For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Tchebysheff's Inequality

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
- Tchebysheff's Inequality (Also Known as Chebyshev's Inequality)
- Tchebysheff's Inequality: Definition
- Tchebysheff's Inequality: Equation
- Tchebysheff's Inequality: Intuition
- Tchebysheff's Inequality in Reverse
- Example I: Money
- Example II: College Units
- Example III: Using Tchebysheff's Inequality to Estimate Proportion
- Example IV: Probability of an Earthquake
- Example V: Using Tchebysheff's Inequality to Estimate Proportion

- Intro 0:00
- Tchebysheff's Inequality (Also Known as Chebyshev's Inequality) 0:52
- Tchebysheff's Inequality: Definition
- Tchebysheff's Inequality: Equation
- Tchebysheff's Inequality: Intuition
- Tchebysheff's Inequality in Reverse 4:09
- Tchebysheff's Inequality in Reverse
- Intuition
- Example I: Money 5:55
- Example II: College Units 13:20
- Example III: Using Tchebysheff's Inequality to Estimate Proportion 16:40
- Example IV: Probability of an Earthquake 25:21
- Example V: Using Tchebysheff's Inequality to Estimate Proportion 32:57

### Introduction to Probability Online Course

### Transcription: Tchebysheff's Inequality

*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 Chebyshev’s inequality.*0005

*This is the second lecture on using inequalities to estimate probability.*0009

*The first one we had was Markov’s inequality.*0014

*If you are looking for Markov’s inequality, there is another video covering that one.*0017

*It is the one video before this one.*0021

*This one is on Chebyshev’s inequality.*0023

*You get some similar kinds of answers which Chebyshev’s inequality.*0026

*The difference is that we use a little more information because now,*0031

*we are going to use the standard deviation of random variable, as well as the expected value or mean.*0035

*In return for using a little more information and doing a little more calculation,*0044

*we get stronger results using Chebyshev’s inequality.*0048

*Let us dive into that.*0052

*Chebyshev’s inequality is a quick way of estimating probabilities.*0055

*It never tells you the exact probability, by the way.*0058

*That gives you an upper and lower bound for the probability but it never tells you it is exactly equal to something.*0061

*It is based on the mean and standard deviation of our random variable.*0068

*We are going to use the Greek letter μ for the mean and we are going to use the greek letter σ, for the standard deviation.*0073

*Let me go ahead and tell you what Chebyshev’s inequality says.*0079

*Suppose y is a random variable and K is a constant.*0084

*K is often a whole number like 2, 3, or 4.*0088

*Chebyshev’s inequality says the probability that the absolute value of 1 - μ is greater than K σ,*0093

*is less than or equal to 1/K².*0101

*That is quite a mouthful, starting with the name Chebyshev’s himself is a little tough to deal with.*0104

*But inequality is a little bit complicated.*0111

*I want to think about the intuition of that first.*0113

*What we are really saying there, first of all, σ is a standard deviation.*0117

*K is a measure of how many standard deviations you are willing to go away you from the mean.*0122

*That is what they are really measuring right here.*0130

*The absolute value of 1 – μ, μ is the mean.*0132

*The absolute value of 1 - μ is the distance, that is the value of A - b is the distance between A and B.*0137

*This is the distance from Y to μ.*0145

*What this is really saying is, how far are you willing to go from your expected value, from your mean, μ?*0156

*The question is, what is the probability of you deviating more than K standard deviations from your mean?*0165

*Maybe, I can try to graph that.*0175

*If we have a certain amount of data here in and it is grouped like that, that is kind of a common distribution of data.*0178

*You have a mean right there in the middle.*0184

*What is the chance the probability of being more than K standard deviations from your mean?*0187

*In other words, how much area could there be out there in the tail of the distribution?*0195

*What Chebyshev’s inequality says, is that there is not that much.*0204

*It is unlikely that the variable will be far from its mean, 1/K² is your balance.*0208

*That is usually a fairly small number, for example if K is 3 then 1/K² would be 1/9.*0216

*The probability would be less than 1/9 that you would be 3 standard deviations from your mean.*0224

*That is what Chebyshev’s inequality is saying.*0230

*It is that, your probability of being many standard deviations from your mean is quite low.*0233

*The more standard deviations you go, the smaller the probability gets.*0240

*There is also a reverse version of Chebyshev’s inequality.*0245

*You just turn around the inequalities there.*0248

*This is the same inequality that I just showed you.*0253

*If you turn that around, if you change this greater than to a less than or equal to, that is the change here.*0256

*We are asking the opposite question, what is the chance that you were close to your mean?*0264

*What is the chance that you will win K standard deviations from your mean?*0270

*Since, the probability of being far away from your mean is very well,*0275

*that means the probability of being close to your mean is very high.*0279

*The answer we get, the probability of being close to your mean is greater than or equal to 1 -1/K², its exact opposite.*0284

*It was 1/K² before, now it is 1 - 1/K² and where we had less than or equal to,*0294

*before, meaning the probability of being very far away from mean is quite low,*0305

*the probability of being very close the mean is very high.*0310

*What we are saying here is that, it is unlikely that the variable will be close to the mean.*0314

*We say close, we mean within a few standard deviations.*0322

*Remember, σ is the standard deviation and K is the number of standard deviations away from your mean that you are willing to accept.*0326

*If your chance of being with many standard deviations of your mean is quite high,*0339

*that is what Chebyshev’s inequality is saying, if you turn it around.*0348

*Let us check that out in the context of some examples and see how it plays out.*0352

*This first example is quite similar to an example we had back when we are studying Markov's inequality.*0357

*You will recognize the numbers.*0365

*The difference is that we are now going to incorporate the standard deviation, before we just incorporated the mean or the expected value.*0367

*Now, we incorporate the standard deviation and we are going to use Chebyshev’s inequality,*0375

*instead of Markov’s inequality, and we get much stronger results this time.*0379

*In this case, we done a survey of students on a campus and we discovered that on average,*0384

*they are carrying about $20.00 in cash with them.*0391

*We also know that their standard deviation is $10.00.*0393

*If we need a student at random, we want to estimate the chance that that student is carrying more than $100.*0398

*We also want to estimate the chance that she is carrying less than $80.00.*0405

*Let me start out by writing down Chebyshev’s inequality, just reminding you what the formula was.*0410

*The probability that Y - μ is greater than K σ, it is called the Chebyshev’s inequality, that probability is less than 1/K².*0417

*Let us fill in what we can use here.*0429

*We know that the μ, that is the average value of the variable or the mean or the expected value, which we are given here is 20.*0432

*And we are also given the standard deviation is 10, that is the σ there.*0442

*We want to find the chance that a student is carrying more than $100.*0449

*Let us figure out how many standard deviations away from the mean, we would have to be.*0456

*The mean is 20, if I want to have more than $100 then that $80 more than the mean which is 8 standard deviations away from mean.*0460

*We would have to be 8 standard deviations away from the mean.*0474

*The probability that Y -20 is greater than 8 × 10.*0478

*As you have to be $80.00 away from the mean.*0492

*According to this, that 8 is the K there.*0496

*It is also equal to 1/K² so 1/8² which is 1/64.*0499

*What that is telling us is that, if we meet a student on campus, we say what is the likelihood that that student has more than $100?*0509

*It is very unlikely, the probability is less than 1/64.*0518

*If you interview all your students, most likely every 64 students you interview,*0525

*at most 1 of them will actually be carrying more than $100, according to these numbers here.*0539

*There is a second part to this problem, we also want to estimate the chance that she is carrying less than $80.00.*0545

*That is going to use the reverse incarnation of Chebyshev’s inequality.*0553

*Let me go ahead and write that down.*0558

*The probability of Y - μ is less than or equal to K σ.*0561

*According to Chebyshev’s inequality is greater than or equal to 1 -1/K², that is the reverse version of Chebyshev’s inequality.*0567

*In this case, our μ is still 20, our σ is still 10.*0578

*In this case, we want 80 dollars.*0584

*We want 80 -20 is 60, we want to be $60.00 away from the mean.*0586

*60 is 6 standard deviation, 6 × 10.*0596

*We will use K = 6 there in our own Chebyshev’s inequality.*0601

*We get the probability that Y -20 is less than or equal to 6 × 10, is greater than or equal to,*0606

*that is what Chebyshev’s inequality tells us here, greater than or equal to 1 -1/ the K was 6, 6² there.*0620

*That is 1 - 1/36, and that simplifies down to the probability is greater than 35/36, because that is 1 -1/36*0629

*If we meet a student and we want to say, what is the chance that she is carrying less than $80.00 in cash?*0646

*The chance is at least 35/36.*0653

*Remember, just like the Markov’s inequality, you never want to just give an answer,*0657

*when you are being asked the question using Chebyshev’s inequality*0662

*because it never gives you a specific numerical answer, it always gives you inequality.*0665

*It will never tell you what the probability is, it will just tell you that the probability is less than this or greater than that.*0671

*In all of these cases, it is very important to include the inequality signs in our answers.*0678

*All we are doing is we are giving upper and lower bounds for the probability.*0684

*We are not saying that we know what the probability is.*0687

*Let me show you where I got each of those steps there.*0692

*We start out with the basic version of Chebyshev’s inequality.*0695

*The probability that Y – μ is greater than K σ, is less than 1/K².*0698

*In this case, our standard deviation is 10, that is the σ, the mean is 20.*0704

*We want to have more than $100, that means we really want to be 8 standard deviations away from the mean.*0712

*A hundred is $80.00 away from 20.*0719

*That is 8 standard deviations away from the mean.*0722

*That is where we got the value of K = 8 there.*0725

*Chebyshev’s tells us that the probability is less than 1/K², that is where we got 1/64 for our probability.*0729

*Actually, our probably is less than or equal to 1/64 there.*0738

*For the second part of the problem, we want to estimate the chance that she is carrying less than $80.00.*0744

*I use the less than version of Chebyshev’s inequality and it tells me*0749

*that the probability is greater than or equal to 1/1 – K².*0755

*You are very likely to be within many standard deviations of your mean.*0759

*How many standard deviations are we talking about?*0764

*The mean is 20 and we want to have less than $80.00.*0766

*80 -20 is 60, that is 6 standard deviations.*0770

*I put that value in for K and put that K in, we get 1 -1/6².*0774

*That simplifies down to 35/36, that tells me that the probability that a student will have less than $80.00 is at least 35/36.*0782

*It does not say equal to 35/36 but at least 35/36.*0795

*In example 2 here, we got students on a college campus have completed an average of 50 units and their standard deviation is 15 units.*0803

*We meet a randomly selected student, we want to estimate the chance that this person has completed more than 95 units.*0814

*Let me write down Chebyshev’s inequality for that.*0823

*The probability of Y - μ is greater than K σ is less than or equal to 1 /K², that is Chebyshev’s basic inequality.*0826

*We are going use the basic version because we are trying to estimate the chance that something is more than 95.*0841

*Let us try to figure out what the relevant numbers here are.*0850

*The average of 50 units, that is the μ, that is going to be 50.*0853

*The σ is the standard deviation, that is 15 units there.*0858

*We have to figure out what K is here.*0868

*We have to figure out how many standard deviations away from the mean are we expected to be here.*0870

*In this case, we want to have more than 95 units.*0876

*95 -50 is 45 which is 3 standard deviations, three × 15.*0880

*That means that our K value is going to be three here.*0892

*The probability that Y -50 is a greater than three × 15, is according to Chebyshev’s inequality less than 1/3² which is 1/9.*0896

*What we conclude here is that the probability is less than 1/9.*0916

*That means fewer than 1 in 9 students will have more than 95 units.*0921

*That is what we can conclude from that.*0929

*We cannot say it is equal to 1/9, we cannot just give 1/9 as an answer.*0932

*We have to say the probability is less than 1/9.*0936

*To recap how we derived that, we started with the basic version of Chebyshev’s inequality.*0941

*The probability that Y - μ is greater than K σ is less than 1/K².*0947

*And then I filled in what I knew here, the 50 is the average number of units, that is the μ there.*0952

*That is where that 50 comes from, 15 also came from the problem because that is the standard deviation.*0961

*I want to figure out what K should be.*0966

*In order to calculate that, I figured out how many standard deviations away from the mean are we interested in?*0968

*95 -50 is 45 and that is three standard deviations because the standard deviation is 15.*0976

*Our K there is three.*0984

*We pop that right there into Chebyshev’s inequality, we get 1/9.*0986

*1/9 is not our answer by itself, we have to say the probability is less than or equal to 1/9.*0991

*That is how we answer that.*0997

*In example three here, we got the scores on a national exam are symmetrically distributed.*1002

*I see you got a small typo there in the distributed, let me fix that.*1008

*Symmetrically distributed around a mean of 76 with the variance of 64.*1015

*The minimum passing score is 60.*1021

*We want to use Chebyshev’s inequality to estimate the proportion of students that will pass this exam.*1024

*Let me draw a little graph of what is going on here.*1031

*We are given that the scores are symmetrically distributed.*1034

*It will look something like this.*1040

*The mean is 76, let me fill that in.*1043

*The mean here is 76, that is our μ =76, right there in the middle of the data.*1048

*The minimum passing score is 68.*1058

*We want to try to find out how many students are going to be scoring above 60.*1061

*Let me put cutoff down there at 60.*1066

*There is 60, that is the minimum passing score.*1070

*We want to see how many students will be above that cutoff.*1073

*In order to do that, we are going to use Chebyshev’s inequality.*1077

*Let me go ahead and set up Chebyshev’s inequality.*1080

*It says the probability that Y - μ is greater than K σ.*1083

*That probability is less than or equal to 1 /K².*1092

*Let us figure out what we know here.*1096

*We know that μ is 76, we are given that.*1099

*Σ is the standard deviation.*1104

*We were not given the standard deviation.*1106

*The variance is 64, the variance of Y is 64 that is not the same as the standard deviation.*1108

*You have to be careful here.*1116

*The standard deviation is the square root of the variance.*1117

*That standard deviation would just be √ 64, that would be 8.*1123

*If I plug that in, σ is equal to 8.*1128

*We want to figure out what K is going to be, in order to use Chebyshev’s inequality.*1131

*That means I want to figure out how many standard deviations away from the mean do I need to be.*1136

*In this case, we are interested in a cutoff of 60, that is the value we are interested in.*1142

*60-76, absolute of value of that is 16 which is 2 × the standard deviation of 8.*1150

*That tells me that the K value that I’m interested in is 2.*1160

*The probability that Y - μ is greater than 2 × 8, according to Chebyshev’s inequality is less than or equal to 1 or 2², that is just ¼.*1165

*Here is a twist that we have not yet seen before with Chebyshev’s inequality.*1184

*You got to follow me closely on this.*1189

*That is telling me the probability that Y - μ will be bigger than 16 in either direction.*1192

*What that is really doing is, that is giving you a bound on the probability of being less than 60 or bigger then,*1199

*let us see 76 + 16 will be 92.*1209

*That is really going 2 standard deviations down below the mean or 2 standard deviations up above the mean.*1214

*Remember, our σ was 8.*1222

*What we are really found is the probability of being in either one of those together is less than ¼, is less than or equal to ¼.*1224

*We are also given that it was symmetrically distributed.*1243

*That means the probability of each one of those is less than ½ of 1/4 which is 1/8.*1247

*In each one of those, the probability of being in that region is less than ½ of ¼ which is 1/8.*1255

*The probability that Y is less than 60 is less than 1/8.*1266

*What we are interested in, is the probability that Y is bigger than 60 because those are the students that are passing the exam.*1276

*In that case, you turn it around and you do 1 -1/8.*1290

*Notice, I'm not worrying about all those scores bigger than 92.*1298

*I’m not worrying about the 1/8 of the students that score bigger than 92, because all the students passed anyway.*1303

*I’m just interested in these 4 students below 60, those students fail the exams.*1309

*That is less than 1/8 of the population, that means more then 7/8 population will pass the exam,*1315

*will get higher than a 60 score on the exam.*1325

*Let me write this in words.*1329

*At least 7/8 of the students will pass the exam.*1332

*That is what Chebyshev’s inequality let us conclude.*1359

*It does not tell us that exactly 7/8 will pass.*1363

*It tells us that at least 7/8 students will pass.*1365

*It could be higher but Chebyshev’s inequality would not give us a specific value.*1370

*To recap what happened here, we start out with Chebyshev’s inequality.*1376

*The probability of Y - μ being greater than K σ is less than 1 /K².*1381

*In this case, our μ was 76, our σ was 8.*1390

*We got that from the variance of 64.*1395

*But remember, standard deviation is the square root of variance.*1400

*The standard aviation is √ 64 which is just 8.*1403

*I’m wondering, how many standard deviations away from the mean are we interested in going here?*1408

*We are interested in a cutoff of 60, because that is the passing score for the exam.*1414

*How far is that from 76?*1419

*That is 16 units away, that is 8 × the standard deviation of 8.*1422

*That means we use K = 2 in Chebyshev’s inequality.*1427

*The probability that you are more than 2 standard deviations away from the mean is less than 1/2² or ¼.*1431

*Here is the subtlety, that ¼ of the population could be 2 standard deviations below the mean*1440

*or that could be 2 standard deviations above the mean.*1448

*Since, we are given that the scores are symmetric, we know that half of them are below the mean and half of them are above the mean.*1451

*We divide that 1/4 into two parts and we find that the probability of being less than 60,*1459

*being 2 standard deviations on the low side is less than 1/8.*1466

*That really came from doing 1/4/2,that is how we got that 1/8.*1473

*The probability of being bigger than 60, in other words,*1478

*the probability of scoring higher than 60 passing the exam is at least 1 -1/8 which is 7/8.*1482

*In the end, we know that at least 7/8 of the students will pass.*1490

*It would not be accurate, if you are given this problem and you just gave your answer and you gave 7/8.*1495

*My probability students sometimes do that.*1501

*I will give a complicated problem and they will just say 7/8.*1503

*That does not really tell us exactly what is going on.*1505

*You really have to say, it is at least 7/8, the proportion of students that will pass is greater than or equal to 7/8.*1509

*That is what Chebyshev’s inequality tells you.*1518

*In example 4, and this is one that is very similar to one we had back in the lecture on Markov’s inequality*1523

*but there is a little more information than it now.*1529

*We are going to get a different answer.*1531

*Be very careful that you do not get this two problems mixed up.*1533

*In example 4, we have seismic data telling us that California has a major earthquake on average, once every 10 years.*1536

*Up to there, it is the exact same as a problem we had back in the video on Markov’s inequality.*1543

*Now, here is the new stuff.*1549

*We have a standard deviation of 10 years, what can we say about the probability that*1551

*there will be an earthquake in the next 30 years?*1555

*This is all exactly the same as the problem we have for Markov’s inequality, except for this one key phrase here,*1558

*with a standard deviation of 10 years, that is the new information.*1566

*That is going to let us use Chebyshev’s inequality, instead of Markov’s inequality.*1571

*Because Chebyshev’s inequality depends on the standard deviation and the mean,*1578

*Markov’s inequality do not use the standard deviation.*1583

*Let me set up Chebyshev’s inequality.*1586

*The probability that Y - μ is greater than or equal to K σ.*1589

*I think when we originally get Chebyshev’s equality is greater than in that place.*1602

*This says, the probability of being that far away from your mean is less than ¼.*1610

*We have to be careful what Y is here.*1617

*Here, what I mean by Y is, Y is the waiting time until the next major earthquake.*1621

*Starting today, how long do we expect to wait in years before there is a next major earthquake in California?*1627

*What we are given here is that, the average waiting time is 10 years.*1643

*That means that my μ is 10 years.*1649

*That also says that the standard deviation is 10 years too, σ is also 10.*1651

*We want to figure out what K is.*1657

*I wrote down 1/4 there, that was kind of looking ahead to starting to solve the problem.*1660

*Chebyshev’s inequality just says that it is less than or equal to 1 / K².*1670

*We need to figure out what K is and I spoiled the ending here but we will go ahead and figure it.*1674

*The question here is how many standard deviations away from the mean are we expected to be here?*1684

*We want to talk about the probability that there will be an earthquake in the next 30 years.*1693

*That means we want to find the probability that Y will be less than 30.*1702

*30 -10 is 20 which is 2 × our σ of 10 here.*1706

*That means that our K is 2, as I inadvertently let slip earlier, K is 2 here.*1717

*I will plug that value of 2 in.*1723

*The probability that Y - 10 is greater than 2 × 10, according to Chebyshev’s inequality*1725

*is less than or equal to 1/2² which of course is ¼.*1737

*That is why I had written down before.*1742

*That is the probability that Y -10 is greater than 20.*1746

*What we really found there is, the probability that Y is greater than 30 is less than or equal to ¼.*1751

*That is not exactly what the problem is asking for, because this is saying, if Y is greater than 30 that means*1762

*we are waiting longer than 30 years for an earthquake, which means we do not have an earthquake in the next 30 years.*1768

*The problem is asking, what is the probability that there will be an earthquake in the next 30 years?*1775

*That means that are waiting time will be less than 30, which is the opposite of it being greater than 30.*1782

*We are going to switch this around, it is greater than 1 -1/4 which is ¾.*1791

*Our conclusion here is, there is a greater than or equal to 3/4 chance that there will be an earthquake in the next 30 years.*1799

*Notice that, I'm being very careful to include the inequality in there.*1836

*I'm not saying that there is exactly ¾ chance, I do not know that.*1840

*I cannot figure that out from the information that we are given.*1844

*What Chebyshev’s inequality does is, it gives me a bound, it allows me to say that there is at least a ¾ chance,*1847

*at least a 75% chance that there will be an earthquake in the next 30 years.*1855

*To recap that, I started out with the generic form of Chebyshev’s inequality.*1861

*The probability of being K standard deviations away from your mean is less than 1/K².*1866

*I filled in the mean is 10, that is this 10 right here is where we get the μ from.*1874

*The standard deviation is 10 as well.*1882

*That is where we get the σ from there.*1886

*I just had to figure out what the value of K was.*1890

*To do that, I remember I was interested in the probability of Y being 30.*1894

*That is why I got that 30 from the problem here.*1900

*Filled in 30 for Y, 10 from μ, and that came out to be 20.*1904

*20 is 2 × 10, that 10 is the σ, that 10 was the μ.*1911

*It is a little confusing because there is a lot of 10 going around here.*1916

*That 20 is 2 × σ, that is where we get our K being 2 which tells us that the probability is less than or equal to 1/2² which is ¼.*1920

*Our probability of Y being greater than 30 is less than ¼.*1932

*That is not what the problem asks, the problem asks what is the probability that*1937

*we will have an earthquake in the next 30 years, which means our waiting time is less than 30 years.*1940

*We will see an earthquake within 30 years.*1946

*It will be less than 30 years until we see an earthquake.*1949

*We have to turn that around, probability that Y is less than 30 is greater than or equal to 1 -1/4 which is ¾.*1952

*I do not just say there is a ¾ chance, I do not just give ¾ as an answer.*1961

*I’m giving a complete sentence which is that there is a greater than or equal to ¾ chance,*1965

*at least a 75% chance that we will see a quake in the next 30 years.*1971

*In example 5, we are looking at housing prices in a small town USA.*1979

*Apparently, they are symmetrically distributed with a mean of $50,000 and a standard deviation of $20,000.*1985

*We are going to use Chebyshev’s inequality to estimate, what proportion of the houses cost less than $90,000?*1993

*Let me do a little graph here because again, we are dealing with a symmetric distribution.*2001

*We will actually find out that this problem is quite similar to an earlier one that we did.*2005

*I think it was examples 3 in this lecture.*2010

*If you remember how to do that one, you might want to try doing this yourself before you watch me give the answers here.*2013

*It works out pretty similarly.*2020

*Let me make a little graph of the housing prices in small town.*2023

*They are symmetrically distributed, I'm going to draw something nice and symmetric here.*2028

*Something that looks like a bell curve.*2033

*We will learn later that this is actually the normal distribution but we have not gotten to that point in the videos yet.*2036

*They are distributed around a mean of $50,000.*2044

*Μ here is 50, I would not bother with the thousands.*2050

*We want to estimate the proportion of houses that cost less than $90,000.*2055

*Let me put in the 90,000 here, at somewhere out beyond 50, that is 90.*2061

*We are told that we have a standard deviation of 20,000.*2069

*I guess that means the distance from the mean, to the cutoff we are interested in is 40.*2074

*That is 2 σ, 2 standard deviations because that is 2 × 20.*2080

*Because we are going to be using that, let me go ahead and put in 2 σ in the other direction.*2085

*2 σ in the other direction which I guess will get you down to 50 -40 is 10 on the low side there.*2090

*That is just setting up a picture, we still need to bring Chebyshev’s inequality.*2099

*Chebyshev’s says the probability that Y will be more than K standard deviations away from its mean,*2103

*is less than or equal to 1/K².*2117

*In this case, we are interested in being 2 standard deviations away from the mean.*2121

*Where I got that was 90 -50 is 40 which is 2 × 20, that is 2 standard deviations there.*2126

*The probability that Y -50 is the mean here, is greater than or equal to K was 2,*2137

*σ is the standard deviation, 2 × 20, that will be 40.*2149

*According to Chebyshev’s inequality that is less than 1/K², 1/2² which is ¼.*2154

*The probability of being that far away from the mean is less than ¼.*2162

*Let me go ahead and fill that in here.*2167

*That is this probability but it is also the probability on the lower end because*2170

*we are told that we have a symmetric distribution here.*2178

*What we know is that, all that shaded region there has combined probability.*2184

*The probability of the shaded region is less than or equal to ¼.*2190

*Since, we know it is symmetric, we know that each one of those tails must be less than 1/8.*2196

*1/8 being ½ of 1/4 there, the probability is less than or equal to 1/8.*2204

*Let me go ahead and fill that in.*2211

*In particular, that Y is bigger than 90, the probability that Y is bigger than 90, according to Chebyshev’s inequality,*2214

*since, we are allowed to split up between the top and the low end, we said that the distribution was symmetrical.*2224

*It is less than or equal to ½ × 1/4 of which is 1/8.*2231

*What was the probability that the house was greater than 90?*2240

*We want to estimate the proportion of houses that cost less than 90,000.*2244

*The probability that Y is less than 90, let me turn that around.*2248

*It is greater than or equal to 1 – 1/8, that is 7/8.*2254

*If we convert that into a percentage, that is halfway between ¾ and 1.*2262

*It is halfway between 75 and 100, that is 87.5%.*2268

*The proportion or the probability, the proportion of houses that cost less than $90,000 in this town is,*2280

*I cannot say it is the equal to 87% but it is at least 87.5%.*2289

*I can say that at least 87% of the houses in this town must cost less than $90,000,*2296

*that is the interpretation that I can put on that.*2303

*At least 87% of these houses cost less than $90,000.*2307

*Let me recap where that is coming from.*2312

*The probability that Y - μ is greater that K σ is less than 1/K².*2315

*That is just the original version of Chebyshev’s inequality.*2322

*In this case, my μ was 50, that came from the mean housing price their.*2325

*The standard deviation, the σ is 20, that was also given to us in the problem.*2333

*I have not figure out what K would be.*2341

*In order to figure out what K would be, I want to know what I was being asked about.*2343

*I was being asked about houses costing less than $90,000.*2347

*We are going to use 90 as our cut off and 90 -50.*2352

*50 is the mean, 90 is what we are interested in.*2358

*The difference there is 40 which is 2 standard deviations, 2 × 20.*2360

*That is where I get my K is equal to 2 there, K =2.*2365

*I plug that in to Chebyshev’s inequality and I get that the probability is less than ¼.*2371

*That is the probability of being 2 standard deviations away from mean, in either direction.*2377

*That includes both of these regions here, both the high region and the low region.*2384

*But I'm really only interest in, how many of the houses are costing too much on the high side?*2390

*Let me cut that region in half and get a probability of less than 1/8.*2398

*That is why that is less than 1/8/.*2403

*That was describing the proportion of houses that cost more than $90,000.*2409

*I want to find the proportion of houses that cost less than $90,000.*2417

*Let us switch that around.*2421

*Instead of talking about 1/8, I will talk about 1 -1/8.*2422

*Instead of talking about less than or equal to, I have a greater than or equal to.*2426

*1 -1/8 simplifies down to 87.5%.*2430

*My answer here is, I do not just say 87.5% is my answer.*2435

*My answer is that, at least 87.5% of the houses in this town cost more than $90,000.*2439

*The at least part is very important part of the answer there.*2448

*That wraps up our lecture on Chebyshev’s inequality.*2454

*You can figure this as companion to the lecture on Markov’s inequality which*2458

*we have on the previous videos is on Markov’s inequality, they go hand in hand.*2462

*Markov’s inequality, you just need to know the mean.*2468

*Chebyshev’s inequality, you need to know the mean and the standard deviation because of that σ there.*2470

*You need to know both for Chebyshev’s inequality.*2476

*You do a little more computation for Chebyshev’s inequality but the trail off that is that you usually get stronger results.*2479

*You usually get some more information about the probabilities for Chebyshev’s than you do for Markov.*2488

*Remember, for either one of these inequalities, you have to give your answer as an inequality.*2495

*You never give numerical value because those numerical values are just lower or upper bounds.*2501

*Your answer will always be that the probability is less than this or greater than that.*2506

*That does it for Chebyshev’s inequality, kind of wrapping up a chapter here.*2512

*We will jump in later on with the binomial distribution.*2518

*I hope you will stick around for that.*2521

*You are enjoying the probability lecture series here on www.educator.com.*2523

*My name is Will Murray, thank you so much for watching, bye.*2528

1 answer

Last reply by: Dr. William Murray

Mon Oct 3, 2016 2:30 PM

Post by Thuy Nguyen on September 30, 2016

Hi, in my class I learned that Chebyshev's Inequality is:

P(|T-mean| >= a) <= variance / a^2.

I believe a = k * standard deviation.

Because variance / (k * standard deviation)^2 = k^2.

Is that right?

Also, does it matter if we write P(|T-mean| > a) vs. P(|T-mean| >= a)?

1 answer

Last reply by: Dr. William Murray

Mon Oct 3, 2016 2:30 PM

Post by Thuy Nguyen on September 30, 2016

Hello, for the college credit example, P(credit > 95) <= 1/9. Isn't 1/9 the combination of both tail ends? Meaning, P(credit < 5) + P(credit >95)?

If I were to sketch the distribution, then the probability of being 3 standard deviation away from the mean on BOTH sides is 1/9.

So why didn't we have to split the 1/9 for the left and right tail ends?

Thanks.