Dale click to view larger table
b) throws two dice
6,12,7,9,4,7,9,6,3, 10
10,9,7,6,9,9,8,9,5,4
4,3,8,4,10,7,2,5,11,6
3,4,7,10, 11,2,6,7,7,12 Make
2,5,8,7,4,10,5,7,7,8
taking the corresponding table of the previous model year
Since the beginning of civilization have been simple statistical forms, as already used illustrations and other symbols in furs, rocks, wooden sticks and cave walls to count the number of people, animals or things. By 3000 BC the Babylonians used small clay tablets to collect data on agricultural production and the goods sold or exchanged through barter. XXXI century BC, long before the pyramids, the Egyptians analyzed data from population and income in the country.
The abacus comes from the need for accurate statistics for the Chinese
The Roman Empire was the first government that collects a large amount of data on population, area and rent of all territories under their control include the trip they had to make Mary and Joseph to Bethlehem for a census count. 
Ascolli Bronze Art: Squadron recruited 30 riders Iberian Salduie (Zaragoza) by the Roman army during the war between Rome with his Italian allies (1991-1989 BC). The statistics of everything that made the service of empire are there.
During the Middle Ages were only a few comprehensive surveys in Europe. Caloringios kings Pepin and Charlemagne ordered to do careful studies of church property in the years 758 and 762 respectively. After the Norman Conquest of England in 1066, King William I of England commissioned a census. The information obtained from this census, conducted in 1086, is included in the Domesday Book.
The registration of births and deaths began in England in the early sixteenth century, and in 1662 came the first significant statistical study population entitled Observations on the London Bills of Mortality ( Comments on the death certificates in London ) written by John Gaunt. A similar study on the mortality rate in the city of Breslau, Germany, conducted in 1691, was used by the English astronomer Edmund Halley as the basis for the first mortality table. In the nineteenth century with the spread of the scientific method to study all the phenomena of natural and social sciences, researchers accepted the need to reduce the information into numerical values \u200b\u200bto avoid ambiguity of the descriptions verbal.
scientist Birth Lamark, France, 1744
Today, statistics has become an effective method to accurately describe the values \u200b\u200bof economic, political, social, psychological, biological or physical, and serves as a tool to correlate and analyze data. The statistical expert's work no longer consists only in collecting and tabulating data, but above all in the process of "interpretation" of that information.
Key Concepts
Raw Data
Latin name for a group of data that has not yet been tested, and which will work to find matches and rules that affect them.
Arrangements rectangular tables that organize the information regarding a particular characteristic. Each column or row is a specific feature of the analyzed information. They are generally used information numbers or short (yes / no, true / false).
Some of its main characteristics are presented in columns:
- Category : The main aspect we are studying, it is usually the first column and never receives the name of "category" in the table, but put the issue to be studied (eg.: Qualification, Liter, Price, Students, Name, etc.).
- Absolute Frequency: The number of times each category is repeated in the raw data.
- Relative Frequency: The ratio of the absolute frequency of this data and total data can be presented in decimal (always less than 1) or percentage. Charts
visual representation of data in a table to better understand how they relate to these data.
The main types of graphs are:
- Polygonal : Line segment connecting the different values, drawing an irregular polygon (origin of name).
- Pictogram : Derivation of the bar graph, only instead a continuous bar drawings get accumulated. If necessary, split the drawings for a more accurate representation.
- Circular or Cake: This chart is the only one not based on Cartesian coordinates, but from the total data represented by a full circle of 360 degrees, each category being represented in frequency relative to the total data.
If this keeps up, he told Robert, we can avoid this kind of mathematics. But it went on, because now it was Albert who wanted to sit-entirely to left.
"But then we have to get up all," said Bettina. I do not see why, but if there is no choice ... Come on, Charlie! And when they returned to sit the thing was this:
course, did not last long.
"I can not take a minute next to Charlie," said Bettina. It really tore the nerves. But as he kept, the other guys had to give. Robert wrote:
"And now enough said. - Do you believe? Asked the Number Devil "Those three have not yet tested all the possibilities. What you seem to Albert sit on the left, Charlie and Bettina in the center right?
"Never! Bettina cried.
"Do not be so," said the old Bettina. Reluctantly, the three got up and sat down like this:
- You see, Robert? Hey Robert, I'm talking about! Surely these three can not think. Robert looked up at the board:
"I think we tried all the possibilities," he said. "I think so too," said Number Devil, "It can not be in your class you may be only four. I fear that there are still few. Even as he said when Doris opened the door. He was breathless.
- What happens here? Is not Mr. Bockel? Who are you? Asked the devil of the numbers. "I'm just here as an exception," said the old man. Your Mr Bockel has taken the day off. You said you could not go. That your class is too moved for him.
"I can say," said Doris, "are all moved. How long is that your site, Charlie? I feel there! "Then gives some seating, Doris," the Number Devil. "I would simply follow the order alphabetical, "she said. A Albert, Betty B, C for Charlie, etc. That would be easier. "Whatever. Intentémoslo.Robert scored on the board:
But others were not at all agree with the order proposed by Doris. In class the devil was loose. Bettina was the worst. Biting and scratching when someone does not want to give your site. Everyone was pushing and prodding. But, over time, that crazy game started to like all four. The change occurred faster and faster, so that Robert could not keep in his notes. Finally, the gang of four had tried all possible orders and on the board read:
Luckily today have not come all, thought Robert, otherwise it would never end. Then the door opened and Enzio, Felicitas, Gerardo, Heidi, Ivan, Karol Jeannine and rushed inside.
- No! Robert shouted. Please, no! Do not you sit! I'll go crazy.
"Okay," said Number Devil, "we'll leave it here. You can go home. No class in the coming hours.
- What about me? Asked Robert.
"You can stay a little longer. The others had run into the yard. Robert looked which put on the board.
"Well, what do you think? Asked the Number Devil.
I do not know. Only one thing is clear: they are growing. Increasingly able to sit. While only two students had the thing still worked. Two students, two possibilities. Three students, six possibilities. With four and are ... a moment ...: twenty-four.
- What if I only had one?
- What nonsense! So naturally, it would have a chance.
-test to multiply, "said the elder.
"Yeah," said Robert. How interesting.
"If more and more people involved in the game, it gets boring and target them. Can also be made shorter. Write the number of participants and an exclamation point behind it:
'is pronounced like this: four pum!
"If we had not sent home Enzio, Felicitas, Gary, Heidi, Ivan, Karol Jeannine and what do you think would have happened?
-A massive confusion, "the devil" They would numbers up to 154 gorging been testing all possible positions, and I can assure you it was something demonic-mind long. Counting Albert, Betty and Charlie had been eleven, and that means eleven pum! opportunities to sit. Do you know how many chances would be? "No one could figure that head. But at school I have always my calculator handy. Secretly, of course, because Mr. Bockel can not stand working with her. And Robert began typing:
- Eleven pum! He said exactly 39,916,800. Nearly forty million! "You see, Robert, if we tried it we would still be here in about eighty years. Long ago that your fellow class would need a wheelchair, and we'd have to-treat eleven nurses to take them back and forth. But with a little mathematics it's going faster. It occurs to me one more thing. Look out the window to see if your classmates are still there.
"I think he will quickly bought ice cream, and now go on the way home. "I guess that will come together to say goodbye.
"No way. How much say goodbye or bye. "Pity," said the Number Devil. I wonder what happens if everyone shakes hands with everyone. - Stop it! Surely that would last forever. Is likely to be a huge number of handshakes. Can eleven pum! if they are eleven.
- Error! "Said the elder. If two, thought Robert, you just need a handshake. With three ... "Better write on the board. Robert wrote:
"Then, with two one, three is three, and four are now six handshakes. -1, 3, 6 ... "Did not know that? Robert could not remember. Then the devil of the numbers painted a few bullet points on the board:
- Coconuts! Robert shouted. Triangular Numbers! - And what followed? "You know:
exactly 55 handshakes. "That still can be calculated," said Robert. "If you do not want to spend so much time calculating, you can do otherwise. Draw circles on the blackboard, thus:
"Then you put one letter in each new circle: A for Albert, Bettina B, C for Charlie, and so on. "Then join the letters with lines:
" It looks bad, right? Each line represents a handshake. You can count them. -1, 3, 6, 15 ... As before, "Robert said. Only one thing I do not understand: can you explain to me why you always block everything?
"That is precisely what demon of Mathematics. It all fits. Well, say better than almost anything. Because you know that numbers have their first hit. And as others have to put a huge attention, because otherwise it is easy to fall in with the whole team. But in general, in mathematics the thing runs well in order. That's what some people hate them. But I can not stand the messy and sloppy, and they are going backwards, do not support the numbers. By the way, look out the window, the patio of your school is a real pen! Robert had to admit it, because the court had coke cans empty, broken comics and snack wrappers everywhere. "If three of you cogierais a broom in half an hour your yard would look much better.
- And who would those three? Asked Robert.
-Albert, Betty and Charlie, for example. Or Doris, Enzio and Felicitas. In addition, we also have to Gerardo, Heidi, Ivan, Jeannine and Karol.
"But you say you only need three.
"Yes," said the Number Devil, "but what about three?
"They can be mixed at will "Said Robert.
"No doubt. But what if not all? If only we had three: Albert, Betty and Charlie? "Then you have to do them.
- Well, write it! Robert wrote:
"And if then comes Doris, what do we do? Once again there are several possibilities. Robert thought. Then he wrote on the blackboard:
"Four possibilities," he said. "But Enzio casually walks by. Why will not lend a hand? Now we have five candidates. Test.
But Robert refused. "Better tell
what you will get demoralized-do said.
"Okay. With three people can only form a group of three. With four people and there are four distinct groups, and five are ten. I'll write:
"There is another rare thing in this list. I have ordered, do as the alphabet, you see. How many groups begin with Albert? Ten. How many of Bettina? Four. And Charlie does not start more than one. In this game up again and again the same numbers:
- Guess what's next? I mean, if we now add a few more, say Felicitas, Gerald, Heidi, etc. How many groups of three would go?
"No idea," said Robert. - Do you still remember how we flow the issue of shaking hands, when everybody was saying goodbye to everyone?
"That was easy, with the help of triangular numbers:
" But not good for our cleaning crews who work in threes.
-No. But what happens if you add the first two triangular numbers? -Sale four. - What if you add next?
-Diez. - And another one? -10 + 10 = 20.
"There you go. - And I have to continue to calculate until the eleventh? This is not your way of doing things.
"Do not worry. Can also be done without calculating, unproven, without ABCDEFGHIJK. - How?
triangle "With our old number," said the elder. - Are you going to paint it on the board?
-No. I'm not thinking any such thing. I would be too boring. But I have my cane by hand. Touched the board with his staff, and there was a triangle, in all its splendor and four colors.
"could not be easier," said the old devil of the numbers. When shaking hands, just the green cubes accounts from top to bottom with two people squeezed hand, with three people three, with eleven 55. "For our cleaning crew need the red cubes. Retell up and down. You start with three people with them no more than a possibility. If you choose four people you have four combinations, with five people and ten. What happens when eleven students? "Then they said 165-Robert. It's really simple. This triangular numbers is almost as good as a calculator. But what are they yellow cubes? "Oh," said the old man, and I know I am easily satisfied. We, the demons of the numbers, always taking everything to extremes. What will you do if the three people who have not enough for the job? Have to take four. And the yellow line will tell you how many possibilities there are, for example, to choose a quartet from eight people. "Seventy," said Robert, because he understood very well how easy it was to get the response of the triangle. "Exactly," said the Number Devil. Not to mention the blue cubes. "There are probably groups of eight. If you only have eight people, I have to think hard. There is only one possibility. But with ten candidates and I can form 45 different groups. And so forth.
"I see you got it."
"Now just let me know what it looks like the yard, "said Robert. He looked out the window, and behold, the yard was immaculate as ever.
"I just wonder what three will now broom. "Anyway, you're not one of them, my dear Robert," said the Number Devil.
- How am I going to sweep the school yard if I have to spend all night fighting with numbers and cubes!
-accepted, "said the old man that you have fun doing it.
- What now? Will you come back soon? "Before I take a vacation," said the Number Devil.
"In the meantime, you can be entertained by Mr Bockel. That was something that he wanted Robert very little, but what choice did he have? The next morning I had to return to school. When he reached the classroom, Albert, Betty and the others were sitting on their sites. Nobody was willing to change your site to others.
"Here comes our math genius Charlie exclaimed. "The good thing about Robert studied even in dreams," she prodded Bettina. - Do you think that is going to count for something? Asked Doris. "I do not think Karol cried. Anyway, Mr. Bockel not support them. And vice versa, "said Robert. For me that again! Before he arrived Mr Bockel, Robert took a quick look out the window. As always, he thought to see the patio. A real lot of garbage! You can not trust the things that dreams. Only numbers. They can be trusted. Then came the inevitable Mr. Bockel, with his briefcase full of twists
