Kahjin’s Weblog


Posted in 未分類 by kahjin on 3月 26, 2008


Mathematical treatment of shape also includes graphical depiction of numerical and symbolic relationships. Quantities are visualized as lengths or areas (as in bar and pie charts) or as distances from reference axes (as in line graphs or scatter plots). Graphical display makes it possible to readily identify patterns that might not otherwise be obvious: for example, relative sizes (as proportions or differences), rates of change (as slopes), abrupt discontinuities (as gaps or jumps), clustering (as distances between plotted points), and trends (as changing slopes or projections). The mathematics of geometric relations also aids in analyzing the design of complex structures (such as protein molecules or airplane wings) and logical networks (such as connections of brain cells or long-distance telephone systems).




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Posted in 未分類 by kahjin on 3月 25, 2008


Coordinate systems are essential to making accurate maps, but there are some subtleties. For example, the approximately spherical surface of the earth cannot be represented on a flat map without distortion. Over a few dozen miles, the problem is barely noticeable; but on the scale of hundreds or thousands of miles, distortion necessarily appears. A variety of approximate representations can be made, and each involves a somewhat different kind of distortion of shape, area, or distance. One common type of map exaggerates the apparent areas of regions close to the poles (for example, Greenland and Alaska), whereas other useful types misrepresent what the shortest distance between two places is, or even what is adjacent to what.




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Posted in 未分類 by kahjin on 3月 22, 2008


Geometrical relationships can also be expressed in symbols and numbers, and vice versa. Coordinate systems are a familiar means of relating numbers to geometry. For the simplest example, any number can be represented as a unique point on a line—if we first specify points to represent zero and one. On any flat surface, locations can be specified uniquely by a pair of numbers or coordinates. For example, the distance from the left side of a map and the distance from the bottom, or the distance and direction from the map’s center.




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Posted in 未分類 by kahjin on 3月 21, 2008


Both shape and scale can have important consequences for the performance of systems. For example, triangular connections maximize rigidity, smooth surfaces minimize turbulence, and a spherical container minimizes surface area for any given mass or volume. Changing the size of objects while keeping the same shape can have profound effects owing to the geometry of scaling: Area varies as the square of linear dimensions, and volume varies as the cube. On the other hand, some particularly interesting kinds of patterns known as fractals look very similar to one another when observed at any scale whatever—and some natural phenomena (such as the shapes of clouds, mountains, and coastlines) seem to be like that.




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Posted in 未分類 by kahjin on 3月 19, 2008




Spatial patterns can be represented by a fairly small collection of fundamental geometrical shapes and relationships that have corresponding symbolic representation. To make sense of the world, the human mind relies heavily on its perception of shapes and patterns. The artifacts around us (such as buildings, vehicles, toys, and pyramids) and the familiar forms we see in nature (such as animals, leaves, stones, flowers, and the moon and sun) can often be characterized in terms of geometric form. Some of the ideas and terms of geometry have become part of everyday language. Although real objects never perfectly match a geometric figure, they more or less approximate them, so that what is known about geometric figures and relationships can be applied to objects. For many purposes, it is sufficient to be familiar with points, lines, planes; triangles, rectangles, squares, circles, and ellipses; rectangular solids and spheres; relationships of similarity and congruence; relationships of convex, concave, intersecting, and tangent; angles between lines or planes; parallel and perpendicular relationships between lines and planes; forms of symmetry such as displacement, reflection, and rotation; and the Pythagorean theorem.





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Posted in 未分類 by kahjin on 3月 16, 2008


Two quantities are positively correlated if having more of one is associated with having more of the other. (A negative correlation means that having more of one is associated with having less of the other.) But even a strong correlation between two quantities does not mean that one is necessarily a cause of the other. Either one could possibly cause the other, or both could be the common result of some third factor. For example, life expectancy in a community is positively correlated with the average number of telephones per household. One could look for an explanation for how having more telephones improves one’s health or why healthier people buy more telephones. More likely, however, both health and number of telephones are the consequence of the community’s general level of wealth, which affects the overall quality of nutrition and medical care, as well as the people’s inclination to buy telephones.





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Posted in 未分類 by kahjin on 3月 14, 2008


More generally, averages by themselves neglect variation in the data and may imply more uniformity than exists. For example, the average temperature on the planet Mercury of about 15o F does not sound too bad—until one considers that it swings from 300o F above to almost 300o F below zero. The neglect of variation can be particularly misleading when averages are compared. For example, the fact that the average height of men is distinctly greater than that of women could be reported as “men are taller than women,” whereas many women are taller than many men. To interpret averages, therefore, it is important to have information about the variation within groups, such as the total range of data or the range covered by the middle 50 percent. A plot of all the data along a number line makes it possible to see how the data are spread out.



We are often presented with summary data that purport to demonstrate a relationship between two variables but lack essential information. For example, the claim that “more than 50 percent of married couples who have different religions eventually get divorced” would not tell us anything about the relationship between religion and divorce unless we also knew the percentage of couples with the same religion who get divorced. Only the comparison of the two percentages could tell us whether there may be a real relationship. Even then, caution is necessary because of possible bias in how the samples were selected and because differences in percentage could occur just by chance in selecting the sample. Proper reports of such information should include a description of possible sources of bias and an estimate of the statistical uncertainty in the comparison.


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Posted in 未分類 by kahjin on 3月 11, 2008


The most familiar statistic for summarizing a data distribution is the mean, or common average; but care must be taken in using or interpreting it. When data are discrete (such as number of children per family), the mean may not even be a possible value (for example, 2.2 children). When data are highly skewed toward one extreme, the mean may not even be close to a typical value. For example, a small fraction of people who have very large personal incomes can raise the mean considerably higher than the bulk of people piled at the lower end can lower it. The median, which divides the lower half of the data from the upper half, is more meaningful for many purposes. When there are only a few discrete values of a quantity, the most informative kind of average may be the mode, which is the most common single value—for example, the most common number of cars per U.S. family is 1.


a small fraction of people who have very large personal incomes can raise the mean considerably higher than the bulk of people piled at the lower end can lower it.の部分はかなり端折ったが、この部分は私が個人的に英語の表現が面白いと感じたところなので、英語のまま理解した方がいいのではないかと思い、直訳は避けた。これってパレートだかパレードとかの法則っていうんだっけ?



How To Analyze Data Using the Average



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Posted in 未分類 by kahjin on 3月 9, 2008


Summarizing Data

Information is all around us—often in such great quantities that we are unable to make sense of it. A set of data can be represented by a few summary characteristics that may reveal or conceal important aspects of it. Statistics is a form of mathematics that develops useful ways for organizing and analyzing large amounts of data. To get an idea of what a set of data is like, for example, we can plot each case on a number line, and then inspect the plot to see where cases are piled up, where some are separate from the others, where the highest and lowest are, and so on. Alternatively, the data set can be characterized in a summary fashion by describing where its middle is and how much variation there is around that middle.

データの集約(Summarizing Data)

私たちの回りは情報で溢れている。多くの場合、その中には、私たちには理解できないようなものが大量に含まれている。データの集まりというのは情 報の重要な側面が見え隠れする可能性のあるいつくかの集約した性質で表現できる。統計学は大量のデータを整理し、解析するための有用な方法が日夜開発され ている数学の一つの分野である。例えば、データの集まりからどのようなことが考えられるのかを調べるために、数直線上にそれぞれの場合分け(数と対応)し た点をプロットすることができる。そして次に、同じような点がどれくらいあるのか?、点と点の間隔はどれくらい離れているのか?や一番高い点や一番低い 点はどれか?などプロットをじっくりと調べることができる。その代わりに、データの集まりの中央はどこにあるのか?やその中央に対してどれくらいのばら つきがあるのか?を記述することによって、データの集まりを集約したある性質を浮かび上がらせることができるのである。


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Posted in 未分類 by kahjin on 3月 6, 2008


A second major concern that determines the usefulness of a sample is its size. If sampling is done without bias in the method, then the larger the sample is, the more likely it is to represent the whole accurately. This is because the larger a sample is, the smaller the effects of purely random variations are likely to be on its summary characteristics. The chance of drawing a wrong conclusion shrinks as the sample size increases. For example, for samples chosen at random, finding that 600 out of a sample of 1,000 have a certain feature is much stronger evidence that a majority of the population from which it was drawn have that feature than finding that 6 out of a sample of 10 (or even 9 out of the 10) have it. On the other hand, the actual size of the total population from which a sample is drawn has little effect on the accuracy of sample results. A random sample of 1,000 would have about the same margin of error whether it were drawn from a population of 10,000 or from a similar population of 100 million.





次は、[9-26] Summarizing Dataをやります。



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