December 9, 2008 at 19:35Task from an old book. (Iron tables)
Wandering around the used-up ruins for a ridiculous price of 35 rubles. bought a book by Hugo Steinghaus, "Tasks and Thoughts." There he found a good set of mathematical problems suitable for teaching computer science. But I do not want to share this discovery with the Habrasociety.
There was also a random sampling problem. The task is classical. I rephrase it a little. But the essence of this does not change.
So ... In the warehouse are 1000 identical boxes. In the boxes mixed nails and screws. Screws are 10 times more expensive than nails. How much can I sell the goods for?
Solution One (Bad) We take the first 10 boxes. We sort, evaluate, extrapolate the result to all boxes. What a bad decision. We do not know the law by which screws and nails are distributed. Maybe with the time of filling the warehouse the number of screws decreased.
The second solution (classic). We take a random sample of 10 boxes, then according to the algorithm. Than this decision is not absolutely. Random numbers are random and can accidentally lie in a bunch.;) Of course, the randomness of this randomness is strictly determined mathematically, but nevertheless, “Case is the Devil’s tool”.
The third solution (cool). The book provides a method using "iron" tables. I myself have never met the term itself, but I am not in the subject. So I'm waiting for your comments.
Iron tables are built in an interesting way. All of you (especially the designers) could hear about the “golden section.” [Golden is a section in which the ratio of the length of a segment to its greater part is equal to its ratio to a smaller one]. If the length of the segment = 1 then the length of the majority w = 0.618033 ..., the number is irrational. Therefore, in the sequence 1 * w, 2 * w, ..., 10000 * w there are no integers, i.e. each number consists of an integer and a fractional part. Sorting our sample by increasing the fractional part, we get an interesting sample of numbers from 1 to 10000. (Here I have to completely rely on Hugo Steinghaus - I haven’t checked it myself yet.)
1. The numbers are arranged in such a way, the slightly different numbers are located far from each other, not too few separated by average distances.
2. This property is preserved if we take not the entire table, but only numbers lying in a certain interval.
Thus, for selection, we must count ten numbers in an interval from X to X + 1000 from an arbitrary place in our tables, where x is an arbitrary number from 0 to 9000 = 10000-1000.
Next, we act according to the algorithm: open, sort, evaluate.
Honestly, this method made a strong impression on me. Not the sampling method using an irrational number, but the underlying property of the golden section of such a perfectly distributed number. I felt like an ancient Pythagorean froze in ecstasy before the divine revelation revealed in numbers.
BUT enough lyrics. How I plan to use the revelation given to me.
I plan to give it to students for analysis. Let him explore it (so far purely numerical and statistical methods). We will draw up the results in a scientific report, bring the children to a conference, etc. Therefore, I WAIT for your reaction (tips, suggestions, and most importantly, maybe the topic is trivial, I just don’t know it). In general, I'm waiting.
How can you use it? - Well I do not know. Count the nails and screws stored in the cabinet or something! :)
Threat (in the morning). For those who do not understand what the trick is.
We construct an iron table up to 10.
We construct a table number - fractional part (number * w)
1-0.618
2-0.236
3-0.845
4-0.472
5-0.090
6-0.708
7-0.326
8-0.944
9-0.562
10-0.180
Sorting by the second column, we get the row
5,10,2,7,4,9,1,6,3,8.
These are by no means random numbers. Compare the distance between 10.9.8
or 5.4.3 . Therefore, a random or root of 2 is worse for sampling - they can give a bunch, but this distribution is not.
But to statistically evaluate whether it is better or not - this is what we have to do.
At the same time, children will learn something.
There was also a random sampling problem. The task is classical. I rephrase it a little. But the essence of this does not change.
So ... In the warehouse are 1000 identical boxes. In the boxes mixed nails and screws. Screws are 10 times more expensive than nails. How much can I sell the goods for?
Solution One (Bad) We take the first 10 boxes. We sort, evaluate, extrapolate the result to all boxes. What a bad decision. We do not know the law by which screws and nails are distributed. Maybe with the time of filling the warehouse the number of screws decreased.
The second solution (classic). We take a random sample of 10 boxes, then according to the algorithm. Than this decision is not absolutely. Random numbers are random and can accidentally lie in a bunch.;) Of course, the randomness of this randomness is strictly determined mathematically, but nevertheless, “Case is the Devil’s tool”.
The third solution (cool). The book provides a method using "iron" tables. I myself have never met the term itself, but I am not in the subject. So I'm waiting for your comments.
Iron tables are built in an interesting way. All of you (especially the designers) could hear about the “golden section.” [Golden is a section in which the ratio of the length of a segment to its greater part is equal to its ratio to a smaller one]. If the length of the segment = 1 then the length of the majority w = 0.618033 ..., the number is irrational. Therefore, in the sequence 1 * w, 2 * w, ..., 10000 * w there are no integers, i.e. each number consists of an integer and a fractional part. Sorting our sample by increasing the fractional part, we get an interesting sample of numbers from 1 to 10000. (Here I have to completely rely on Hugo Steinghaus - I haven’t checked it myself yet.)
1. The numbers are arranged in such a way, the slightly different numbers are located far from each other, not too few separated by average distances.
2. This property is preserved if we take not the entire table, but only numbers lying in a certain interval.
Thus, for selection, we must count ten numbers in an interval from X to X + 1000 from an arbitrary place in our tables, where x is an arbitrary number from 0 to 9000 = 10000-1000.
Next, we act according to the algorithm: open, sort, evaluate.
Honestly, this method made a strong impression on me. Not the sampling method using an irrational number, but the underlying property of the golden section of such a perfectly distributed number. I felt like an ancient Pythagorean froze in ecstasy before the divine revelation revealed in numbers.
BUT enough lyrics. How I plan to use the revelation given to me.
I plan to give it to students for analysis. Let him explore it (so far purely numerical and statistical methods). We will draw up the results in a scientific report, bring the children to a conference, etc. Therefore, I WAIT for your reaction (tips, suggestions, and most importantly, maybe the topic is trivial, I just don’t know it). In general, I'm waiting.
How can you use it? - Well I do not know. Count the nails and screws stored in the cabinet or something! :)
Threat (in the morning). For those who do not understand what the trick is.
We construct an iron table up to 10.
We construct a table number - fractional part (number * w)
1-0.618
2-0.236
3-0.845
4-0.472
5-0.090
6-0.708
7-0.326
8-0.944
9-0.562
10-0.180
Sorting by the second column, we get the row
5,10,2,7,4,9,1,6,3,8.
These are by no means random numbers. Compare the distance between 10.9.8
or 5.4.3 . Therefore, a random or root of 2 is worse for sampling - they can give a bunch, but this distribution is not.
But to statistically evaluate whether it is better or not - this is what we have to do.
At the same time, children will learn something.