigger because it is taller than this area.0625

In this area have been chunk out of it.0634

You cannot just divide it by half.0640

That is not going to be a good strategy.0642

That would not give you the right area.0647

How do we deal with this?0650

Well, there are two ways of dealing with this these weird z scores and how to go from these weird z scores to probabilities that we do not know yet.0651

That are not nice than my 34, 13 ½ .0660

There are two ways of doing it.0665

One way is by looking at up on a table. 0666

Often there are tables in the back of your text book or even in the back of an EP statistics 0670

like Princeton Review book or something that show you the transformation from weird z scores like .5 and .67, 1.9 probability. 0675

That is one way of doing it using z tables.0690

The second way of doing it is by using Excel or your calculator.0694

If you do not have a fancy TI something calculator, it should also come with similar functions to Excel.0703

I’m going to show you those two methods of how to go from weird z scores to probabilities of their z scores and vice versa.0710

How to go from the probabilities, like weird probabilities like 50%.0720

We do not know where like 51% but we do not know what this value would be for 51%.0725

how to go from those weird probabilities into that weird z scores.0733

First, let us talk about the method by using the tables in the back of your butt.01225.1 Usually it is the first table you will see back there, table A or something like that and let us break it down.0739

A lot of tables looks somewhat like this one here might look slightly different than probably roughly similar.0752

And what it shows you up here is exactly what probability is plotted down here. 0760

What it shows you is the probability that shown certain on the negative side.0765

Everything below the z score and this is what we call the cumulative probability because you are accumulating it as we go right.0771

It is adding up all the probabilities on this side.0783

This is showing you the cumulative probability at the z score.0786

The table entry for z is the probabilities lying below z. 0793

Here the z scores and these are the probabilities.0801

Now for the weird z scores, what would really be helpful is if we had z scores of - .34, -.341, -3.42, -3.43.0804

We had all of these decimal places.0827

It would be really nice if we had all these different z scores.0829

That would probably be a skinny list of a whole bunch of z scores and a skinny list of a whole bunch of different probabilities.0833

That would be a very inefficient use of space.0843

What a lot of tables do is they put the z score up to like here, it is up on the tens place in this side and then we put the hundreds on this dimension.0847

In order to find the probability for the z score -3.45, you have to find -3.4 and then go to .05.0861

It is like you add this and you stick it on there if you added it would not work because it is negative.0876

Here that would be -3.45 and at -3.45 the cumulative probability is .0003.0887

Notice that it is a very small probability, it is not 0 but it is very, very small.0902

And let us do another example. 0907

Let us say we wanted to know the probability at the z score -2.48.0909

We go to -2.4 and then also go to 8 and that would be our probability .0067 less than -.1%.0919

What if we wanted to find out the upper side?0933

We are like your only giving me the cumulative probability on the lower side. 0938

What if we wanted to find out this probability at 2.48?0943

What would we do then?0955

Because the normal distribution and standard normal distribution to be is perfectly symmetrical 0956

what we would see is that if we know it for the negative side, we can just flip it over on the positive side.0965

The negative side over here at -2.48 is a probability of .00066 then this 2 is also .0066 because it is perfectly symmetrical.0970

You do not need to have both the positive and negative sides.0986

Oftentimes tables might just give you the positive side or the negative side.0991

Sometimes, they will give you both, but you do not actually need both, you could just figure it out from there.0997

Let us talk about how to do that with Excel.1007

For Excel they are nice functions that Excel prewritten out for us so that we do not have to actually look it up on a table.1009

For normal distributions, when you know the mean and the standard deviation, there are ways to go from the raw score to the area underneath the curve.1018

So basically it is very similar to that picture we found that z score.1035

What it will give you is this cumulative probability.1040

That entire area below that value.1045

The normal distribution, you need to enter in the mean and the standard deviation.1050

In order to go from the score into the probability you would use the normdis function and1057

you would put in the score, but you would also put in the mean, the standard deviation.1066

and there is another thing, you also have to put in shrew in order to get the cumulative probability because this is sort of asking cumulative probability.1072

and that will give you the probability of that score. 1086

The cumulative probability of that score.1090

Norm inverse is just the inverse of that.1092

Here we give it the probability and it is been stuck out the score. 1097

Here you give it the probability, the mean, and the standard deviation, and it spits back out to you the score.1101

Notice that these two are inverses that is why one of them is called norm in because in one, you get the probability and the other you get the score. 1112

These are our 2 flipping around values.1127

For standard normal distribution, you do not need to enter in the mean and standard deviation because all standard normal distribution means are 0.1132

The mean of 0 the stdev is 1.1145

It makes it a lot simpler for us. 1148

The only difference between a normal distribution and standard normal distribution is this one little letter. 1152

All you have to do is remember to enter in norm s dis and here you could just put in the score, forget everything else and it will spit out the probability.1158

For norm set in, it is exactly the opposite where you put in the probability and a spit back out of the score.1172

Another handy little functions might be the functions standardized.1188

Here you put in the raw score, the mean, and the standard deviation, and it will give you the z score. 1192

The standardized function simply uses the formula that we have talked about earlier where 1204

in order to give you z score, it takes the raw score or x – mule ÷ stdev.1213

Given that, let us look at a few examples and do them in Excel.1226

The distribution of SAT scores for Math for incoming students to the University was approximately normal. 1230

That is again important, it has to say approximately normal.1237

watch out for that.1241

I do not know if you have tricky instructors or tests but sometimes they might give you a problem that looks like a normal distribution problem, 1242

but it does not say it is normally distributed.1247

With the mule of 550 and a standard deviation of 100, what percentage of scores with 400 or below.1251

Note that 400 if you transform it into a z score would not it be nice even z score like -1 or -2 is actually like -1.51260

That is going to be a little bit difficult for us to use just the empirical rule.1276

That is where we have to use something like either the table in the back of your book or Excel.1279

Here I just have an empty Excel sheet and one thing I’m going to do is I’m going to use my columns 1286

to denote probability raw score z scores just because that can help me keep track of what I’m doing.1300

I’m also going to write down my mean and standard deviation just to help make things a little easier.1308

550, 100, and my probability where x is less than 400, that is where I’m going for.1316

My raw score is 400 and my z score.1332

I can just do this in my head it is just the distance 150 ÷ 100.1344

It is on the negative side so it is -1.5.1354

But just to practice using Excel, let us use our standardized function.1357

Excel will guide you into what you exactly need.1367

You need your x value for raw score.1370

The mean and the standard deviation.1374

I’m going to close my parentheses and I will get -1.5.1379

The other way you could do this in Excel is do the formula because you know that you need the raw score – mean / stdev.1384

You will get the same thing but I just wanted to illustrate for you that is what the standardized function is doing.1400

Let us see how that now you need the probability, I could find the probability in 2 ways.1406

One, I could use this standard normal distribution or I could use the regular normal distribution.1418

Let us do both.1423

First, I know that is the norm dist, I do not need the inverse one because I have the raw score, I need the probability.1425

Let us start with norm dist.1434

Here I know I need my x value and I know I need the mean, stdev.1437

I’m just going to ask you it is cumulative?1445

Let me just write should.1448

That gives us .0668 and that makes sense if we think about where -1.5 was, it is in between -2 and -1.1451

If it was at -1, it would be like 16%.1468

If it is all the way at -2, it will be like 2%.1472

6% sounds like it is in between 2% and 16%.1476

We could also do this by using norms set dist.1483

For that, all you need is the z score.1493

I will put in my z score and I should get the exact same thing because the z score corresponds with the raw score.1496

They are right above each other.1505

Probabilities at that point should be exactly the same.1508

Now let us talk about the different types of problems.1519

Now that you know z score and standardized normal distribution, what kinds of problems might come in your way?1522

The first sign of information you do not always need these numbers because the first set of information 1529

is the mean, stdev, probability, raw score or x, the z score or z.1535

That is the first set.1546

Anyone of these things can be missing and you could find it.1548

What is missing here?1552

Here we have that same prompt, it says what percentages of scores where 400 and below?1554

That is percentages of scores.1561

Or it might ask what percent of scores fall below the z score of -1.5.1563

Here we see that they are both about percentage of score.1575

What might be missing is dist.1579

If you have either one of these, and this 2, you could find it no problem.1582

Here is another set of problem, same prompt but what is missing here?1588

Here it says what math scores separates the lowest 10% than the rest or what z scores separates the lowest 25% from the rest?1595

Here you have the probabilities and you have the mean of stdev but one of these 2 things might be missing.1605

I think of this as the score missing problems.1613

We have probability missing problems and we have score missing problems.1616

The last set looks like this.1621

Note that this prompt does not give you the stdev but it does give you the z score and the raw score so that you could find the stdev here.1625

Or it gives you the percentages and the raw score and then you find the stdev.1637

Here something like a stdev is missing.1645

You could also find the mean to be missing as well and they would look similar to this problem.1648

As long as you have some combination of the other values in place, you could actually figure out what is missing.1654

Now let us add in the shape analogy that we did in the previous lesson.1666

I’m going to skip over a lot of this because it all stays the same.1671

They are only parts where they drive your attention to is right here.1674

Before we could only look at our probabilities and raw scores but now we have extended our repeater. 1677

We could look at probabilities to raw scores to z scores and even find the missing mean and stdev.1685

That is all done to a combination of the z scores formula as well as either the tables or Excel.1693

Let us do some problems.1705

Here is example 1.1707

In the US, the distribution of deaths due to heart disease 289 deaths per 100,000 stdev of 54 and cancer a mean of 200 stdev of 31 are roughly normal.1709

We know that we could use our normal distribution stuff.1728

In California, 254 deaths are form heart disease and 166 deaths are from cancer per 100 residents.1730

Which rate is more extreme compared to the rest of the states, the average for the US?1740

We have California’s death rate from heart disease and from cancer.1748

We want to know which of these are really extreme?1756

One way that we could find out is by finding out the z scores.1759

I am going to get my Excel out.1766

Let us first start with the US.1777

The US mean for heart disease. I will use this row for heart disease and this row for cancer.1780

The US mean is 289 and the stdev is 54.1791

For cancer, it is 231.1809

We want to know how extreme the California deaths are.1813

It is hard to compare with just the numbers because even though 240 is less than 289, and 166 deaths is less than 200, 1817

we are wondering how far away from the mean are you?1833

One way that we could do that is by using z scores because z scores will give you the distance in terms of the stdev.1838

Because these populations have very different standard deviation, that is worth knowing.1845

Here is California, heart disease and cancer.1852

In California the mean is 240 and 166.1865

What we might want to know is the z score.1875

What is the z score?1879

I might just put standardized and put in my x, put in the theoretical mean that I want to compare it too and my stdev.1882

Obviously, I could also do my (x – mean) ÷ stdev.1897

Here we see that the z score for heart disease is about 1 stdev away, -.9.1908

How far down is cancer?1918

How much less is cancer?1921

How more healthy are Californian in terms of cancer?1925

For here I will put in my regular formula, just so that we practice that too1929

My cancer x – theoretical population ÷ stdev.1935

I want it in terms of standard deviation step.1944

We find out that cancer and this is probably because stdev is smaller and actually sort of out is farther than heart disease.1947

The cancer rate is more extreme in a positive way.1962

It is more extremely low than heart disease, although they are very close.1968

Here the trick is find rates in terms of stdev or z scores.1977

Example 2, heights of male college students in the US are approximately normal, estimate the percentage of these males that are at least 6ft tall.1995

It will help if we just sketch this out briefly.2012

Here is my mean as 70.1, my stdev is 2.7, what percentage of these males are at least 6 ft tall?2016

What helps is if you transform this 6ft into inches, that is 72 inches.2027

Where is 72 inches?2043

That might be something like this.2045

What percentage of males are at least 6 ft tall?2048

I want all of these people because all of these people are at least 6 ft tall.2052

They might be 7 ft tall but they are at least 6 ft tall.2058

That is what I really want.2061

What might be helpful is instead of my raw score, if I could find my z score and then I can look it up on the table in the book.2063

Or I could use Excel.2072

I’m going to use Excel to help me.2075

Let us find the z score for this.03449.5 In order to find z score, let me write the formula here.2086

In order to find my z score for 72 inches that would be 72 which is my (x – mean 70.1) ÷ stdev 2.7.2095

Because I want to know how many jumps away, my jumps are 2.7.2115

Now I will pull up my Excel and I could just put in (72 – 70.1) ÷ 2.7.2127

I find that my z score is .70.2144

In order to find the area, the cumulative area, remember cumulative area means this side.2151

Excel is going to give me this side but that is now I want.2159

I want this side.2163

I might put in 1 – this area in order to get that area.2165

1 – and will just put in my norms dist.2172

Thankfully Excel gives you a little hint if you are a little off.2190

What I get is .24, about 24% of these males are at least 6 ft tall.2195

This area is about 24%.2216

If I want to write it all out I would write my probability where x is greater than 72 inches is .24.2223

This is example 3, in a standard normal distribution where P(z score) < .41 = .659, what is the mean and stdev?2235

Actually this is a tricky question, before you go often trying to find the z score and all the stuff, note that it says standard normal distribution.2249

Every standard normal distribution, mean is 0, stdev 1.2257

Example 4, find the percentage of values in a standard normal distribution that fall between -.1.446 and 1.46.2266

This is nice if we would sketch this out so that we know what to expect.2283

Here is 0, and this is a standard normal distribution that is why I know that it is 0.2288

Here is 1, here is where 1.46 is.2295

A little bit less than1/2, probably a little bit too much less than half.2308

I want to find this area right in between.2314

Usually we look at the area between -1 and 1, and we know that 68%.2328

We know that it is a little bit more than 68%.2340

It is not quite like 95%, it is not quite that high but it is somewhere in between 68 and 95%.2342

Let us try to figure this out.2352

One way we could do it is by using our Excel or by looking it up in our book.2354

In our book and in Excel, our problem is that they will give you the cumulative distribution.2363

They will give you this area.2369

What we really want is this area.2372

What do we do?2376

One thing we might want to do is just use our ability to reason so this is 50% of that curve.2378

If we take 50% and take away this area, we could use norms dist and put in -1.46 then that should give us this area right here.2388

This whole area is this.2405

This area is in blue.2414

That should give us the red area.2418

Here I’m going to put in 50% - norms dist – 1.46.2421

I want to take that area and multiply it by 2 because I want the other side too and it is perfectly symmetrical.2442

We do not have to do any work just by multiply by 2.2452

Then I would get .8557 so a little bit more cloes to 86%.2455

That makes sense.2465

Since it is more than 68%, it is less than 95% right?2466

80 and 86%.2470

My answer is where -1.46 my z falls in between this values, .86.2475

That is example 4 and that is it for the standard normal distribution.2496

Thanks for watching