For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Variance & Standard Deviation

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
- Definition of Variance
- Variance of a Random Variable
- Variance is a Measure of the Variability, or Volatility
- Most Useful Way to Calculate Variance
- Definition of Standard Deviation
- Example I: Which of the Following Sets of Data Has the Largest Variance?
- Example II: Which of the Following Would be the Least Useful in Understanding a Set of Data?
- Example III: Calculate the Mean, Variance, & Standard Deviation
- Example IV: Calculate the Mean, Variance, & Standard Deviation
- Example V: Calculate the Mean, Variance, & Standard Deviation
- Example VI: Calculate the Mean, Variance, & Standard Deviation

- Intro 0:00
- Definition of Variance 0:11
- Variance of a Random Variable
- Variance is a Measure of the Variability, or Volatility
- Most Useful Way to Calculate Variance
- Definition of Standard Deviation 3:44
- Standard Deviation of a Random Variable
- Example I: Which of the Following Sets of Data Has the Largest Variance? 5:34
- Example II: Which of the Following Would be the Least Useful in Understanding a Set of Data? 9:02
- Example III: Calculate the Mean, Variance, & Standard Deviation 11:48
- Example III: Mean
- Example III: Variance
- Example III: Standard Deviation
- Example IV: Calculate the Mean, Variance, & Standard Deviation 17:54
- Example IV: Mean
- Example IV: Variance
- Example IV: Standard Deviation
- Example V: Calculate the Mean, Variance, & Standard Deviation 29:56
- Example V: Mean
- Example V: Variance
- Example V: Standard Deviation
- Example VI: Calculate the Mean, Variance, & Standard Deviation 37:29
- Example VI: Possible Outcomes
- Example VI: Mean
- Example VI: Variance
- Example VI: Standard Deviation

### Introduction to Probability Online Course

### Transcription: Variance & Standard Deviation

*Hi and welcome back to the probability lectures here on www.educator.com, my name is Will Murray.*0000

*Today, we are going to talk about variance and standard deviation, two very closely related concepts.*0006

*Let us check those out.*0011

*Let us jump right in with the definition of variance.*0013

*The variance, you want to start with the random variable Y.*0016

*The variance of that random variable is by definition, it is the expected value of the quantity Y - μ².*0019

*Where μ here is the mean of the random variable, also known as the expected value of the random variable.*0028

*That is the definition of variance.*0036

*We use 2 different notations for that.*0038

*Sometimes, we use V of Y for variance, and some×, we use σ² of Y.*0041

*Or let us just say σ², we are not even including the Y.*0046

*Those mean exactly the same thing.*0050

*If you see σ² and if you see V of Y, those are exactly the same.*0052

*It is just 2 different notations for the same concepts.*0058

*That is the first thing you want to get straight.*0061

*Σ² and V of Y are the same thing.*0063

*What that is really measuring is you are measuring how far the variable Y deviates from its own mean.*0067

*Remember, μ is the mean of the variable.*0076

*It is a measure of how much the variable wobbles around and how far it deviates from its own mean.*0078

*Another way to put that is to say it is a matter of the volatility of Y.*0087

*How volatile is the variable?*0093

*Let me give you some mini examples here. *0095

*Let us suppose that that is the mean of the variable.*0099

*A variable that does not deviate that much would have a small variance.*0103

*Another variable that could have the exact same mean, it could get to that mean by doing much more serious wobbles.*0110

*It can wobble much farther from the mean, that variable would have a much higher variance.*0121

*This would have the same mean on both of these variables but this would have a very small variance.*0128

*And then this variable would have a very high variance because it is wobbling around so much,*0139

*it is a very volatile random variable.*0145

*This is high variance here.*0148

*That is kind of the intuitive idea of what variance means.*0152

*It intuitively means how much is that variable wobbling around? How much is your dataset varying?*0159

*The most useful way to calculate the variance is using this formula.*0167

*The variance, you get it by calculating the expected value of the function Y² *0173

*and then it looks like I'm talking about the same thing but I'm not.*0179

*You subtract off the expected value of (Y²).*0183

*You might think that these 2 things are the same but they really are not.*0188

*Remember, the expected value of Y is also known as μ so this is μ².*0193

*The expected value of Y² here, in general, it is not equal to the expected value of (Y²).*0197

*That is not something that you can say that those 2 are the same.*0207

*You have to be very careful there.*0211

*The expected value of Y² is something you want to calculate separately.*0213

*We will see in the examples how you calculate that separately.*0217

*I got one more definition for you, before we jump into the example which is standard deviation.*0223

*The way you get to standard deviation of a random variable is you start with the variance and then you just take its square root.*0229

*That is the definition of standard deviation.*0237

*It is by definition, the square root of the variance.*0239

*That is what this := means.*0245

*This is the definition here, we are defining it.*0247

*Sometimes, we use σ to represent the standard deviation of a random variable.*0251

*The first thing you notice about standard deviation is that it is defined directly from variance.*0256

*If you tell me the variance, I know immediately the standard deviation.*0264

*If you tell me the standard deviation, I know immediately the variance.*0268

*You can just go back and forth from one to the other.*0273

*Essentially, they measure the same thing.*0276

*If one of them is big then the other one is going to be big and vice versa.*0279

*The standard deviation is also a measure of how much the variable wobbles around, how volatile the variable is.*0284

*You can use variance to keep track of that, or standard deviation to keep track of that.*0292

*Essentially, those are 2 different ways of measuring the same idea, standard deviation and variance.*0297

*Essentially, the reason we have 2 different notions for the same idea is that in computations,*0305

*It is sometimes more useful to use one of the other but they are not really measuring 2 distinct concepts there,*0311

*as mean and variance really are measuring 2 distinct concepts.*0318

*Let us see how we can calculate some variances and also I want to kind of test whether you have a good intuition for these.*0324

*We are going to start out with that in the examples.*0332

*First example, I'm giving you 3 datasets and I'm saying which one has the largest variance.*0335

*You can choose A which is a bunch of a 123, B is 11, 11, 13, 15, 15, C is 20, 21, 22, 23, and 24.*0341

*You might want to think about that for a little bit to see if you have a guess as to which one*0353

*has the largest variance, before I jump in and answer it for you.*0357

*Let me try to answer that.*0361

*I'm not going to calculate the variance on each one, I’m just going to graph them and I hope that when I graph them,*0363

*if you have a good intuition for what variance means then on you will understand quickly*0369

*which of these sets of data has the largest variance.*0377

*In this first one, we got 5 copies of 123, 5 copies of the same point there.*0380

*The mean of that set of data would just be a line going through all of them.*0389

*Let me make those dots a little bigger there so you can still see them even after I draw the mean.*0396

*The second one, the second set of data here, we got 11 those are much smaller numbers 11, 11, 13, and then 15 and 15,*0402

*let me draw the mean for that.*0421

*The mean runs right through the middle there.*0423

*There is the mean of that set of data.*0426

*I’m drawing my points a little bigger here.*0428

*For the third one, the 20, 21, 22, 23, and 24 here, I will graph that set of data.*0431

*We have slowly increasing set a data there.*0439

*I will draw the mean of that set of data, that goes right through the middle there, there is the mean right there.*0447

*The question is which of the sets of data has the highest variance?*0455

*If you look at these, I hope that it is clear from the graphs because if you look at the first set of data,*0459

*there is no variance at all because all those points are exactly on the mean line.*0466

*In the second set of data, there is a quite a high variance because most of the points are quite far from the mean lines.*0474

*In the third set of data, there is a lower variance because some of the points are still quite close to the mean line.*0488

*The highest variance that we see there is in the middle set of data because the points there are farther from the mean line.*0500

*I did this intuitively, I did not actually calculate the variance on any of these 3 sets of data.*0508

*But I can tell just by looking at them that the data points are farthest from the mean for set B there.*0515

*That is clear to me that that is going to have the highest variance, *0527

*just by knowing kind of intuitively what variance and tails, and I do not even have to calculate the variance on those.*0531

*Let us go ahead and look at the next example.*0541

*It is another multiple choice question.*0543

*Which of the following would be the least useful in understanding a set of data, knowing the mean and the standard deviation,*0545

*knowing the mean and variance, or knowing the standard deviation and the variance?*0552

*The key point here is to remember that the standard deviation is always equal to the square root of the variance which means.*0557

*If you tell me the variance, I can figure out the standard deviation easily and quickly.*0577

*If you tell me the standard deviation and the variance, you have told me some redundant information.*0582

*You have not told me anything new.*0588

*As soon as you tell me the standard deviation, I figure out the variance on my own.*0590

*Or as soon as you the variance, I figure out the standard deviation on my own.*0594

*This C here, this is redundant information.*0599

*On the other hand, if you tell me the mean and the standard deviation,*0611

*those are independent pieces of information which means you have really told me two useful pieces of information there.*0616

*You tell me the mean, even after you tell me mean I did not know the standard deviation.*0631

*Even you told me the standard deviation, I did not know the mean.*0635

*These are two useful pieces of information, you told me 2 different things about a dataset.*0639

*The same thing for part B, if you tell me the mean and variance, you have told me 2 pieces of independent information.*0644

*Those are both useful independently of each other.*0654

*The mean and variance are both useful to know independently of each other.*0663

*Even if you tell me one of them, I still do not know what the other one is.*0667

*When you tell me the other one, I get something useful.*0669

*The standard deviation and the variance, if you tell me one I can calculate the other.*0672

*That is something redundant.*0678

*The question asks which is the least useful in understanding, that would definitely be C then.*0680

*Because essentially, if you tell me the information in part C, I really only know one thing about my data set.*0685

*If you tell me any information in part A the mean and standard deviation, I can calculate the variance so I know everything there.*0691

*If you tell me in part B the mean and variance then I can calculate the standard deviation on my own so I get something a lot more useful there.*0699

*Let us keep going.*0709

*In example 3, we are going to look at your probability class again.*0710

*This is actually very similar to an example we had in the previous lecture*0715

*where we are calculating your semester average for your probability class.*0719

*You might want to check that out if you have not just looked through those lectures.*0723

*In this example, we are going to take 2 midterm exams.*0729

*They are going to count 25% each of the semester grades.*0733

*The final exam counts for 30% and the homework counts for 20%.*0737

*It turns out that the score 79 of both midterms and on the final, and on the homework.*0741

*We are very consistent here, scoring 79 on everything.*0748

*We want to calculate the mean, the variance, and the standard deviation of our scores using the weights above.*0752

*You might be able to guess a lot of the answers to this question ahead of time.*0759

*In the next question, we are going to mix it up using different numbers.*0764

*In the next question is an extension of this one.*0768

*That is why I want to work through this one first and you will see how all the numbers come in.*0771

*Let me go ahead and start by calculating the mean there.*0778

*The mean is the expected value of Y and we did calculate one very similar to this.*0781

*I have different numbers but it was the exact same principle in the previous lectures.*0789

*You might want to check back in the example in the previous lecture where we calculated one like this.*0793

*The expected value of Y, just by definition, I gave you the definition in the previous lecture.*0800

*You add up all the possible values Y can be and then multiply each one by the probability.*0805

*In this case, you are just scoring a 79 on everything.*0812

*This Y is 79 for everything, P of Y × 79.*0816

*There is only one value which is the 79 because that is how you scored on everything.*0824

*That 79 × the total probability, the total probability of any experiment is 1, that is just 79 is your mean.*0829

*That was fairly easy to calculate, that is not all surprising.*0840

*If you score 79 on everything then your average should be 79.*0843

*The σ² is the variance, what was the mean if we calculated the first.*0847

*The variance is, the σ² remember is the expected value of Y - μ².*0853

*I try to find the expected value of the function of our random variable.*0864

*Let me remind you for the formula for that.*0868

*The expected value of a function of a random variable, by definition is you add up all the possible values that random variable can take.*0872

*The probability of each one × not the value itself but G of that value.*0881

*In this case, you look at all the possible values P of Y.*0887

*In this case, G of that variable is Y - μ².*0894

*It is (Y - 79)², the 79 is what I figure out up above.*0900

*All the possible values, all the values in this case were 79.*0911

*Y -79², the only values we have here are 79 so 79 -79², of course is 0.*0918

*The variance here is 0.*0929

*That should be intuitively clear because you are completely consistent the whole semester, you scored 79 on everything.*0931

*You know it did not vary at all from your mean of 79.*0939

*Finally, let me calculate the standard deviation.*0943

*That is very easy once you figure out the variance.*0950

*It is just the square root of the variance, V of Y.*0953

*Our V of Y that is the same as σ², that is √0 which of course is still 0.*0959

*Your standard deviation is also 0.*0967

*Again, that is not a surprise if you scored 79 on everything all semester *0969

*then you have not deviated at all from your mean which means your standard deviation is 0.*0975

*I will emphasize it in the next example, we are going to mix up the numbers so *0980

*we will have some more tricky numbers to calculate that would not be so obvious, I hope.*0986

*Let me remind you where all of these numbers came from.*0992

*To calculate the mean, we calculated the weights on each possible score.*0995

*What every possible score here was 79, it was just all the weight went on 79, *1001

*total probability of 1 on 79, that was your mean of 79.*1007

*Calculate the σ², by definition, σ² is the expected value of Y - μ².*1012

*I’m using this formula that we learned in the previous lecture, in the previous video,*1020

*which was the expected value of a function of a random variable is just the sum *1025

*of the probabilities × the function of each value.*1031

*That was what I plugged in here, Y - μ² here, that is the G of y.*1034

*In this case, the μ was 79 so I just plug in 79 here.*1044

*Since all the Y’s are 79, we are just adding up 79 – 79², of course that comes out to be 0.*1050

*The standard deviation is just the square root of the variance, that is √0 which just comes up to be 0.*1059

*In example 4, we are going to revisit this example except, I’m going to mix up the numbers on you,*1066

*so you should get a more interesting answer and less predictable answer.*1072

*In example 4, again, we are in this probability class, this is very similar to the previous example.*1076

*Same weight, we got two midterm exams with a 25% each and a final exam counts for 30%,*1081

*and the homework is going to count for 20%.*1090

*This time, instead of scoring 79 on everything, you are going to score 60 on the first midterm, *1092

*80 on the second, 80 on the final, and 100 on the homework.*1098

*I want to calculate the mean, variance, and the standard deviation of all of the scores using these weights.*1103

*This one should come out a little more interesting.*1109

*In fact, these are the same numbers that I used in an example on the previous lecture,*1112

*when you are first learning to calculate expected value.*1117

*You might want to go back and check on that example from previous lecture *1120

*because I may be doing the same calculation to get the expected value here.*1124

*Μ is the unexpected value of Y.*1128

*By definition, the way you calculate expected value is you look at all the Y values and*1137

*the weighting or the probability on each one × that value.*1143

*Here, our Y values are 60 because we scored 60 on something, *1147

*we scored 80 on a couple different things, and we scored 100 on something.*1152

*Let us figure out the probabilities or the weights on each one.*1159

*We scored 60 on the first midterm and that was 25%, 25/100.*1162

*We scored 80 on the second midterms, 25/100, and also on the final.*1171

*Let me put a 30/100, fill that in there as well.*1177

*Both of those get multiplied by 80 because there is a two different test on which we scored 80.*1182

*We scored 100% on the homework.*1189

*The homework was 20% of the grade, 20/100 × 100.*1192

*If you just run the numbers through there, I’m not going to do labor that on my own calculator because I did this ahead of time.*1200

*You get 79 as your average for the semester, that actually comes out to be the same average that we got in the previous examples.*1206

*The μ is the same as in the previous example but the standard deviation will definitely not be the same, and the variance will not be the same.*1216

*Let us calculate those.*1225

*I’m going to calculate the variance using two different formulas.*1227

*I’m going to use the original definition of variance then I’m going to use a clever formula that we found out later on.*1230

*Let me calculate the variance.*1238

*The first way I’m going to use is the original definition of variance which is the expected value of Y - μ².*1241

*Now, the expected value of a function of a random variable, that means you calculate it just like the expected value of Y ×*1252

*the probability of each value except that instead of calculating Y itself, you calculate that function of Y.*1261

*Y - μ², μ was 79, I will go ahead and fill that in.*1269

*That is coming from up above, μ is 79.*1276

*I have to calculate all my possible values -79².*1282

*I'm going to calculate the same probabilities as above 25/100 except instead of 60, I will have 60 -79².*1291

*Just like above, I have 25/100 + 30/100, that simplifies to 55/100 × 80 -79² + 20/100 × 100 -79².*1307

*If I simplify that down, I get 25/100.*1328

*60 -79 is -19, since I’m squaring, I will make it a positive, 19².*1333

*Let me make that bigger there so you can read it.*1341

*+ 55/100, 80 -79 is just 1 × 1² + 20/100.*1346

*100 -79 is 21².*1357

*If you work this out, I did this on my calculator.*1364

*I get 79/100 and that simplifies down to 179, that is the variance.*1367

*That was using the original definition of variance but we have another way to calculate variance.*1382

*Let me remind you what that was.*1387

*The original definition was the expected value of (Y - μ)², that was our original definition.*1390

*But we also have a clever formula that I want to remind you of.*1401

*It was the expected value of Y² - the expected value of Y, the whole thing².*1408

*Let me calculate that one because that was fun to workout here.*1418

*Of course, we should get the same answer if I do the arithmetic right.*1422

*Let me calculate V of Y using that new formula.*1427

*It is E of Y² - the expected value of Y.*1431

*Of course, we already figured that out,².*1438

*E of Y² is the sum of Y × P of Y × Y².*1441

*It is all those same probabilities and I'm going to fill in the values of Y² everywhere, 25/100.*1449

*My first value of Y was 60, I will put 60² + 55/100.*1458

*That was corresponding to the value of Y being 80, so 80².*1466

*20/100 × 100².*1473

*Now, I still have to subtract off E of Y².*1479

*This E of Y, that is the same as the μ, that is the mean that we calculated earlier.*1482

*Let me subtract off 79² because that was the mean that we calculated earlier.*1487

*If you on run these numbers, I did run them on my calculator, we got 6420 -79², that simplifies down to 179.*1495

*It is the same thing we got above but it is a slightly different way of calculating.*1514

*I think it might be a little bit easier but you can make your own decision there, and it does check.*1517

*We have the mean right there, we have the variance right there.*1525

*Finally, we have to find the standard deviation.*1534

*That is a very quick because once you know the variance, the standard deviation is just the square root of the variance.*1541

*Σ is just by definition, the square root of the variance.*1549

*In this case, we already calculated the variance that is 179.*1559

*179, the² root of that is let us se, 13² is 169, 14² is 196.*1563

*It is a little closer to 13.*1571

*It is approximately equal to 13, there is definitely some decimals there that I'm not bothering to work out there.*1574

*If you want to be really accurate, say the standard deviation is the √ 179 and that will certainly be accurate.*1582

*If you want to get a quick estimation, I estimate that at about 13.*1590

*Let me remind you how we calculate each one of these.*1595

*The mean was actually something we calculated before in the previous lecture.*1599

*I will just do that quickly, looked all the values of Y, took a probability of each one, and then add them all up.*1605

*The tricky one was this value of Y being 80 because there were 2 different things we scored 80 on.*1613

*80 on the second midterm and 80 on the final.*1620

*That is why I took 25% for the second midterm and 30% for the final, that is where that came from.*1623

*If you run your arithmetic on that, you get a mean of 79.*1631

*The variance, there is 2 different ways you can calculate that, using the original definition here or using this formula.*1634

*I calculated it both ways here because I want to check and make sure that we got the same thing both ways.*1642

*It is also a good practice to see both methods.*1648

*If you want to calculate the expected value of Y - μ², it is just like the expected value of Y except that you replace Y with Y -79².*1652

*Same probabilities here, that 25, 55, and 20.*1664

*Instead of 60, 80, 100, we do each one of those numbers (-79)².*1669

*And then, if you work the arithmetic out there, it comes out to be 179 for the variance.*1678

*That is one way of calculating the variance.*1683

*The other way of calculating the variance is using this formula.*1685

*The expected value of (Y²) - the expected value of (Y²).*1689

*It seems like those things are the same but they are not, do not be confused by that.*1697

*The expected value of Y² is that is the same as μ.*1703

*That is the 79² that I filled in right there.*1707

*The expected value of what (Y²) is that same formula with all the probabilities 25, 55, and 20.*1712

*All the values of Y except we are squaring each one, 60², 80², 100².*1720

*I ran that through my calculator, simplify down to 179.*1727

*The standard deviation is very easy to calculate once you know the variance because by definition,*1735

*it is just the square root of the variance.*1741

*We just take √179 and I did not run that through my calculators, I just estimated that to be about 13.*1743

*But of course, that is not 100% accurate.*1750

*The accurate value would be √179.*1752

*The interesting thing about this example is that we have a very same mean on this example,*1758

*as we had on the previous example, example 3.*1763

*Example 3 was also calculating your scores over the semester.*1767

*But in example 3, you got that mean of 79 by getting a 79 on everything all semester.*1771

*You had a mean of 79 and no variance at all.*1778

*Here, we still ended up with a mean of 79 by much more variable set of scores *1782

*which is why we had a much more interesting variance now, we have a positive variance.*1788

*Last time, we just had a variance of 0.*1793

*In example 5, we are going to roll one dice and Y is going to be the number showing on that dice.*1798

*We are going to calculate the mean, and variance, and the standard deviation of Y.*1805

*Let us jump right into that.*1811

*The mean, μ, also known as E of Y, those things mean the same thing.*1814

*Remember that means you calculate all the possible values of Y that you might see in the probability of each one.*1825

*When you roll a dice, you can see a 1, 2, 3, 4, 5, 6.*1832

*The probabilities of each one, not ½, that is for sure, it is 1/6 + 1/6 × 2 + 1/6 × 3 + 1/6 × 4 + 1/6 × 5 + 1/6 × 6.*1841

*The easy way to do that is to factor out 1/6 and then add up 1 + 2 + 3, up to 6.*1859

*1 + 2 + 3 up to 6 is 21, 21/6, and that simplifies down to 7/2.*1870

*That is the mean, that is the expected value if you roll one dice.*1878

*You are not going to see a 7/2 because no side on a dice has 7/2.*1883

*But, that means if you roll the dice many times then the average of all the answers that you see will be 7/2.*1889

*To find the σ², I’m going to use I think the new formula σ² divided by the variance.*1899

*We have a definition for variance but I’m not going to use the definition, we will use this new formula that we have.*1908

*The expected value of Y² - (E of Y)².*1914

*I think that is going to be a little more easy to calculate.*1920

*Let us calculate first the expected value of Y² which means we do *1923

*the same calculation as above except that to we are going to look at Y², instead of Y.*1929

*We actually did this one as an example on the previous lecture.*1937

*I think it was the last example on the previous lecture.*1941

*You can go back and look that up, if you want.*1943

*I will do the same thing as I did above here, I will list out my probabilities 1/6, 1/6, 1/6, 1/6, 1/6, and 1/6.*1945

*I list out my numbers 1, 2, 3, 4, 5, 6, except I’m supposed to square each one.*1960

*Let me square each one of those.*1968

*Again, it is useful to factor out the 1/6 on all of these so we get 1/6 × 1 + 2² is 4, 3² is 9, 4² is 16, 25, 36.*1975

*If you add up those numbers, we actually did this on the previous lecture which I remember doing.*1990

*I remember that I got 91/6, that is not the variance yet, that is the expected value of Y².*1997

*In order find the variance, we have to subtract something off.*2006

*Σ² or V of Y, this is calculating the variance now, is E of Y² as a quantity, - E of Y by itself².*2011

*We get 91/6 and then we calculated E of Y up above, there it is, it is7/2 except we have to square that.*2032

*That is 91/6 - 7/2 is 49², not 47 for sure, 49/4 is 7/2².*2043

*It looks like a common denominator there will be 12.*2059

*I have to multiply 91 by 2, I got 182.*2064

*After multiply 49 by 3 which gives me 147.*2068

*182 -147 is 35 and I divide that by 12, that is my variance.*2073

*That is my variance if I roll one dice.*2085

*Let us figure out the standard deviation.*2089

*It is easy to find the standard deviation after you find the variance.*2096

*Find the variance first and then you can find the standard deviation very easily,*2100

*just by doing the square root of the variance, that is the definition of standard deviation.*2104

*We just get the, not very enlightening here but √ 35/12 or the standard deviation.*2110

*Let me recap how we calculated each one of those.*2124

*For the mean, we just use the standard definition for expected value.*2128

*That means you list out all the possible values you might see and the probabilities of each one, multiply and add.*2133

*Our possible values when you roll a dice are 1, 2, 3, 4, 5, 6.*2140

*1/6 probability of each one, that is where these 1/6 come from.*2144

*If you simplify the numbers there, you get 7/2.*2148

*For the variance, I was going to use this formula.*2153

*It is not the original definition of the variance but a very useful formula.*2157

*We find the expected value of Y² - the expected value of Y, all².*2161

*As a preliminary step here, I calculated the expected value of Y² which looks a lot like the expected value of Y.*2169

*You list out all those probabilities and all the values of Y except you use Y² for each one.*2177

*I² each one of those numbers, that is the only difference from the calculation above *2184

*is we are squaring each one of those numbers, as we go through.*2189

*We get a different answer, 91/6, that is E of Y².*2192

*That is easy now to drop into our formula for variance which is E of Y² - the expected value².*2200

*That is our formula for variance which we can write is V of Y or σ².*2211

*Those would mean exactly the same thing.*2214

*91/6 comes from above, the 7/2 comes from up here, that was our mean that we already found.*2217

*Then it is just a matter of squaring 7/2 and reducing the fractions down to 35/12, *2225

*that is our variance for that experiment.*2231

*The standard deviation is easier because once you know the variance, the standard deviation,*2234

*by definition is the square root of variance.*2239

*We get that with √ 35/2.*2244

*We got one more example here.*2250

*When a flip a coin 3 × and we are going to let Y be the number of heads, *2251

*and we want to calculate the mean, the variance, and the standard deviation of Y.*2257

*What I'm going to do here is list all the possible outcomes of this experiment.*2263

*There are better ways to do this kind of example, after we learned about some probability distributions later on that.*2268

*That is a couple of lectures down below.*2277

*Unfortunately, at this point we are reduced to listing all the outcomes.*2279

*After learn about the binomial probability distribution, we will have a better way of calculating these.*2283

*But in the meantime, let us just look at all the possible outcomes that can happen with this experiment.*2287

*I’m going to list the outcomes, I’m going to list Y for each one.*2296

*Because we are going to need that later when we calculate the variance, I'm also doing it as Y², that would be useful.*2300

*When you flip a coin 3 ×, you can get a head-head-head, head-head-tail, head-tail-head, head-tail-tail.*2307

*I’m trying to list these in some kind of a binary counting order.*2319

*I get tail-head-head, tail-head-tail, tail-tail-head, or tail-tail-tail.*2323

*I’m going to list the values of Y for each one of those.*2334

*Y was the number of heads we see each time.*2338

*There is 3 heads there,2 heads, 2 heads, 1 head, 2 heads, 1 head, 1 head, and 0 heads.*2341

*I’m going to need Y² later on.*2350

*Let me fill in the Y², 9, 4, 4, 1, 4, 1, 1, 0.*2353

*I'm in good shape to calculate my expected value, my variance, and standard deviation.*2363

*The expected value of Y by definition is the same as the mean.*2370

*If they ask you the mean, the expected value, that is the exact same thing.*2377

*You can calculate one for the other, they mean the same thing.*2383

*It is all the probabilities.*2386

*The probabilities are something I'm going to have to figure out but the probabilities of each one of these outcomes, the probabilities are all the same.*2388

*The probability is 1/8 for each of these.*2406

*Let me just factor that 1/8 and then I will add up all of the Y that I see 3 + 2 + 2 + 1 + 2 + 1 + 1 + 0.*2410

*And I hope I have not made a mistake there.*2424

*3 + 2 + 2 is 5 + 1 is 6 + 2 is 8.*2426

*I think I have left something out there, let me go back and check there.*2434

*3 + 2 + 2 is not 5, 3 + 2 + 2 is 7 + 1 is 8 + 2 is 10 + 1 is 11 + another 1 is 12 + 0 is 12.*2439

*12/8 and of course that simplifies down to 3/2.*2450

*That is the mean or the expected value of Y, in this experiment.*2458

*That is not surprising, if you flip a coin 3 ×, I'm not saying you expect to see exactly 3/2 heads.*2463

*I’m saying that the average over many iterations of this experiment will be 3/2 heads.*2470

*On average, I will see 3/2 heads per time that you run this experiment.*2477

*Let us find the variance now.*2483

*In order to find the variance, first I’m going to find the expected value of Y².*2487

*The reason I'm doing that is because we have this nice formula for the variance*2492

*which says that σ² which is another way of saying the variance.*2499

*V of Y is also a way of saying the variance, it is the expected value of (Y²) - the expected value of Y by itself and then².*2506

*That is a good way to calculate the variance quickly.*2518

*But in order to do that, you have to know what E of Y² is.*2522

*That is the same as the same probabilities but we are looking at Y² instead.*2526

*That is 1/8 × all the values of Y², 9 + 4 + 4 + 1 + 4 + 1 + 1 + 0.*2534

*Let me see if I can add those up better than I did for the mean.*2548

*9 + 4 + 4 is 17 + 1 is 18 + 4 is 22 + 1 is 23 + 1 is 24, and 0 does not change it.*2552

*24/8 gives me exactly 3 there.*2564

*I did not box that because that is not the variance yet, that is just E of Y².*2571

*We have not found the variance yet.*2577

*In order to find the variance, we are going to drop in this other formula down below.*2579

*E of Y² is 3 – E of Y by itself was 3/2.*2583

*We calculated that above, I² that.*2589

*I get 3 - 9/4, 3 is 12/4 so 3 -9/4 is ¾.*2593

*That is the variance is ¾.*2604

*Finally, we have to find the standard deviation but that is easy once you know the variance *2610

*because the standard deviation is just the square root of the variance.*2616

*By definition, it is the square root of the variance.*2621

*That is the V of Y right there and that is the square root of what we just figured out, ¾.*2624

*I can simplify that a little bit to be √3/2.*2636

*I hope you watch a few more lectures and you learn about the binomial distribution.*2644

*We will calculate the mean and variance and standard deviation of the binomial distribution.*2650

*You might want to come back and check this problem because that is one of the very quick ways to get to these answers.*2656

*But we have not learned that yet, in the meantime, we are just working things out from scratch.*2661

*Let me remind you what we did.*2665

*We listed all the possible things that can happen, all the possible outcomes that can occur when you flip a coin 3 ×, *2667

*which is just all strings of heads and tails, strings of 3 heads and tails.*2675

*The probability of each one of those is 1/8 because you got to flip a coin 3 ×, *2680

*you got a 50% chance of getting the right flip each time.*2685

*There are 8 of them in total because there are 2 outcomes for each individual flip.*2689

*2 × 2 × 2 is where that 8 come from.*2695

*We want to calculate, Y was the number of head.*2698

*I listed the number of heads that we get from each one of those outcomes here.*2702

*Because we are going to use it for calculating the variance, I also listed Y².*2707

*That is what these numbers are here.*2713

*To find the mean, you add up the probabilities of each one × the value of Y.*2715

*Since, the probabilities were all equal, 1/8 everywhere, I just added up all these values of Y and I multiply by 1/8.*2721

*We get 12/8 is 3/2 is the mean.*2731

*Not surprising, if you flip a coin 3 ×, on average, you expect to get 3/2 heads.*2734

*Of course, you get a whole number of head that will be exactly 3/2.*2740

*In the long run, you will average 3/2 heads per experiment.*2744

*E of Y² is a preliminary step towards finding the variance.*2749

*It is because the variance formula is this E of Y² - (E of Y)².*2755

*We calculate E of Y² first, that is the same, you list the same probabilities but instead of looking at Y, you look at Y².*2763

*Instead of 3, we have 9.*2771

*Instead of 2, we have 4.*2773

*Those add up to 24/8 which simplifies down to 3.*2776

*I will not box on that because that is not our final answer.*2779

*To calculate the variance, σ² is one notation for the variance.*2782

*E of Y is another notation, we are calculating the same thing either way there.*2787

*We have this nice formula to calculate it.*2792

*V of Y² was the 3 and E of Y is the 3/2 and then if you just simplify that down, you get ¾.*2794

*Finally the standard deviation is easy, you just take the² root of the variance, assuming we already calculated that.*2805

*It is √ ¾ reduces down to √3/2.*2810

*As I mentioned, we have an easier way to calculate all these, *2815

*after we learn about the binomial distribution and all its important properties.*2818

*But that is a couple of lectures down the road.*2823

*In the meantime, we are just calculating things from scratch.*2825

*I hope that clears things up here, this is the end of our lecture on standard deviation and variance.*2829

*This is part of the probability lecture series here on www.educator.com and my name is Will Μurray, thank you for watching, bye.*2836

1 answer

Last reply by: Dr. William Murray

Mon Sep 15, 2014 6:18 PM

Post by Christian Faya on September 13, 2014

In example V, when calculating the mean, I'm confused as to why we multiply the p(y) times the number showing on the die. To my understanding, each face on the die has the same probability, so why multiply the value of each facet of the die? Maybe I'm not understating this particular question. I had no trouble understanding the other problems, but this threw me off. By the way the lectures have been of great help!