1. Promo Video: welcome Another. Professors make statistics just too difficult. You see a lot off difficult science, and maybe you don't have any clue what they actually mean. And you see difficult formulas and you don't see any connection at all between everything within statistics about me. I am now a private teacher for 13 years, and I love it. A lot of students come to me on they think me with bottles of wine with all kind of things you can check my reviews on. I'm sitting on $2 golden now to read more about me. But most important, it's not about me. It's about you because you guys have to finish statistics. So I'm here to help you. What I'm gonna do in this course is to explain statistics in an easy way. I'm going to cut the shit. I'm just going to explain to you what really matters in statistics. So and I know exactly exactly what things professors ask so I can explain them to you. At last, I'm going to give you study aids, which will the minimize your study time so you just will finish your studies quickly at last. But not least, this course is especially designed for people who just want to pass the course we want to finish. So if you want to become a statistics geek or want to become a statistics professor, I know everything about statistics. This is not a cause for you. This the course for you if you just want to graduate and just want a Boston chasing and just want to be dominated. Actually, what I see with a lot of my students is the in the end. They just want to deposit, but in and they start liking this subject and possibly the high grade higher than they expected. So that's always good, looking forward to see you winning the course. 2. 2: At the end of this video, you will know what out of degrees of freedom and what is a standard deviation. Because of statistics. You hear a lot about standard deviation at first degrees of freedom. I will explain it to you in a little example. For example, you have free numbers with the ever words equals. Then now the first moment you can in feeling for free, for example, you can feel in an aide, or you can feel in a seven for 10 or 11 on the second number. You can also still feeling for free. You can feel it. Then you can feel it. Nine. You can fill in eight, but now the last number you can not feeling for free is this. One was for free for free. You can like feeling every number what you want, but for the last one, you have to put the number 11 or otherwise. If you don't put the number 11 the Edwards will not be 10 because 11 plus eight plus 11 is sturdy. Divided by three equals 10. But if I would put a number like eight, you will get 11 plus eight plus eight equals 27 then the effort will be nine. So then the effort would not be correct. Just a short explanation about degrees of freedom. And you will see this bag a lot in statistics. And that's about this. I go to the next subject of this video. What is the standard deviation? Exactly. Standard deviation. You will write down for population as the s Sigma. Or as as for the sample as example. I got this numbers. I got a number 12 tree animus. 567 You could calculate yourself with the mean will be like in the middle. So we'll be X bar will be for a mean off x Different notations for different studies, but actually mean of X are mean if x or the egg by egg. Sexually the same. So what's the ever exist of the mean? It will be around around still the six and A until two. So the effort distance will be six minus four. He calls, too. So, as will be around to it will not be exactly two because there's a lot off, um, adjustments in the formula itself. For meta medical purposes. It's not important that you know It's just important that you know, for now, like the as or the Sigma will be around to notice that Sigmund as will never be, uh this same if you calculate them but they will be really the same. What will be the sigma if you have, for example, this data said 32 and 36 Think about it for 10 seconds. Really good. The signal will be again off the mean will again be in the middle. So the X bar, but before and the average distance 6 to 4 equals two. And is this tool so that sigma equals around too? 3. 2: At the end of this video, you will know how to calculate the standard deviation as well as for the sample. As for the population and what you see here directly in the formula is like this formula is not for the standard deviation. It's for a standard deviation squared and extended deviation squared is also called variants. So what's important? And we're going to calculate at the end of this video something and what we actually going to calculate is the very and so the standard deviation square. So what we need to do at the end is take the square root of that and we get a standard deviation as first. How you will see the formula in most off the courses is like the above one you see, like the standing ovations correct equals one divided by end minus one. The minus one is the simple adjustment. So the formula for the whole population will be exactly the same. Only it will be for the population the miners want will disappear. Ah, phone. Um or what you will see is summation sign was I explained earlier. So if you don't know that exactly, please refer to the earlier video about it times the frequency and a frequency you don't see back in that many textbooks. But it makes it much easier and quicker to calculate. So I want to include it here. But if you don't see it, just no, If you just see this formula textbook, just know even with the frequency in it, you can also use it and get exactly the same answers with calculations will be quicker. Um, times X minus X bar. So the mean off X sometimes you see like mean off eggs and that's squared with the formula you see below. This formula is a quicker formula, which you can use to calculate it. So in this case, I'm going to explain you this formula so you can calculate it quicker. I will be moved out both one and put his formula you on top. And as an example, I will give you a data set because this thing off to 22 and 666 So the mean off this data set will be you can calculate yourself to Pristoop is 26666 divided by six equals four. But you can also see like two and six. I like around four. So therefore, it must be really the middle one, So that must be to me now. We're gonna put in this in the formula. Um, I'm going to copy the formula for you. I'm feeling all the blanks, like the eggs bar equal for next week. You see here the end. Think I didn't explain it yet, but the end isn't number off numbers in this case it 123456 Uh, your simple size is six big, so the end will be six will fill it in six and six. Hey is like an X bar. Square means like end times expire square so year I have to fill in the time sign. Otherwise it will say 64 square, which is different at six times four square. Last thing we have to fill in is the combination off F frequency times X square. What's the frequency like? There's three times to. So instead of like adding to square pless to square, please to square, I can also say it's the same s three times to a square next to his is three times a six so se times six square. So and it's like something which will be three times four plus three times 36 equals. Fill it in in your calculator and you will see it equals one under 20. So the last thing I cannot feel in this piece because 120 this query disappeared with important we were calculating the s Square. So I put it back, fill it in and a calculator one divided by fix Mine is one 120 minus six times for square in bracelets equals 4.8. So that square equals four point ain't so what must be s square equals 4.8 as to go to s. I need to take the square root opposite off the square and the square root off 4.8 equals two point 19. So that corrected every distance for the mean is 2.19. It's not exactly that at what This is why I mean, because that's too because to was to like four plus two is six and four minus two is two. But you have to calculate with a corrected one and it will be 2.19 4. 2: in this video, you're going to practice with calculating the standard deviation off a population data sets . So the question here is, what is the standard deviation off the following population? Data said three times to three times that, then and especially for you, I really well, the formula year again for you. So please use this to calculate it. Yeah, I put in the new, which will be off course, the X bar. Like the efforts off the population. Now, the thing you have to do is to grab some favor and stop the video and make the question and afterwards check with me. If you did. Ah, did a good job. So stop the video. 54321 Stop the video. Did you stop the video? If not, still stop it and try to do it for yourself. That's the best way to learn on. Anyway, if you stopped it, you tried it. Really? Well done. We're gonna check together. If you did a good job or if you can still improve. Also, maybe it was still too difficult for you. So I will give you the example to do this. So you got the formula as first What? What is the end? The end is amount of numbers. 123456 Again So we can field is in in the formula. Have a copy it again. The end will be six. Next thing we have to calculate is the efforts. So two plus two plus two, it's six 10% of Stannis 30. So to get it will be 36. Divided by six will be six. So the Evers mu equals by coincidence. Also six. So I can feel in their two equals one divided by six. Grab food off F Times X Square minus six times six square. Now the last thing we need to calculate is estimation. Off frequency times X Square equals there's three times a two so three times to square and there is three times a 10 3 times, then square. Yet its summit. So three times four plus three times 100 will be a lot I will need to calculated to to calculate it Have you will be state 112 so I can feel in this want to No. The last thing is the standard deviation squared or variants equals just fill in the world Formula One divided by six times. She itself minus six times six square 16. So the standard deviation off the population Little Sigma To go from Sigma squared two sigma, you have to take this crab boat. So you also have to take the square root of 16. So it equals four. And it's exactly the every distance from the mean. Because six plus four equals 10 and six minus four equals two is really a coincident because most of them is never exactly that with in this case it is. 5. 3: after this video, you will have a feeling about what is correlation. Now you see a I made for your three examples one about ice cream, salt and temperature so you see, like on the excess, the temperature on the wild. You see the ice cream, so use a little dose. So this day it would have been like a cold day, and not so many had come sold on this day. Warm day and quite a lot of ice cream self. And what you'll see here is that if you put a line through those points, the line will be positive. So this is gold. There will be a There is a positive correlation between temperature and ice cream sold. Go. The line goes up, but go to the second example you see, like the temperature and winter clothes floats soul. So not close, but ca lot's when the clothes sold. And of course, if the temperature goes up, probably it will be summer. And because it's summer, people don't buy the winter clothes. So if you put a line through this point, the link will go down and this will be called a negative correlation. Ah, at last if you have the temperature and Ah, and the house is sold, it doesn't matter that much because people will buy a new house in winter or in summer or in spring. Temperature doesn't really have an influence and like you will see, is if you see all these dots you cannot really make a line through with so that no correlation. So in the first case, a postive correlate because the land goes up a negative correlation because the language is down. If you cannot really make a line, no, up or down, there will be no correlation you below. I put like some correlation examples. Some more examples. Yeah, it's the again, a bit example. Off the temperature and the ice cream sold, it looks the same and see a positive correlation. It will be, in this case, 0.8. What you see in this picture is that the correlation goes for minors. One do one, and if you have one or minus one, you have a perfect correlation because if the at one the temper rises the ice cream rise for sure, if you have a correlation of zero up with a, probably if the temperature rises that ask you will rise to. But it doesn't have to be because here is a little dot and here is also a little bit. If you put a line between those two dots, the line rule go like this. See, it's negative, but in general the line is off course. Still positive. If the correlation comes lower like 0.4 you see, it's still positive, but does much more and bigger negative connections in it. Ah, last one and become zero. There is no correlation or you cannot really make the line. Yeah, line becomes negative and here also so see, here you have to zero point for he was the minus zero point for he's a 0.8 minus 0.8. What you see in the next go it doesn't matter. Like how positive connection is all those 123 lines going up at all to see the correlation one. As soon as the line goes down, you get the negative correlation. Mine is one. Notice if the language exactly straight. There's no correlation. Adore. You cannot say it. Zero, but you can also not say it is one. It's like nothing if you look at the lower line any you see some connection because in the 1st 1 you get together some coil there, some connection between you have the same temperature. Ice came solves you can like this, but it's not a straight line. Correlation is just about a straight line like here, too. That's a connection, because there is a circle. But the circle is not a straight line. So the correlation will be zero. Because if you have the 1st 1 here, gonna put a straight line through which you can put it like this put it like this, like this, like this, like everything. And then this strange form becomes exactly the same as this little about thing. 6. 3: at the end of this video, you will know how to simply remember the co variance video at first is really important. You think what its goal variants gold variance is nothing difference as a part of a formula you need for correlation and more statistical calculations. So don't take time in knowing exactly what it's called variant because it's something really difficult. You don't need to know it just need to know what is correlation what our artists, statistical calculations in the future. What do you exactly calculate for that? But not the co variants? How's it, um, to explain? It's good to you. I'm first going explain you again. What is the variance? You see the formula for the variants. It's a standard deviation, like I explained to you before squared. And in this formula, you see that one divided by n minus one, and his mind is one is the simple correction factor. So if you are calculating the standard deviation for the population, the mine is one disappears. Furthermore, the advance of eggs I wanted he down in different way. Friends of X is exactly the same as the covert Hands off X and X. We're going down. X square is exactly the same. S x times x x bar square like the mean of X is the same honest eggs bark times, eggs by. So the next thing is the Covance of X and y is nothing different as I made, especially some colors for you. Like this green color comes back here, it will be the eggs. But the red eggs becomes a why. So it's not been different as frequency times, extent why minus end times the mean off X times that mean off. Why you didn't see this time before? That doesn't mean anything else. Is that mean why? Look at the next video. I will give you an example with this formula and then you're ready to go. 7. 3: in this video, I'm going to give you an example how to calculate the sample variance. You see, a couple of different data points to together with one to get it with one feet together with five, 35 and four with six for with six. And I have this free dots and the question is, what is the goal? Very ins of the three dots. You see that war formula? What's important? Getting of the formula, You see the end. So the end equals. Maybe you will be thinking six. But actually you have, like, fate points the two together with one C together with five and the four together with six. So the end equals three important next to it. You see the frequency off X? Actually, it should be the frequency off eggs. Why, just a frequency doesn't matter so much. And all these things have a frequency of one because there's only 1.2 point 11.3 point five and $1.4.6. So in this case, it's easier to you can do all these things times one with its a bit of works. It can also remove the frequency out off the formula next to it. You see the mean off X and a mean off. Why already calculating for you mean of X equal? C and I mean, why equals four. So now you can feel in the formula and the last thing you need to know is what is the summation off X times? Why? No, it's less difficult than you think. Because the mater examines why is two times one plus three times five plus four times six. Those calculated together, like just two times one C 54 times six equals to 15 24 if I add them will be 11 41 and I can feel in all the things for Formula one divided by sleep in mine is one times 41 minus three times three times four. I feel that the into your calculator you get one divided by two is 0.5 times 41 minus three times to dance for you get 2.5. So the government's off. Those 123 points is 2.5. What does they say? Actually, actually, nothing but you will. We will need it in the next videos to calculate correlation 8. 3: At the end of this video, you will have calculated the Cove Arians off a population yourself. Here. Still the sample Simple formula. I'm gonna change that. The bid to the population For me it's a like Oh, Vince X, y not. It doesn't matter if its simple of population it's still Sekou, Vince X y But this means also co Vince X, y s ex wife is not a Stana. Deviate is the co variance because it has two para meters the x and the y. But as soon as I wide not that s but the sigma means it's from the population. So what changes in the formula? The minus one is like the simple adjustments. So if you have the population, you don't need it. So we re moving the frequencies in this case ever removed them again because, yeah, everything is just happening one time Ah, yes, The X times Y minus and x bar mean off x mean if I But in the case off population, you were wide them down like the mean off x times that mean off why it is exactly the same . You calculate exactly the same. But just how you wind it down. Now, please stop this video and make this exercise yourself with this day that I blow that the points here for you. So stop the video in 54 Treat Toe What? Stop the video. I hope you stop the video. If you were like or no, it's too difficult. It doesn't matter. Just sit down. Relax. I'm going to show you how to do it. Or you can just check if you did the good thing. At first, you have to calculate the let's start with the 1st 1 the end. And it's like the previous example if free data set so the n equals C. Furthermore, we need the mu of X and the mule. Why New of X one plus three plus five equals 9 90 Fight about three equal 30 presumably six equal six and 65 by three data points he goes to so we can feel in these two points so you get the goal variants off eggs. Why equals one divided by three times Sigma X qy minus three times three times to three times to three times to. Next thing is to calculate the summation off X times y equals this case one times zero x times. Why again Extends 13 times zero The last time. Five times six. We have to So many. This equals five times six is 30. The other ones are nothing. So the answer will be 30. Now we can feeling this inside this formula. So you will get This will be 30 e And putting this all your calculator you will get 30 minus three times, two times two equals 12 divided by three four to the co variants of ex wife. In this case, the Population Co Vince equals four. 9. 3: how to calculate the correlation Now in this video, I'm going to show it to you. But first, remember the difference between population sample like the population power meetings are in Greg letters the sample in just normal letters and now correlation we can add relation is denoted with our or Greek are the pronounce as oh going back to the exercise calculating correlation you have here to formulas First formula for the population second formula for the sample first for the population and for example. And this piece off the formula is the go variants off did population and this piece is the go variations off the sample. So just remember, governments of the sample is doing is calculated with a sample adjustment the minus one and the governments of the population is not calculated with that adjustment. So he started the formula Ecause one divided the end and here it starts with one divided by an minus one below you see here the standard deviations off the X data and off the wide it and you have to do with times he's artist Exactly here the same If you look good, performers are exactly the same only like the input is in the one formula. All the input for the population and hear all the input is for the sample. In the next video, I will give you some calculation example. 10. 3: at the end of the video, you have seen an example of how to calculate the same sample correlation. So he But you see that little formula and what you see, what we need for a formula is the Cove Arians and the standard deviation off X understand deviation of why, as input, we use this data and it's exactly the same data as we used before for calculating the variance. So these three points to remember for you and the co variance in that video just look back at it. If you don't remember it, it's 2.5. So I'm not going to calculate the co Vance again. Girlfriends. I can just fill in here in the Formula 2.5. Now what you see is we need also the standard deviation off X and a standard deviation of why remember the formula for the variance off X standard Deviations square equals one divided by end minus one with a minus one as the sample a correction factor times the summation of X square miners and ex everts square mean of X square. So first going to calculate all the X class sometimes just easy to do it directly here to square Well before three square, it will be nine and foursquare will be 16. So if I now feel all this in, I get one divide by n minus one and will be in this case one. So three minus one times. Uh, four plus nine is 13 plus 16 is 29 minus three times everyone's of Exley Square. And if you feel that in only a calculated you can do it, just do it for the sake of checking. If you feel it in good, you get the answer one so X squared equals one as X is the same as the square root off s square so it will be squared off. One it goes one for what? You can do exactly the same. What changed in the formula is here you will have a white and you will have white bar ever . It's off. Why so? And if you feel that it will in practice yourself, it's a good breakfast. You must get the answer. Parents of why equal seven and the standard deviation will be the square root of seven. Now I can feel it. This two numbers square root of seven and the one in the main formula. One times square root of seven, if I feel, is in him a calculator. 2.5 divided by in between braces one times the square root of seven. I get the answer. R equals zero points. 945 There's a really strong correlation because it's almost at the one. Remember, Correlation go from minus 1 to 01 and where zero is no correlation at all. Minus one is a negative correlation and one is completely positive Correlation and see it here also in this picture, this points, these three points are really close to the line, to the correlation is really strong. 11. 3: Yes. I haven't exercised for you. The question will be what is the co variance off the following population data You have like tea boys. The point 2.1 2.1, 3.5 and 4.6. Hey, it's exactly the same data as the last example. Only in the last example It was not the population data. It was simple data. So probably because now you got population data. You will get an auto answer, please. What I put here for you is the formula for the correlation for the population. And I put here notice no good. Not a formula for a variants off the population. But for the parents off the sample that they don't need to use this formula. So think what has to change in this formula. What do you have to change as first? I will just help you with this. It will be the variance off X square. Robi One divided by n the minus one will disappear. It's a simple correction factor. And instead of the eggs bar, you will use the Greek sign. Mean that's all. They stop the video and then we're gonna continue after you made the exercise. So 54321 And stop the video. Welcome back at the video I'm gonna give you now. I hope you did a good job and you made the exercise life perfect. If not, it doesn't matter at all. You're gonna make it to get first. We need this. Three things you need go very ins off the population we didn't calculated before. So we have to calculate it now. I first realized the formula. It looks just the same as above. So it's really easy to rewind it. Using the formula with stating there above instead of x square, you have the X times that why and instead off the mu square, because we deem you off X. In this case, you get new X times, the you off Why and I re within the formula now what I can do is just filling all the all the numbers. It's easy to put behind the X Times y for this part for information x times y extensive eyes. In this case, to this case, three times five is 15. In this case, four times six is 24. And if you send them up, you get for 9 11 and you get 41. So this piece will be 41 one divided by N robi. Want to think of also one divided by three mine and would be free? The efforts of 23 or four will be three and evidence of 15 or six will be four. Next, I'm going to calculate variants of eggs. We need a standard deviation with first parents and we take the square root so long divided by 123 Any three in this case again equals Just ride down with a green collar Wanted about it but free you get the the summation off X ext So x squares or two square will be 43 square will be nine and foursquare will be 16 And if you send them up you will get 29. But this But these will be 29 minus three times C square. You have to do the same. Exactly with why so one square becomes 15 square becomes 25 becomes 36. So, variance, why will be one divided by again? And I want to free times. If you sum up this three numbers, you get six and six is 12 uh, 62 minus and will be three again and ever it will be four. I will. I calculate everything on my calculator, and I will get his answers to third for 2/3. And for the cove Arians, get one through Sirte. So to calculate the correlation, it will be 1 2/3 divided by this crowd off to third times the square root off 4 to 30. Remember, we just calculated variants. We variance is the square squared standard deviations who have to take this square road to get a standard deviations and the ends of will be. I see your 0.945 which will be exactly the same. It's a simple correlation. So, like for Cove Arians, it makes a difference to calculate the the government for the sample or for the population for standard deviation. It makes it different to calculate the standard deviation. For example, afford it population before the correlation. In the end, it doesn't matter. But for your teacher, it's important that you use the right formula so otherwise you can get like deduction off points, so always do it this way. Use a separate formulas for the population used to separate formulas for their sample. You will get always your full points 12. 4: welcome. At the end of this video, you will know exactly what is regression. What is the intercept and slope off a regression line. So first I'm going to show you what I made already for Umno. Any little graph you see a strange country with the temperature possibilities 0 to 3 degrees on what you see is like if the temperature rises, the mounds of isil rises to its normal. In this county, you see like different dots. For example, See, just got here. It was one day with one degree and 20 Eisold. If you have this little dot on this day, it is like a little bit more than one degree. And it's like around 35. I sold on this thought. It was three degrees and around 30 I sold. What you can't see is that the correlation is quite high because you see here the line and the dogs are quite close to it. So it will be a correlation of 0.9, I guess. But it's not that important for now. For now, just important. What is the regression? Regression is there's only distorts. The line is no here yet and out of creation off the best fitted line like this is not the best fitted line through the points. Also know like this that's fitted line must be going as close as possible through all the points. So this would be the best fitted line, and that is regression and waited. Line belongs to kind of formula here I put why expectation? It's like the why better put I sold. And this little hat means expectation and some other studies. They don't write a little hat. They put the assault embraces and put it e from expectation in front of it. So both a possible for now I put ice old hat equals and you see here the 20 and 20 is the constant or the intercept. It's exactly this point where the line goes through the why excess and that it's 20 you see in the next off formula, plus 10 times T and T stands for temperature, and how how I get to this turned its a temperature rise with one. The isil right on every words by 10 is not always because you see a point in the authorized by 20 but in every words. If you just look to this line. The temperature rise by 10 if you go from 1 to 2 temperature guys returning in, probably if you guys from 2 to 3 temperature will goto 40 degrees. And that's the way out of formula works in the next video. I'm going to explain to you like making this line like this is a bit arbitrarily now. The points. A quiet, perfect five, but most of time they're not that perfect. So you have to use statistical formulas in order to calculate a line. I'm going to explain you completely how to calculate slope, how to calculate it, constant or intercepts? Note, Khan said. The intercept exactly the same. Only in some studies. They call it constant in order status, they call it intersects. 13. 4: At the end of this video, you will know the formulas for the slope and for the intercept off regression line. So remember regression line just through older points. Not at first. A straight line is in in mathematics is why is a please be IQ be X in, which is the constant on it Intercepts and B is the slope. So he, for example, if you have 10 here, A will be then. And if you want to do right, you go like five. For example, to 15th and to 15 in the slope will be five. So the line will be. Why equals 10 plus five X is that's It's almost the same. You just denoted a little bit different, he said. Why you want out? Why expectation? Because you're not sure that the points are probably not all on the line. Like what you see. It's like a bit higher orbit low so you don't write down why. But why with a head and the head means expectation. Or again, like explaining last video expectation off why depending on your course equals B zero, so the A the constant intercept is now called B zero and slow it's called B one. So just know why expectation equals these air. Hopeless. Be one times X and that's all you have to remember. Next thing. Recall the formula for the correlation coefficient. It is the co variance divided by the standard deviation off X times. The standard deviation off. Why no For the B one, the formula, the slope of the aggression line is almost the same. It's also the co Vince off X and y divided by the sample standard deviation off X, but not times a standard deviation of why, but again, time sustaining the deviation off X. And a lot of times you see them in studies like as X Y divided by as X square would be off course exactly the same. It s X as X, that's all. And in the next video, we're gonna practice with death next to it. The B zero you calculate with either formula B zero is why average the mean of why miners be one. So you have to calculate the slope first and for the answer in here times X average and thats all if you're good in, if you're good in math, you can remember the formula Like why expectation equals B zero plus B one times X like feeling why expectation is busier. Place be one times x average Onda and B zero equals I ever was minus beat one X efforts minus B one. Thanks. Average this side exactly the same just was easier for you. I always remembered it this way. But if you're good in just commending formulas, please just learn it by heart and you don't need to. They get off this spot. 14. 4: At the end of this video, you will have seen an example of how to calculate the regression slope and the intercept or constant. First, I give you here the two formulas and as example, I will use this numbers again, the same numbers as before. So that 2.1 the 3.5 and 4.6 I won't down the co variance is 2.5. You can calculate it yourself and like find it out in the previous videos where explained to calculated and the standard deviation off X equals one. Um, what I can do now is just fill it in in formula equals 2.5, divided by one times one, which will be exactly 2.5. So if it's about temperature and I sold, if the temperature rises with one that I saw, it will be rising with 2.5 if the temperature will be arriving. But not with one with 10. Then that I sold were we rising with 2.5 times, then will be 25 25 more to get a complete formula. Now we can put in this in the formula it would be Is there a plus B one B one is 2.5 times X. To get the complete formula, you have also cooperated. B zero equals Why ever it's for minus B 12.5 times x are ever it's free equals four minus 7.5 equals minus 3.5. So I expectation equals minus 3.5 plus 2.5 times x. Maybe this looks a bit strange in my example, because I give an example about I sold and I saw cannot be negative. And now you see if the temperature will be zero degrees, The I saw will be negative because expect days of the isil of minus 3.5. Now this regression line is only Vela's in the area with our regression points, so it will be this area. 15. 4: Hello, is this stuff afforded your statistics teacher and this video? I'm going to explain you one off the exercises. This excites about regression on. Most important. Before watching this video, I make the little PdF foul. You can download it, and there are all the exercises together with all the answers, which I'm going to explain in this video. Feel awfully. So it's not that important to What's all this? Exercise videos is more important to just make the exercises yourself on. Check the answers and as soon, if you don't fully understand it, please go to specifically death video. You need to understand the exercise and then go on back to the pdf. I'll go to the next exercise. Try it yourself. And if you don't, if you don't find out like how to make it, um, just what's again a video about that exercise. If you need more exercise about a topic, please please mail me and I will create more exercise on the topic specifically for you. Let's start. He is exercise one. Calculate the regression slope off the following simple data said here, over here, see some information like that as ex white. The governance because eight as standard deviation toe efforts of x 10 and the efforts off Why across 20 Remember, for my last explanation, the slope is to be one s. It's why covert is divided by variance off s standard deviation square. I'm just going toe. Why down b one equals equals it Divided by to square eight divided by four equals to so that aggression slow always has the formula. Why equals a X plus? Let's see how I did it before the zero Please be one x April be X and the beat. We can already feel it will be too a plus two x Now what point has to be on the regression ? Slow is the efforts points X bar and the Wiebe. This is like together one co ordinate which has to be on the slope tohave feeling this two numbers the expo for the X and the Y before the why I can calculate the A. So why 20 a bliss two times then is the same. US 20 is a plus 20. Whatever I said a equals zero. So the world would be guessing slope will be Why equals zero plus two x and the zero. You don't really need the wider Nancy can. Just wide. Only regression, Slobo be. Why is equals two X? Thank you for watching this video. Looking forward to seeing you in the next. But first and sure you practice with all the exercises. Thank you. 16. 4: Hello. This is storm of Order, your statistics teacher. And in this video, I'm going to show you how to the next exercise. It will be exactly the same kind of exercise, like exercise one about creating a regression slope by hand, but it's always good. If the 1st 1 didn't work out, you can practice it again. So let's see how it goes. And now, so calculated progressive slope off the following Simple data co variance equals minus 16. This time the variants, too, so will be to square will be four. Expect that and why by 20. Remember, the formula B one equals Cove Arians, divided by variance or standard deviation. Square equals minus 60 divided by for so the answer will be minus four. It's the regression slope, calculated guesses. Slope Slope will be minus four. Let's continue to the war regression equation. It's not an equation with is one of create the regression equation for you. A plus the X, the B equals in this case minus four. So we get why is a minus for X and over here we can fill in the X Bar and the Y bar because the line has to go to the co ordinates both points. So do I. Buy equals 20 equals a minus four times 10. So would you get over here? Is 20 equals a minus 40 Both sides. Well, maybe easier to write them down on the other side. A minus 40 equals 20. And now on the left side I want to have a So I add 40. A minus 40 plus 40 becomes a can do with it Keep it in balance. So on the right side I also have to add 40 A equals 60 So finally you get there Will regression equation Why equals 60 minus four x? Thank you very much. If you want more videos on on this subject, see the next to exercise I created for you. If after once you want more, please send me a mail and I will create more for you. Thank you. 17. 4: Hello. This is still have order your statistics teacher. In this video, I'm going to explain next exercise and what we have to do a little bit more work. So first and sure, you understand the 1st 2 exercises, and now we're gonna combine it together with the information about previous exercises. Actually, there's, like, two tree exercises in one first calculate the standard deviation or variants, then calculate the co variance. Then we have all the information we need. We can go on into calculating the regression slope, So let's go over here. You see, the exercise could also find a pdf, of course. And furthermore, I won't down for you to formulas we need. No, we gonna fill them in as, um as first you need a formula for beat. One equals a square off ex f B squared is monster. Over here, we have to square the right side to just get rid off the square root. It makes it a bit easier. Now we're gonna fill it all the numbers. What we need over here is the some of all that excess. And we need the X bar. So if you see 357 and nine. He just add them together and divide them by four Tree plus five. Got my calculator over here. Seven plus nine equals 24. Divided by four across six. This x by over here will be six. And this, of course, to next week. We need over here. Why bar? So you see why do over here, too, to calculate its exercise first, wider death formula for this formula to calculate a slow, I will need the S X y and the parents of eggs. And then I avoid this two formulas. And over here I see more things I need to calculate. So it's like the, um upside down matters. I started with the most important things, and I just go down and see what I need That way he needs to remember less. I think so. Makes it much easier for yourself. So let's calculate why Evert seven plus nine plus 16 plus 18 equals 50. Divided by four is 12.5. This one over here is 12.5. All the excess square. Remember, from the previous videos, I would just show you one time or go t 57 and nine, 357 and nineties on the excess Xs Square will be 93 times three is 9 25 49 81 just every over left side number squared. And if you want a summation of them, I just need to send them. So I feel it in my calculated nine plus 25 plus 45 for five story 49 plus 81 filling in the calculator press okay and get 164. So did some X squared equals 164 for this part next to it. Here I need a simple off x times Why it's easier to wife now. Here extends Why three times 77 foot is 21 5 times nine is 45. But order ones. I use my calculator 112. If I go too fast, just pause a video calculated yourself over. Little steps in between can also learn from them. Give you 162 if and his numbers 21 plus 45 plus 112 plus 1 62 equals treat 40. So did is one over Here is 3 40 No, I have no information to calculate and variants and the go various as last. I need the Ansari. The end maybe could make the mistake that you think and its eighth. No. And it's four because I want to treat four passed off observation. So the end here equals four. He's his way to calculate late IHS formulas. Let me rearrange it a little bit for you to go more space on screen is to calculator for left to right Like that's too little groups in here. Is this group one divided by three times You get like this 1 164 miners and was four times and the X bar is six square like this, you can just feel it in your calculator. Easiest way is 1st 1 divided by three equals. You get life, this number and now I use this as an answer. I just press that times and I opened the bracelets 164 minus four times six square and I close to basis Every calculator, it works like this. Maybe not on your phone calculator, but please buy Calculate off at least $10 or 10. You work okay and you get your answer equals 6.667 It's like continuously. Six to turn to be exactly go very ins. Weaken. Do exactly the same. So if you got like Aaron here, stop the video and do with first yourself or the co variance. Otherwise, just watch one divided by three again, the big guy over here equals to 40 now miners four times, six times 12.5 I'm already getting excited. So feeling this again and you want divided by t equals and I do times brace with open treat 40 minus four times six times 12.5 and I press equal and I get answer over here. Toe 13 Builds treaty treaty 31. Do it. I get this two numbers. Let's go back to the most important formula over here to be one. I can just fill in those two numbers into the B one. Formula B one equals 13. Treat Treat treat treat divided by 66 667 equals 56.667 You will see the answer will be 1.9994 Although I said maybe you get like a form or four less depends on how many trees and sixes you fill in over here. Although I said it's around it too, especially if you have the possibility and your calculated to put like to third and wanted inside of it. You get exactly the answer to knowing to be one. We're gonna do the same steps as we did in the last video. Now we know why equals eight um B X Plus A, in which the b want equals two became filling two averages so x Evert's equal six in the y , 12.5 12 points. Five equals two times six plus a 12.5 equals itself, plus a I interchange both sides. So I will get over here 12 plus a equals 12.5. So a equals How do I get to a I suspect on the left side, minus 12. So have to subtract. Also on the right side, minus 12. Get equal 0.5. So the complete regression equation will be Why equals two x plus 0.5. A lot of steps look good to this video again. If you didn't manage, try to do the next exercise last one yourself. It's a lot of reputation also from the former exercises, but that's good keep shopping that way. Finally, you will know everything about statistics. Thank you so much for watching this video and looking forcing in the next one. 18. 4: Hello. This is Stone from Ford in your statistics teacher. And in this video, I'm going to explain to you the next exercise. Let's go quickly through it. And so Awfully. So you understand everything? Let's see. Over here calculated regression equation. The falling co ordinates you get like recording 8.1 dot 100. So X equals one and the y equals 100 five and 23 are billowing together seven and 16 nine and 80. So, like you see, actually, this exercise is the same as the last one, only different numbers. So if you don't, If you didn't get this first step, stop this video and try to do it yourself. Otherwise just wants everything and see where you maybe made a mistake. I will do everything for you so you can just sit down and relax. First thing is regression equation. Why don't regressing equation y equals be explicit A. For this, you will need the be and B you can calculate with this formula the co variance off X Y divided by the variance off eggs, ex or otherwise says Senate deviates of X squared Those two formulas those two inputs you need and he is he like the formulas for it. Notice this formula over here. It's for the variants. If I remove the square, this will be disclosed. Did we need a very And so let's not use the square road and then squared again is like two months time and effort. So what we need over here, it's first. We need end. It's like one to treat four observations. So n equals four in both cases is right before here with then next to it. For these two formulas, we need X square, the symbol of X square. Next two X, I put here X square, put 15 times 5 25 49 and 81. And if I send those numbers, I get this sim off X square, which equals one plus 25 plus 49 plus 81 equals 156 next to it. So this guy over here is 156. All this piece Ah, the end over here equals four again X bar. Evidence of X will be one plus five plus seven plus nine. This phone number added and divided by four equals 5.5 going to that next formula you got here and again for get here. The summation off X times. Why, so I will make here. X White column One times 100 is 100 five times 23 According to my calculator, 115 7 times 16 112 nine times 18 equals 1 62 And if I sem all these numbers, I will get 100 plus 1 15 Bliss 1 12 plus 1 62 for 89 So the sim off X y equals 489. This little piece over here, the end again equals four expired we calculated Ready 5.5 and the Y bar to calculate it to 100. Plus, it's calculate the efforts off this four numbers plus 23 plus 16 plus 18 divided by 4 157 divided by four 39 0.25 Given other collar 39 don't 25 have to calculate this, too. Make it a bit smaller to give the bid more space, and I can just feel it into my calculator. I get like first I coagulate. This part one divided by three, equals zero point safety. 53 times. Embrace it. Open 156 minus four times 5.5 square, close to bracelet and I will get its answer. 11 2 TERT, or 11.66666 seven For the co variance, I was exactly the same. So one divided by three equals times. It's important to do the equal in between here, like first calculate this thing and then used again used the answer button or something. Do you see? Like ants? Every calculator is different to put that in as a number. Otherwise, just put in Singapore 50 50 53 times, braced with open for 89 minus four times 5.5 times 39 points. 25 equals minus 124 Adult. It's t t See gold brands can be negative. Normal variants of standard deviation is always positive. Having just two numbers, I can fill them in in the main formula over here, B one equals minus Want to 4.8. Treaty tree, divided by 11.6667 equals divide 11.6 minus 10 point seven. So I get minus 10.7 or minus stump 1699999 it will be minus 10.7 and next step is to find a in this equation fill in evidence why and efforts of X y equals minus 10.7 x place. Uh, a why Everts equals 39.25 equals minus 10.7 x plus x everts equals 5.5 times 5.5. Plus a little thing I can calculate over here minus 10.7 times 5.5 equals minus 58.85 on the left side. Still have it. 39 for 25 plus a begin. I switched sides and now I go into the A. How do I get here? But plus 58 points, 85 do exactly the same thing on the right side to keep into balance and then get 39.25 plus 58.85 98.1. Feeling this in into the or do you know the creation Get as final answer? Why equals minus 10.7 eggs bliss. 98 point What? You made four exercises. Now about this subject. If you still need more exercises, let me know Andi, I will make more for you off course. If you have problems with a standard deviation or calculatingly co variance, go back to the exercises about standard deviation and co variance and better too exercise first there, but on otherwise, just continue with all my lessons and you'll be yes, the physics expert in no time. Thank you so much for watching my exercise videos. 19. 5: Hello? Yes, your teacher for 40 speaking. And after the end of this lecture, you will know exactly what are linear transformations. Now what I prepare for you. Is he a little question on? I will use it as an example to explain it to you. Question In the some of the efforts, temperature will be 25 degrees Celsius with a standard deviation off 10 degrees Celsius. Maybe you're from a country where you cells is Maybe you're from a country where you use five night. Let's suppose you're from country from Celsius and you explain the 25 degrees cells with a standard deviation of 10 degrees Celsius to so month from a country with fire tonight. So that person will not understand when you talk in Celsius. So the examples you meet so month from the U. S. A. We don't know what decrease cells usar with only know degrees five tonight How many degrees fire And that will be the efforts and the standard deviation First, what you need is a little formula and give it here to you. The grease fire night equals 1.8 times degree Celsius plus 32 and now the rule the formula for linear transformations. So linear transformation, death, my linear transformation. I can explain to you two degrees efforts in degrees fire night and the standard deviation in degrees fire night. So if w. It's April's be X in this case, W and X compete everything in this case, the X will be. Actually, he creates Celsius and w will be the great Fire night. Then expectation off re expectations. The W expectation in degrees fire and ice is the same as five night average equals a plus B times expectation off X and it's the same as evidence off X order, I said Celsius average quite logical. Here you see the formula. So if you see the formula, w is a plus B X, it doesn't really fit because if you look good too with you, see the A, you see the be so you can agree right formula to 32 place 1.8 Celsius and now you can see the A will be 32 to be will be 1.8 and the X will be degree Celsius. So in this case the eight will be 32 and the B will be 1.8. Now we can feel in the formula ever it in fire night equals a plus B times and Everts Celsius. So it was a fire night will be a 32 the 8.1 and ever Celsius waas 25 degrees. And if you put this into your calculator, you ever get the answer 77 degrees fire night in this case to the average of finite will be 77. Do Greece next to it. I'm going to explain to you the standard deviation. Now, first you see standard deviation or the were you in this case, fire night again equals B with the B is in between stripes in between strives unexplained shortly to you. If you have like free in between stripes, it means three absolute absolute is exactly the same as three means like the three think what's in between the stripes? 1,000,000,000? Positive. If you have minus three, it also becomes three because it's minus three. But I made positive, becomes three. So it is just a positive number. Nothing changes. There is a negative number becomes positive. Important to understand In this case the number is already positive to be nothing will happen with it would take care of it when the number is negative. Also, remember, a standard deviation is ever its distance from the mean on distance cannot be negative because it can go both directions. So let's fill this in standard deviation off. That's a movie is it makes for space standard deviation off W So stand deviation off five nights or otherwise, we're going down the S off f equals B. It would be we just find found 1.8 made postive but stays 1.8 times standard deviation off X during the Greece equals 18. So the answer to this question how many degrees fired will be the efforts and the standard deviation efforts will be 77 degrees and the standard deviation 18 degrees. 20. 5: this is your teacher. Don't afford it and he But I have exercise about linear transformation, especially for you. So quietly. Please Greatest exercise. Stop this video and afterwards I'm going to explain it to you to stop the video in 543 to one Stop and welcome back to the video. Now it's time to explain the video to you. If you read the question, you will see the Evert salary for employees 20,000 euro a year. With a standard deviation of 10,000 this year management. He walks the following bonuses. So this will be the effort Sally Summary Everts And this is the standard deviation off the celery this year. Management rewatched the following bonuses for employees Christmas bonus off 1000 euro variable bones off 10% off MPRI salary The first thing you have to do is to make a little formula out of this because here already put for you. If w is April be X, they have to make a formula in the four w is a pres b x There were you will be. That bonus equals a It is just amount. Everyone gets thousands plus be pay attention be, isn't percentage, 10% is the same as 10. Divided by 100 equals 0.1 0.1 times the salary they already own. I would just call itself if this was a bone for you. And now you think you can make the exercise toe, please stop the video and continue make the exercise. So just stop it. No. Welcome back. Now, I'm gonna explain you, uh, the last part of it in this case, the bonus will be that w is a place be times X, then the expectation off in this case, the bonus. So bonus Everts equals a plus B times celery average expectation off X. So the bonus ever it's will be a In this case, a is thousands plus B is 0.1 times the salary Evert and the 70 Everett's. It's 20 thousands. So it will be 1000 plus a tent of 20,000 three thousands. You owe nice bonus. The second question be what is the standard deviation off the bonus standard? Deviation off W W is the in this case, that bonus equals B, but absolute number of to be the B equals 0.1 is already positive so we don't have to change it any more times a standard deviation off the salary. So as bonus equals 0.1 times standard deviation off the salary is 10 thousands. So it will be thousands. It is may be a dot coma is a comma. 21. B: Hello. At the end of this video, you will know the difference between population and sample as first. If you have the war, population of the world is like five 1,000,000,000 off people with is like too much to inquire in an inquiry. You're not gonna ask a simple question to five billion people to know something about how people in the bill think about something. What you will do instead is you will pick some people from the United States who picks on people from Europe will pick some people from a easier on. Then you get like a sample off. For example, 100 people which you ask your questions. There's 100 people will be your sample and the five billion off people will be the world population. In the statistics, you will have the mean off the population and the mean of the sample and your white him down in a different way. The mean off a sample you were down, just as which will be the mean some other studies You were wide them down as X bar and the menu a white is the mean of eggs. But in a population you will use a Greek letter and which will be the mu off X next to it. You have also a standard deviation which will be the s both ex standard deviation. I come later toe what is exactly a standard deviation. I will explain you completely in the population. It will be the Greek as the smoking as off X like it. See before there's also the big Greek s with you Look also like sigma and but that is the summation sign. There's nothing to do with standard deviation eso that has a couple of more para meters like you know Uh, that better be that Be that be I'm going to explain to you later what exactly means But it was especially important if you look to the population there's all Greek letters If you look to the sample they'll American and European letters Thank you for watching this video 22. B: At the end of this video, you will know toe with measurement level Terrible's belong So First, what's important to know it's measurement levels. You can group them into 1st 2 categories. 1st 1 consists off birds or groups of numbers, and the 2nd 1 consist out off numbers. And if because it's out of numbers, you can already called the scale variables. But the scale wants you can the hide out of two sub categories. But first, let's go to the words or the group of enormous, which you can divide into little question. Is there an order? Is there an order in answer possibilities? If no, you get it? No, me, no available, no comes from the word name. What are no mean available? For example, one equals French. Two equals equals Holland's and to United States. I put one on friends, but I could also have put Holland on number one or the United States and number one. It doesn't matter because it's just ah counties where are given a number, The second possibility if there is order yes or at our group off numbers, because groups of numbers always have in order to become organ of Amal's ah as AH example, education level. For example, one means no education to lower then high school three. High school for Matula of five masa degree, he will see like Masari is high rise. Bachelor is higher as high school. It's hide and load in high school and his hired and no education. It would not make sense to put like master on one. Ah, no education onto conventional auntie. To go down to numbers, you get into the skill and, ah, two different kinds, off scale variables. Why are they already called scale? Because in edit for education, for calculation purposes, no minnow order no and scale like the three main groups. And we didn't skill there is interval interval or radio. But all the coca late calculations in between interval and ratio are exactly the same. And what this interval interval it's when zero is. Actually it means something. For example, zero degrees Celsius is a kind off them preacher opposite toe zero meters, which is really nothing. Um, you can also say like zero degrees Celsius is not equal to say, oh degrees fire night because they were the case. Finally, there's something else selfishness, but zero meters is exactly the same as zero mouse because both are really nothing. So ratio means zero is nothing and in the volatile zero is something. 23. B: welcome in this video, I'm going to explain to you the difference between mean the moat and a medium. What's really important at first I will start with the mean is the mean is the same as the efforts or the expectations of those three subjects are exactly the same. So some cities you see the question? I was the mean in order. What is the efforts and what's the expectation? For example, you have the numbers 1113 for eight like you see here And what, how to calculate. And, Evertz, you just sum up all the numbers. One plus one plus one plus street, please. For this Ed in defied him by the total number off numbers. In this case, six. If you do it on your calculator, you will see that the answer will be exactly three. So this will be the mean average our expectation, the next two things You have our moat, the motives quite easy. The motive just did number. What's the most, uh, most picked in this case is one because they're t once only once the only one for only 18 So in this case, the mode will be the one, Um, what's the median? The median is a number in the middle. Now here. There's a little problem with number in the middle because there are in total six numbers. So what's numbers in the middle? The middle line will be like over here. So you got, like, two numbers in the middle of one and the three of them median will then be tree plus one divided by two equals two. Even if the two is not inside there, the median can still be, too, because the middle number is the one in the tea divided by two equals two. I thank you for watching this video. 24. Introduction video: Congratulations. You made your first step in passing your statistics. Course you made it to my introduction off this descriptive statistics course First, what we're gonna do in this course especially what you're gonna do first. What's important is not started scores yet. Which you first need to do is check what subjects will be in your exam. Because I want you to focus on your exam. You have to pass your exam. That's the most important. Not what I'm going to teach. You know it's important what you professor want you to know. So first, check this as soon If you check it, you can stop. You can start learning for my course just like go to all the subjects and skip the ones which are not in your exams. It saves your time and it saves your focus because I want you to be focused on what you what really matters for you. Next step is to make exercises. I made a lot of exercise for you and will still be at exercise to discourse more and more and more. And what I tell you in the exercise, I ask you a question and then I ask you maybe give you a bit of explanation and then I tell you to stop the video. And please, please also really stop the video and make the exercises. Even if you don't know it. Please stop the video for 10 seconds and just think about one. Don't I know yet? And after you noticed You can just continue with looking to my explanations after you've donors really important make the test exam. I don't have a test exam for you because every statistic example be different. So what you need to do is go to your professor, go to your blackboard website, ask your fellow students if there isn't just exam and if there is, you can start making the test exam and see what point you still need to learn. Last but not least, exercise more like the bones. You don't know yet exercise them or and make another test exam. And then finally you're ready to graduate the subject and be don't with statistics forever . Or maybe you go to the next statistics to subject notice. This subject is only for the descriptive statistics. Afterwards, I will add another course and new to me on inferential statistic hypothesis testing a lot off. Septic will be both. So you will need to finish both courses. Thank you very much. 25. M: welcome in this video. I'm going to explain something about a summation sign or the big stigma notation on the summation. Sign your wide it like the Big Sigma. And for example, we have this data said 246 the first day temperature will be two degrees second, therefore, and the third a six. So the mean will be four. Um, if you have to summation sign, I could put here. I want information off X. If you want to calculate, it equals, uh it's the same s eggs Juan Bliss X to place x tree. And if you feel in the numbers two plus four plus six, you will get 12. So the summation of eggs will be in this case 12. Of course, this summation signed doesn't have to be only like, what's a summation of eggs? It can also be what's estimation, off eggs plus two. Now we get to place toe plus four plus two plus six plus two equals is the same as for plus six plus eighth with equals 18. Of course, this will be different as consummation off X plus two, because the summation experts to will be two plus four plus six uh, plus two equals 12 plus two equals 14. What you see more often in the statistics will be the summation off X minus X bar, which equals like the mean off X ah squared eso. The X bar is the same as a mean of eggs. Isn't this case for With will be in this case to minors four square plus four miners, four square plus six minus four square And this summer and the same out of this with equals two minus four minus two square plus zero square plus to square which will be the same as for policy. Hopeless four equals eight. Thank you for watching this video. 26. M: Hello. Welcome at the video about explanation, marks and combination. For example, you get to explanation Mark how to calculators in mathematics. It's not that difficult. It just the same as three times two times one equal six. Or, if you get four explanation moxie yet four times, two times, two times one equals 24. So just counting down till one and then at the product off all the numbers times each other . Um, next to object, I'm going to explain to you which were also used. The explanation. Other combinations. What are combinations? Combination? Actually, how many possible ways can you pick two out of five numbers and put them from low to high? If I get this question, if you look here, you see all the possible answers. Because there's only five new ministers. For example, one in 2131415 20 to 14 to 15 but not five and two with because 52 are the same nemesis. Toe five. And if you count him all, you will see in a total that 10 members. There's ah little formula to calculate that you write it down like this. Officially, five above two means, like, how many possible ways can pick two out of five numerous and put him from low to high. Bam! And you calculated your write it down like this five and see combinations are to, uh, please. In most courses, you can use the calculator, so if you can use it also, please use it. If not, you have to calculate it by heart. I see a lot of studies, actually that on the formula. Uh, you you see this formula, but the formula sheet, I mean, but you can also use the calculator. So then please don't use the formula for if you have to learn it by heart and have to use it. Um, the first, remember, is the end, and the second number is the Are. So you see, like the little formula, which will be five. Question mark invited by to question mark five minus two. Question one equals five. Question mark is 1 25 times four times, three times, two times one to question. Mark is two times one is two and five minus two equals three. So three question mark is three times to sixth in one ST. Six equals 1 20 divided by to mouth equals. Then like you see above the 10 numbers and the answer will also be 10. Thank you for watching this video.