For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

### Correlation

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
- Summarizing a Scatterplot Quantitatively
- Correlation Coefficient ( r )
- Trees vs. Forest
- Calculating r
- Relationship between Correlation and Slope
- Example 1: Find the Correlation between Grams of Fat and Cost
- Example 2: Relationship between r and b1
- Example 3: Find the Regression Line
- Example 4: Find the Correlation Coefficient for this Set of Data

- Intro 0:00
- Roadmap 0:05
- Roadmap
- Summarizing a Scatterplot Quantitatively 0:47
- Shape
- Trend
- Strength: Correlation ®
- Correlation Coefficient ( r ) 2:30
- Correlation Coefficient ( r )
- Trees vs. Forest 11:59
- Trees vs. Forest
- Calculating r 15:07
- Average Product of z-scores for x and y
- Relationship between Correlation and Slope 21:10
- Relationship between Correlation and Slope
- Example 1: Find the Correlation between Grams of Fat and Cost 24:11
- Example 2: Relationship between r and b1 30:24
- Example 3: Find the Regression Line 33:35
- Example 4: Find the Correlation Coefficient for this Set of Data 37:37

### General Statistics Online Course

### Transcription: Correlation

*Hi and welcome to www.educator.com.*0000

*Today we are going to talk about correlation.*0002

*First let us go back in and just briefly review summarizing scatter plots quantitatively*0007

*and talk about all the other things we have talked about scatter plots.*0013

*Then we will talk about eyeballing the correlation coefficient or what we call r, persons r.*0018

*Actually if you have a set of data that looks a particular way often you could sort of ballpark where the correlation coefficient falls.*0026

*We already talked about precisely calculating it.*0036

*Then we are going to go back and talk about the relationship between r and b1 or slope of our regression line.*0039

*First let us talk about summarizing a scatter plot quantitatively.*0049

*We did not deal with shape.*0056

*We just looked at it and maybe that is pretty good.*0058

*We will talk about shape in the next couple of lessons, but for now we are going to leave shape alone in terms of quantitatively calculating it.*0062

*We did look at how to precisely calculate the trend or by looking at the regression line,*0072

*that middle line in between all those lines that summarizes the middle of all those points.*0080

*And that middle line really gives us the relationship between X and Y, because it is the function that gives us if we have x, we get y and if we get y we get x.*0087

*We can get the relationship between those two variables.*0101

*Finally today we are going to talk about how calculating strength and not just looking at it as pretty strong right,*0106

*but instead we are going to actually calculate the correlation coefficients r.*0115

*That idea is simply, how pact around the regression line is our data points.*0120

*Are they tightly packed?*0130

*Is it a strong correlation like it strongly packed around that regression line?*0132

*Or is it very loose?*0138

*Is it dispersed?*0139

*It is not really sticking close to that line, then we would have low strength or low correlation.*0141

*For instance I am just eyeballing it and there are a lot of data.*0152

*You might have no relationship between two variables, and in that case, the spread looks something like this where there is no real line in there.*0157

*It is just sort of this cloud of dots.*0168

*Remember each of these points is a case.*0172

*Each of those cases has two variables.*0178

*One variable x is on it the x-axis and the other variable we will call it y is represented on the y-axis.*0183

*That point represents x here and y here.*0192

*In this case there is no relationship between X and Y just because you know what x is.*0199

*Let us say we know x is here.*0204

*Do you have any certainty as to where y might be?*0209

*There is some y down here and some y up here.*0213

*Even more so what about if we got X was here do we have any reason to say y is in a particular place?*0218

*No not really.*0226

*Because of that a line would not help us here.*0228

*A regression line does not actually summarize this very well, and it is because the correlation coefficient is very low.*0233

*There is very low strength.*0240

*There is very low adherence to the line.*0241

*Moving out a little bit further, you see that this one is starting to have more of a elongated shape.*0245

*This is still a fairly low correlation, but you can see that they are starting to be a linear relationship between X and Y, namely as x goes up, y also goes up.*0255

*This is what we call a positive correlation.*0270

*There is a relationship between x and y that is linear and positive.*0273

*Notice that y on the other side of this is the exact opposite where it is the same shape,*0280

*but it is been almost like flipped around like we put a mirror here and looked at the mirror reflection.*0286

*In this case it is the same shape cloud but now as x goes down y goes up.*0294

*Here we see the opposite relationship between X and Y and we call it a negative correlation.*0305

*Because of that the signs act accordingly.*0312

*Here the sign for this slope that is negative is -.4.*0316

*Here for slope that is positive as x goes up y goes up, as x goes down y goes down, that is a positive number.*0325

*We could easily just by looking at the correlation coefficient immediately know what kind of roughly what kind of relationship x and y have.*0336

*Notice that as we go out even further, not only do the numbers get bigger and bigger out from 0 but the numbers correspond to how whiny the data are.*0346

*How much they correspond to a line.*0361

*It is not really about having more dots but it is about how much do all those dots fit to a line.*0364

*That is what we often call collect fitting the data to a regression line.*0373

*We want to see is it a good fit? Is it a bad fit?*0378

*Correlation coefficient gives us the strength of that it is really strong fit or is it very week and loose.*0381

*The actual maximum for a correlation coefficient is 1 and the minimum -1.*0390

*That is as far as it will go.*0398

*You cannot have a correlation of like 1.1.*0400

*We will talk a little bit about y.*0407

*Here what you see is that it might have the same number of points of all of these, but it is just that there is very, very little variation from the line.*0409

*There is not a lot of variation out from the line, whereas coming like .8 you could see it is better than .4 but it is not quite as whiny as 1.0.*0423

*That is one way to just very quickly eyeball correlation coefficient.*0435

*You can just look at a data and it is elongated a little bit.*0440

*Maybe it is .4 but if it looks like tighter eclipse we will use .8.*0445

*If it looks very close to a line perhaps it is close to 1.*0453

*I want you to notice something.*0466

*Correlation coefficient, other than caring about positive and negative slope, it does not otherwise care very much about slope.*0467

*For instance, look at all of the situations, these are all lines.*0477

*They are all very lined, they are maximum lined.*0482

*Notice that these lines have positive slope but they all have the coefficient correlation of 1.*0486

*It does not matter whether x changes y changes very quickly or as x changes 1 y changes very slowly.*0497

*A slope does not matter, except for just the positive or negative part.*0507

*The same thing with the negative slopes, even though the slopes are all different they all have a correlation coefficient of 1.*0514

*There is an exception to this rule and it is this line right here, the perfect horizontal line.*0527

*Let us think about what the equation for the regular horizontal line is.*0533

*In a regular horizontal line it does not matter what x is y is always the same.*0539

*Let us say here like y=3 that would be like a horizontal line or y=-2 or y=.1.*0547

*Those are all example of horizontal lines.*0556

*Let us think about in the case of a horizontal line.*0561

*It is a perfect prediction because we know where the x is.*0564

*You could tell me whatever x you want.*0570

*I can exactly tell you the y because y is always negative.*0572

*y in this case is 3.*0577

*It is perfect prediction and it is perfect lining this but the correlation coefficient is 0.*0578

*We will try to figure out y as the horizontal line as we figure out the formula for correlation coefficient*0589

*and that hopefully that will become more clear.*0599

*There are many, many ways in which data can have seemingly no linear pattern*0606

*or very weak linear pattern because that is what the correlation coefficients tell us.*0613

*If you see our data have a 0 as its correlation coefficient do we know that it looks just like a cloud?*0618

*No, in fact it can look like anyone of these crazy shapes down here.*0627

*All of these distributions, all of the scatter plots have a very, very weak correlation because remember correlation just means how whiny is it.*0634

*This one is whiny.*0642

*Even though some of these are very, very regular shapes correlation coefficient cannot tell you that this has an interesting shape.*0647

*All it tells you is whether it coheres to that regression line or not.*0656

*Although these are very interesting set of data for instance here there is a certain 4 rough cluster and even though we could see and eyeball it,*0661

*The correlation coefficient would not tell us that.*0672

*Or in this case, this sort of our data set but even here the correlation coefficient would not tell us that either.*0675

*Are all of these data the correlation coefficient is very close to 0.*0683

*I want you to see there are many ways in which you can have a correlation coefficient of 1 or -1.*0689

*There are many ways in which you can have a coefficient of 0.*0696

*Just because we get the correlation coefficient does not mean we can see the shape of the distribution.*0703

*That is often useful to do a scatter plot anyway even for ourselves just so that we know what the numbers are probably going to be describing.*0709

*Let us say we have this graph and this shows us this nice correlation.*0721

*It was probably pretty high like r=.8.*0728

*It is closer to 1 than 0 but not quite 1.*0736

*This is a pretty good correlation and you might have two variables here.*0740

*For instance, perhaps this gives us the z scores for some variable like we are looking at twins and then we want to know does the intelligence of one twin,*0748

*does the IQ of one twin helps us predict the IQ of the other twin?*0759

*Maybe it is true.*0765

*Maybe that does seem to be the case.*0767

*Here we might put something like the intelligence of twin 1 on this axis and then we will put the intelligence of twin 2 on the z score from their IQ score.*0771

*We will put that on the y-axis.*0783

*When we have the scatter plot it is very important that we could toggle between the 3’s, the individual little dots and the forest, the big overall pattern.*0786

*When we will we look at correlation coefficients we are looking back.*0801

*We are sort of getting a bird’s eye view and looking very far away and trying to see the overall pattern and that is the forest.*0804

*It is really important to remember what are my trees?*0816

*What are my cases?*0820

* It is important to remember what this dot mean.*0822

*That is what I mean by the trees like you want to remember what are your cases?*0826

*What are your variables?*0832

*That is always step one of looking at a scatter plot.*0834

*In this case, it is not that each of these dots represents just one twin it is these dots represent a set of twins, a pair of twins.*0838

*This represents both twin 1 who is a little bit below average and twin 2 was actually a little bit above average.*0848

*Let us pick out another one.*0862

*Let us say this one.*0866

*This twin has a little bit above average and guess what, so is their twin.*0868

*Their twin is also little bit above average.*0878

*Each of these dots actually represents 2 people in this case, a set of twins.*0881

*You want to be able to switch your perspective and to zoom in and see the trees but also zoom out and see the forest*0888

*and try to estimate things like correlation coefficient or even try to estimate the regression line and try to eyeball where that might be.*0896

*Okay, now let us get to the business of calculating r.*0909

*You could think of the correlation coefficient as roughly that average product of z scores for x and y.*0913

*Let us recap a little bit what the z scores are.*0922

*z scores are just giving you how many standard deviations away you are but we do not want to know it in terms of the raw numbers.*0927

*We want to know it in terms of standard deviation.*0936

*We do not want to know, like how many feet away, but we want to know how many standard deviations away.*0941

*We could think of the standard deviation as jumps away from the mean.*0947

*How many of those jumps away are you?*0952

*That is the z score.*0954

*Here is how we calculate r.*0957

*The average product of z scores for x and y.*0958

*Let us put the z scores for x and y and multiply them together because we are getting the product.*0965

*The product is z(x) × z(y) and I’m going to sum them together and then divide by n-1.*0971

*Later on we will talk more about y and -1 as more frequent.*0987

*You can roughly see it is about the average and mostly because we are jumping from samples to populations we need to make a little bit of correction.*0992

*This formula of adding something up and dividing by n is an average and the thing that we are averaging are the product of the 2 z scores.*1005

*Now for all of these formulas you can think of these little z scores as you can double-click them and if you double-click what is inside.*1017

*Each z score let me write this in blue, so each z score is the distance away from the mean,*1030

*but not the raw distance and I want it in terms of standard deviation jumps away from the mean.*1038

*That would just be something like my y - y bar for mean and so that distance divided by the standard deviation.*1045

*Here I will just use little s and also for z(x) that is just x - x bar.*1058

*That is the raw distance away from the mean but divided by x standard deviation.*1069

*I will put a little x to indicate the standard deviation of x’s and a little y there to indicate the standard deviation of the y’s.*1077

*I’m going to multiply those together and add them up for every single data point that I have.*1085

*If that is my twin data for every single set of twins that I have.*1092

*When we divide all of that by n-1 and n is my number of cases.*1098

*How many twins how you got?*1105

*How many sets of twins have you got?*1107

*If we want to do although it goes without saying this just implicitly have an i that goes from 1 all the way to n*1109

*because it is for every single one of my data points that I need to do this.*1117

*Furthermore, we can double click on each of these little standard deviation.*1125

*Now how do we find standard deviation?*1135

*A standard deviation is the square root of the average distance away from the mean.*1137

*The average distance away.*1146

*The square root of average squared distance which is average distance away.*1148

*S sub y and this is think about the distance we already know how to do distance because we have already done it.*1154

*Its average squared distance because remember its sum of squares over n-1.*1168

*It is sum of squared distances because if we just got the sum of the differences then we just get something very close to 0.*1179

*We want that and we divide the n-1 because that sum of squares is very small so we need to correct for that by going from samples to populations.*1193

*That is what we do by n-1.*1203

*Because we want the standard deviation and not the variance we are going to square root this whole thing.*1205

*Same thing for s(x) it is a same thing except we substitute an x here instead of y.*1211

*I forgot to put my little sigma notation because I want to do this for every single y.*1220

*Although it looks sort of complicated if we write the whole thing out*1227

*but if we wrote the actual n or double-click diversion of s sub y in there it might look very crazy.*1232

*What you do have to remember it alternately less is the main idea you want to get out of today and you want to take a moment to think what z score.*1242

*Once you unpack z score you want to take a moment to think what standard deviation and hopefully you will be able to unlock those things as you go.*1256

*Then you do not have to remember all of that stuff at once you can just remember them one at a time.*1265

*Now that you know the formula for correlation coefficient let us talk about the relationship between correlation and slope.*1272

*We already know that b1 and r have the same sign.*1280

*If B1 is negative r will be negative.*1285

*If b1 is positive r will be positive and vice versa.*1288

*We already know that they have the same sign and because of that they already slant in the correct way.*1292

*Remember r does not have any thing about y’s and run in it.*1300

*All it cares about this is how much like a line it is.*1305

*B1 and r have a very strict relationship where r when you multiply it by the proportion of standard deviation*1309

*of all y over the standard deviation of x as long as you multiply r by this and you can almost see rise over run then you get this slope.*1321

*Let us just think about this in our head and let us say r is 1 it is always 1 then whatever this proportion is that will perfectly get us b1.*1335

*Also if r is 1 as always 1 these 2 have a very similar standard deviation.*1349

*The spread of y is very similar to the spread of x then we should have perfect correlation of 1.*1357

*In that case you would be able to sort of say that makes sense if y is varying in a similar way as x then they should have correlation version of a slope of about 1.*1368

*If y is changing more slowly than x for every x you only go a tiny bit of y.*1385

*In that case this number would be smaller than this one and then that would give us less rise more run.*1399

*Something that looks sort of less slanted.*1411

*Something like this versus a slope of 1.*1415

*Something a little more shallow and that make sense less rise more run.*1424

*On the other side if for every y you go will go a little x then that would look something like this more rise less run.*1431

*This gives us this perfect relationship between r and b1.*1446

*Using that information let us try to solve this problem.*1454

*Example 1, here are the 3 pizza companies that we have looked at before, Papa John's, Dominoes and Pizza Hut.*1458

*It says find the correlation between grams of fat and cost.*1466

*I think these are for whole pizza and let us make this 17.50.*1475

*Let us make this $18 and $20 because this is really cheap to have $1.75 pizza.*1485

*It would be ridiculous to have 100g of fat in one slice of pizza.*1492

*If you look at the examples provided in the download below we can use the data in order to find correlation coefficient.*1498

*In order to find correlation coefficient I will break it down into the component pieces and the big component pieces*1513

*and the big component pieces I’m going to need are the z scores for x and the z scores for y.*1519

*I will say that the score for fat and that the z score for cost.*1525

*Z score for fat and z score for cost.*1529

*In order to find the z score I would need to put in the difference between this and the average.*1541

*One thing that might be easier is if we actually just create a column for averages because we are probably going to need this again and again.*1549

*Let me go ahead and get those averages.*1564

*I’m just getting the average cost, as well as average grams of fat.*1571

*I’m going to color it in a different color so that we know that this is the entirely different thing here.*1578

*We have that it would be easier for us to find the score for fat.*1586

*Here we want to get x of fat - the average and probably want to lock that in place and we want to divide that by standard deviation*1591

*and the nice thing about Excel is that it already has the function for standard deviation.*1620

*This one will give us the n -1 version so I can just lock that data down in order to move.*1624

*I probably want to copy it over to E later so I’m just going to unlock the B part.*1645

*As long as they in the same column, as long as I stay in column D it will use column B.*1652

*If I move over to Column E it should use column C.*1657

*Let us try that.*1663

*Here we see that the z score is -1, that is 0 and 1 and that makes sense.*1664

*Your z score is totaled together shared roughly equal 0 because you are getting how many distance away on the positive side.*1669

*How many distance away on the negative side, and they should balance out if you really have the mean.*1677

*Let us check this formula yet it is using B3 that it has average that is getting that Standard deviation perfect.*1683

*Once I have that I can actually just copy and paste this over here.*1692

*Here we see now it is using C and this average and getting the standard deviation of this data.*1700

*We see roughly the negative side as to the positive side.*1711

*We have these individual z scores, now we need to get the z scores for fat multiplied by the z score for costs.*1719

*That is real easy, this times this for every single data point or case that we have and we have 3 cases here.*1729

*The 3 different brands of pizza.*1736

*Once we have that instead of the average actually we could just get the average all at once because we could put it in-one formula.*1739

*We could just sum these together, sum those together, and we want to divide by n -1.*1751

*In this case, it is 2.*1762

*If you wanted to put in a formula you could put in counts -1, but I'm just going to put for our purposes 2 here.*1765

*We get a very, very high correlation where it is very, very close to 1 as cost goes up fat goes up.*1774

*As cost goes down fat goes down.*1788

*They have a very positive correlation and it is very whiny it.*1792

*Here it is very closely to the line.*1796

*Here we could see that this data is very highly correlated.*1802

*It has a strong correlation.*1813

*We do not have a lot of points, but apparently they fall very, very close to the line.*1815

*Previously, we found that the regression line for this data is this.*1827

*I believe that in that case, the cost is 17.50 that previously is $18.00 and $20.*1834

*Previously in the regression we already found this so check that the relationship between r from the previous example and b1.*1844

*It is asking us is this really true that b1 in this example, and we are not going to do this formula proof but just to see for ourselves.*1856

*Is b1 really equal the proportion of r times the proportion of the variation from y over the variation from x.*1867

*Is this relationship really true?*1878

*While we already know b1 .125 and we already know r.*1882

*We know r this is .94 and so we know this .94 multiply by s sub y over s(x) does all of this equal .125.*1894

*Let us see.*1915

*That is not too hard and then move that up here.*1916

*We have r over here I'm just going to find s sub y s sub x and multiplied by it.*1922

*I will just create another column for standard deviation, and let us get the standard deviation for x and the standard deviation for y.*1935

*Now you know that this r × the standard deviation of y over the standard deviation of x and that is going to be equal to .125.*1948

*That relationship holds here we have the b1 and r over to this side so we know what these things are.*1968

*There you have it.*1988

*We see that the relationship between r and b1 holds.*1989

*There is sort of a little bit of y for all a lot of run and we know that this line is pretty shallow and that makes sense.*1994

*This is a pretty shallow slope.*2005

*There is little rise over run and because of that is the fraction less than 1.*2006

*Example 3, the mean score on a math achievement test for community college was 504 with a standard deviation of 112.*2018

*For the corresponding reading achievement test the mean was 515 and a standard deviation was 160.*2026

*The correlation coefficient is very high.*2035

*Use this information to find the regression line.*2038

*Here we see that we have the correlation coefficient, but we but they do not give us the data.*2041

*Can we still do this?*2049

*Yes, we can because there is a relationship between the correlation coefficient and the standard deviation.*2051

*There is a relationship between the correlation coefficient and slope at all and we need to know are the standard deviation in order to find this.*2058

*B1 = r × s sub y / s sub x.*2066

*We actually know s sub y and s sub x and r so we could find b1.*2073

*Once we know b1 and we have the point of averages.*2077

*We have point of averages, which is x bar and y bar.*2083

*In fact, let us say this is x and let us say the reading is y so here we have 504 – 515.*2090

*We could get the slope and we can have one point of averages and we could find the intercept.*2103

*Let us go ahead and r is going to be .7 and s sub y which is the reading one is 116.*2111

*S sub x is 112.*2122

*We can find b1 and I’m just going to use a little bit of space down here to just do the calculations.*2127

*Feel free to do this on your calculator.*2134

*.7 × 116/112 and that is .725.*2137

*I have here .725 as my slope.*2148

*Once I have my slope I could put that into my slope intercept line 4.*2154

*My y is 515 and I'm looking for the intercept.*2160

*I add that to .725 × x which is 504.*2173

*When I go ahead and solve that in here and let me go ahead just solve that in here that is going to be 515 -.725 × 504.*2186

*I will get 149.6.*2194

*My b sub 0 = 149.6.*2210

*We have these two ideas we can now find the regression line.*2216

*A regression line in order to predict y is going to b sub 0 or the intercept 149.6 + .725 × x and that is our regression line.*2221

*Here we see that this slope is less than 1 y is more run.*2237

*More shallow slope and you do not need to have all the points in order to find the regression line.*2247

*Example number 4, find the correlation coefficient for this set of data and this set of data is provided for you on the download below.*2259

*If you go ahead and click on example 4 that data is all there.*2269

*Previously we looked at the data and we thought this is pretty good, pretty found linear correlation.*2275

*Let us see if our eyeballing was actually right.*2282

*I’m just going to move this one over a little bit because we are not going to need that as much.*2285

*Let me shrink this down a little bit.*2296

*It always helps me think about I’m trying to find correlation coefficient I know it is the average product of the z scores.*2298

*I need to find that the z scores.*2311

*I need to find that the z scores for the student faculty ratio the SFR.*2315

*I want to find the z score for cost per unit CPU.*2323

*Let us go ahead and do that.*2329

*In order to do that it is often helpful if you have the mean and standard deviation.*2331

*How do we find the mean and standard deviation somewhere.*2335

*Here let me just get the means here and move this over by one column just so that I can write mean and standard deviation.*2340

*Sometimes it will get confused as to like what we are doing and it is often helpful to write these things down.*2361

*I like to put it in a different color because that helps me know this is not part of my data.*2370

*Let us get the average mean of all of our data here.*2377

*The data for the student faculty ratio, as well as the cost per unit.*2384

*Subtlety that same data and find the standard deviation because we are often going to need that for z score.*2391

*It is just useful to have it in advance.*2400

*We have the mean and the standard deviation.*2405

*Here I’m just going to put a little divider here for now so that I can move this down.*2408

*Notice that it gets from row 7 to row 34.*2416

*Let us find the z scores.*2422

*Now that we have mean and standard deviation it should be really easy.*2424

*It is just the difference between my point and my average all divided by standard deviation.*2427

*I want to lock down that mean and average and as long as it is in the same column it will always use but I do want to use E when I move over.*2437

*I’m not going to lockdown the B part I’m just locking down the row.*2453

*I guess z score of -1.556.*2458

*If my z score calculations and my mean and all that stuff are correct.*2465

*I should roughly have z scores that are both positive and negative, and they should roughly balance out.*2471

*Let us take a look at our data and it seems like half of them are negative and roughly half of them are positive.*2481

*They should balance out.*2489

*Once I have that I could actually take all of these guys and drag that over.*2491

*Let us check one of these formulas here.*2499

*This one it gives me this point at the deviation or the difference between this point and the mean divided by its standard deviation.*2501

*Perfect.*2510

*Once we have that I know I need to multiply and get the product of these z scores.*2513

*Z of s(r) × z(CPU).*2518

*Let us see what we could do here.*2528

*Here I’m just going to multiply this times this for every single one of my points.*2532

*Once I get down here and I know I need to find the mean of these points.*2545

*I’m going to find but I do not want to use just the formula for mean because that is going to divide by n.*2550

*We are going to divide by n -1.*2558

*When I split it up into adding all of these up.*2560

*I am going to sum them all up and divide by count instead of counting all of these.*2563

*I’m just going to use the same points here I’m going to say count all of this and subtract one and put all in my green parentheses.*2584

*We get a negative slope that is pretty high.*2601

*It is you know above .8 and so let us take a look at our data to see if that makes sense to us.*2608

*We certainly understand y it is negative.*2615

*It makes sense that r is negative and we did not think it was pretty good.*2619

*We did think it was pretty strong and if it does end up being pretty strong .6 or .7.*2623

*That is correlation coefficient see you next time on www.educator.com.*2634

2 answers

Last reply by: Kristen Gravlee

Fri Sep 28, 2012 12:05 AM

Post by NASER HOTI on January 23, 2012

it will be very helpful if we have handouts of these lecture...