For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### 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
- Roadmap
- What is a Normal Distribution
- Possible Range of Probabilities
- What is a Normal Distribution
- 'Same' Shape: Illusion of Different Shape!
- Types of Problems
- Shape Analogy
- Example 1: The Standard Normal Distribution and Z-Scores
- Example 2: The Standard Normal Distribution and Z-Scores
- Example 3: Sketching and Normal Distribution
- Example 4: Sketching and Normal Distribution

- 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

### General Statistics Online Course

### Transcription: 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 of*0139

*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 x*0254

*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 distribution*0556

*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 stdev*1094

*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 out*1261

*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 could*1723

*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 www.educator.com.*2072

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

0 answers

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.

1 answer

Last reply by: Professor Son

Wed Aug 15, 2012 2:12 PM

Post by Ryan Mulligan on January 26, 2012

Very clear explanation....cheers!