 Hi, I'm Zor. Welcome to a new Zor education. I would like to consider yet another statistical problem where we will try to establish certain probabilistic distribution of the random variable based on some statistical data. This is so-called task C, as I call it. The task C is actually when the random variable has continuous distribution within certain reasonable boundaries. And we know the statistics from the values of this random variable which it took through a certain period of time. Now, as you see, we have lots of numbers here. So, I will actually do the practical part of this more than theoretical. The previous lecture was discrete distribution. I was actually doing a lot of theory of probability stuff with variables, means, etc., etc. Here I will just do exactly the same, but I will just present the numbers as they are. I assume that all the manipulations which are needed to produce these numbers are known to you, and you can use obviously Excel spreadsheet, which I did to calculate these values. So, let me tell you about exact practical situation. Well, everybody is talking right now about climate change, about rising temperature, rising level of sea level. And I just wanted to verify whether these claims true or not. So, I found the website where information about the sea level was provided. It's University of Hawaii. I referred to this site in the notes to this lecture on Unisor.com. So, what I did was I took two points. One point is in California, and another is Atlantic City in New Jersey. So, one is Pacific Ocean Coast, another is Atlantic Ocean Coast. Now, this website, which I just talked about, provides daily levels of the sea level. I think it measured at noon every day, almost every day. I mean, there are some gaps, but statistically, it's insignificant. So, at noon every day, in California, it's from 1933 to 2014. And in Atlantic City, it's from 1911 to 2014, so it's basically a century. And then I just made whatever the necessary statistical manipulation with these data I could do, including the distribution of the probabilities among certain intervals, which I will just explain what it is. And let me just basically jump to the end of it, the conclusion. What's interesting is that there is no statistically significant rising of the level of the ocean in California. It's practically on the same level throughout the whole period from 1933 to 2014. At the same time, the Atlantic Coast in Atlantic City, New Jersey, there is definitely statistically significant rising of the water. Approximately, it rose by something like 400 millimeters in a century. So, it's about four millimeters a year, something like this, four or five millimeters a year. Whether it's significant or not, it's not up to me to decide. Whether the trend will or will not continue in the future, nobody knows. And if anybody tells you that they will definitely know that this will continue, that's not the fact. There is no theory which would basically provide the basis for this conclusion. But whatever did happen up to 2014, that's the date where I have, it definitely did happen and the water was definitely rising slowly but surely during an entire century. So, I understand people who are assuming and that's reasonable that this process will continue, well, until it slows down and maybe will stop completely or maybe nobody knows actually. But I do understand people who are claiming this. But at the same time, I would like you to pay attention to the statistical results for California Coast. There is absolutely no increase in the level of Pacific Ocean, at least in that particular place. All right, so what did I do with the data? So, as you see, here are decades, basically. It's 10 years, 10 years, 10 years. Well, except the first one, it's 13 years because it's the beginning of the data. So, it's 13 years and then everything else is 10. So, what I'm doing is I'm averaging results during a decade. I have results for each day, or almost each day within this period of 10 years, which amounts to more than 3,000 elements of data, which is significant amount for statistical research. And then I am calculating certain statistical parameters for each particular decade based on daily information which is available. So, by the way, this spreadsheet is also on my website and you can download it from the website. The reference, again, is in the notes for this lecture. So, the first column is the average for a decade. As you see, the average in the first decade, around 1920, was 1890. It's in millimeters relative to certain level, whatever the beginning of their scale is. And as you see, gradually it's increasing from decade to decade, monotonically, actually. I mean, that's kind of unusual, but there is a monotonic increase in the average. Now, well, can it be an accident? Well, in theory it can if the distribution is very wide. Now, is this a wide distribution? Well, it's very easy to check. I got the standard deviation within each of them. And as you see, standard deviation is practically the same throughout the whole history. And if you will take 2 sigma, for instance, 2 sigma is approximately like 300 or 320, whatever. It's definitely less than the difference between these two averages, right? Which means that this average is statistically significantly different, in this case, greater. Because it's above the 2 sigma around this value, right? So, what I can definitely say that there is a statistically significant with certainty greater than 95%, that there is an increase during this century between the average decade temperature in the beginning of the 19th century and in the beginning of the 20th century. Statistically significant with the certainty of greater than 95%. So, that's why standard deviation. Now, then, within each decade, the sea level obviously was not like a straight line. It's a little bit above average, a little bit below average, etc. So, I can actually calculate the slope of this distribution within each decade. And obviously, since my average goes from decade to decade up and up and up, I should expect that there is a positive slope, which means slight increasing within each decade. And yes indeed, as you see, this is positive, this is positive, this is positive. There is only one decade, which seems to be a little bit negative, but doesn't really significantly disturb the whole picture. As you see, all the slopes in all these decades are positive. And in one case, it's really very, it's much bigger than others. You see, this is like 5% slope, which is a lot. This is one and a half, two and a six, one and a half. I mean, all others except these two are relatively on the same level. By the way, the slope has diminished. As you see, this, this and this, it's less than whatever it was in the 1960s, 70s and 1980s. What's next? Well, next is minimum and maximum value within each decade. Now, as you see, the first decade has minimum and maximum from 966 to 2801. And then all these minimums are rising and all the maximum are, well, not exactly monotonically, but still you see that they are rising. So again, it's just a confirmation that the sea level is rising. And then I made certain calculations to get statistical distribution among different levels. You see, this is the level from minimum to 1460, then from 1460 to 1970, etc., etc., up to 3,500. Because these are basically, these values encompass my minimum and maximum. Minimum is 966, maximum is 3392. So I got it from like 900 or 950 up to 3,500. And then I have put the number of statistical observation which falls into each of these categories. So as you see, most of the statistical distribution in the beginning of the 20th century was in this area from 1460 to 1970. A little bit less, well, not a little bit, half of that, was in the next interval. And then as the time goes from decade to decade, the whole volume is shifting from this to this. So the number of values which are falling below, let's say 1460, is actually going down to zero. And this number below 1970 is also generally speaking decreasing, while this number, which is the next category, is increasing. So we are observing the shift of the distribution of probabilities from lower level to a higher level. So it's more frequently occurring now in the higher category than the lower. And same thing happened with this one. This is even higher category and it also shifted upwards. So the observations are moving to the right towards the greater C levels, higher C levels. And then the last category is not actually very populated at all. So basically most observations are concentrated in these three categories. And there is a definite shift from this category to these two during these centuries, sorry, decades. So that's the story about New Jersey. I did not put the numbers, there are similar numbers for California coast. But the numbers are completely different in their quality, which actually means that the average is relatively the same. And again, you can download it from my website, which I am referring to in the notes in Unizord.com. And you will see that basically the average stays relatively the same. Standard deviation, as in this case, is more or less the same. Slopes are in some cases positive, in some cases negative, but very, very small slopes. And there is no general observable rising of the sea level in California. Then again, minimum and maximum are relatively the same as we move from decade to decade on the California coast. And the distribution is also relatively the same. Well, now, can I make any kind of a projection for the future? No, absolutely not. But I do observe the situation as it is right now. That the level of the waters in the ocean in Atlantic City is rising about four millimeters a year on average. And in California, it's not really observed at all. Is it related to global warming, climate change? I don't know, nobody knows. And there are so many factors which are actually influenced, for instance, the rising of water in Atlantic City. The direction of the winds, for instance, like general direction, which might have changed. I mean, you can always say that this is a climate change. Yes, climate is always changing. Is it related to human activity? Well, everything is related in some way or another. The question is by how much? The Gulf Stream is slightly changing its direction, as I've read in one of the articles, which also contribute probably to this rising of the water. So there are many factors, and I'm not going to fantasize about which is exactly the most important, etc. But whatever it is here is just pure mathematics, pure statistical calculations, which we can definitely say it confirms that in one particular spot in Atlantic City and Atlantic Ocean, the waters are rising. And it's confirmed again absolutely statistically correctly that there is no similar activity on the Pacific Coast in California. Well, that's about everything which I wanted to present to you. What I think is very important is for you to basically repeat these calculations yourself. They are not difficult, and if you know how to use Excel, that would be even simpler. So just download the raw data from the site which I provide on Unisor.com website, download them, put them into a spreadsheet, and try to manipulate with this to get the average, to get the standard deviation, and some other things, whatever you can. It will take some time. It took some time for me, by the way. But I think it's very important to have your own personal understanding of what are these processes and how they're happening. Don't get involved in any kind of speculation about the reasons, because it's a very, very complex subject. But you can definitely say that statistically something is confirmed, or statistically something is not confirmed, or statistically confirmed the opposite or something like this. So statistically it's confirmed, for instance, that there is a rising water in Atlantic, and there is absolutely no such effect in Pacific Ocean. That's about it. So with this recommendation, I would like to finish this lecture. I wish you really spend some time and familiarize yourself with the techniques which allow to calculate all these numbers. It's really very, very helpful, and you will probably see in a different light whenever somebody is just putting in an article some statements which probably cannot be confirmed statistically, and they represent somebody's subjective opinion, which is fine, but subjective opinion should be treated as a subjective opinion. This is objective. So I would urge you to always think about whether there are some facts and statistical calculations which really prove this or that particular subjective opinion. Or maybe it's just an opinion based on some graph, and as you understand, the graph can be presented in many different ways with the scaling factors, etc., which basically distort the impression the numbers do not lie if they are calculated correctly. The visual presentation, like a graph or something, it can be distorted using some scaling of the graph. So I would urge you to very carefully approach these opinions based just on the look of the graph. Numbers are much more important. And for this, you need to really understand how they are derived. Well, that's it for today. Thank you very much and good luck.