数模论坛

 找回密码
 注-册-帐-号
搜索
热搜: 活动 交友 discuz
查看: 6049|回复: 2

数学专业英语-The Normal Distribution

[复制链接]
发表于 2004-5-6 09:43:36 | 显示全部楼层 |阅读模式
< ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
< ><FONT face="Times New Roman" size=3>We shall begin by considering some simple continuously variable quantity like stature.We know this varies greatly from one individual to another ,and may also expect to find certain average differences between people drawn from different social classes or living in different geographical areas,etc.Let us suppose that a socio-medical survey of a particular community has provided us with a representative sample of 117 males whose heights are distributed as shown in the first and third columns of Table 1.</FONT></P>
< ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman"><FONT size=3> <p></p></FONT></FONT></P>
<P ><FONT face="Times New Roman" size=3>Table 1.Distribution of stature in 117 males </FONT></P>
<TABLE  cellSpacing=0 cellPadding=0 border=1>

<TR>
<TD  vAlign=top width=91>
<P ><FONT face="Times New Roman">Absolute </FONT></P>
<P ><FONT face="Times New Roman">Height</FONT></P>
<P ><FONT face="Times New Roman">(m)</FONT></P></TD>
<TD  vAlign=top width=108>
<P ><FONT face="Times New Roman">Working </FONT></P>
<P ><FONT face="Times New Roman">measurements </FONT></P>
<P ><FONT face="Times New Roman">with origin</FONT></P>
<P ><FONT face="Times New Roman">at 1.70(x)</FONT></P></TD>
<TD  vAlign=top width=96>
<P ><FONT face="Times New Roman">Number of </FONT></P>
<P ><FONT face="Times New Roman">men</FONT></P>
<P ><FONT face="Times New Roman">observed(f) </FONT></P></TD>
<TD  vAlign=top width=96>
<P ><FONT face="Times New Roman">Contributions</FONT></P>
<P ><FONT face="Times New Roman">to the sum </FONT></P>
<P ><FONT face="Times New Roman">(f x )</FONT></P></TD>
<TD  vAlign=top width=120>
<P ><FONT face="Times New Roman">Contributions</FONT></P>
<P ><FONT face="Times New Roman">to the sum of </FONT></P>
<P ><FONT face="Times New Roman">squares (f </FONT><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">)</FONT></P></TD></TR>
<TR>
<TD  vAlign=top width=91>
<P ><FONT face="Times New Roman">1.58</FONT></P>
<P ><FONT face="Times New Roman">1.60</FONT></P>
<P ><FONT face="Times New Roman">1.62</FONT></P>
<P ><FONT face="Times New Roman">1.64</FONT></P>
<P ><FONT face="Times New Roman">1.66</FONT></P>
<P ><FONT face="Times New Roman">1.68</FONT></P>
<P ><FONT face="Times New Roman">1.70</FONT></P>
<P ><FONT face="Times New Roman">1.72</FONT></P>
<P ><FONT face="Times New Roman">1.74</FONT></P>
<P ><FONT face="Times New Roman">1.76</FONT></P>
<P ><FONT face="Times New Roman">1.78</FONT></P>
<P ><FONT face="Times New Roman">1.80</FONT></P>
<P ><FONT face="Times New Roman">1.82</FONT></P>
<P ><FONT face="Times New Roman">1.84</FONT></P></TD>
<TD  vAlign=top width=108>
<P >-<FONT face="Times New Roman">6</FONT></P>
<P >-<FONT face="Times New Roman">5</FONT></P>
<P >-<FONT face="Times New Roman">4</FONT></P>
<P >-<FONT face="Times New Roman">3</FONT></P>
<P >-<FONT face="Times New Roman">2</FONT></P>
<P >-<FONT face="Times New Roman">1</FONT></P>
<P ><FONT face="Times New Roman">0</FONT></P>
<P ><FONT face="Times New Roman">1</FONT></P>
<P ><FONT face="Times New Roman">2</FONT></P>
<P ><FONT face="Times New Roman">3</FONT></P>
<P ><FONT face="Times New Roman">4</FONT></P>
<P ><FONT face="Times New Roman">5</FONT></P>
<P ><FONT face="Times New Roman">6</FONT></P>
<P ><FONT face="Times New Roman">7</FONT></P></TD>
<TD  vAlign=top width=96>
<P ><FONT face="Times New Roman">1</FONT></P>
<P ><FONT face="Times New Roman">3</FONT></P>
<P ><FONT face="Times New Roman">6</FONT></P>
<P ><FONT face="Times New Roman">8</FONT></P>
<P ><FONT face="Times New Roman">13</FONT></P>
<P ><FONT face="Times New Roman">18</FONT></P>
<P ><FONT face="Times New Roman">19</FONT></P>
<P ><FONT face="Times New Roman">14</FONT></P>
<P ><FONT face="Times New Roman">14</FONT></P>
<P ><FONT face="Times New Roman">9</FONT></P>
<P ><FONT face="Times New Roman">5</FONT></P>
<P ><FONT face="Times New Roman">4</FONT></P>
<P ><FONT face="Times New Roman">2</FONT></P>
<P ><FONT face="Times New Roman">1</FONT></P></TD>
<TD  vAlign=top width=96>
<P >-<FONT face="Times New Roman">6</FONT></P>
<P >-<FONT face="Times New Roman">15</FONT></P>
<P >-<FONT face="Times New Roman">24</FONT></P>
<P >-<FONT face="Times New Roman">24</FONT></P>
<P >-<FONT face="Times New Roman">26</FONT></P>
<P >-<FONT face="Times New Roman">18</FONT></P>
<P ><FONT face="Times New Roman">0</FONT></P>
<P ><FONT face="Times New Roman">14</FONT></P>
<P ><FONT face="Times New Roman">28</FONT></P>
<P ><FONT face="Times New Roman">27</FONT></P>
<P ><FONT face="Times New Roman">20</FONT></P>
<P ><FONT face="Times New Roman">20</FONT></P>
<P ><FONT face="Times New Roman">12</FONT></P>
<P ><FONT face="Times New Roman">7</FONT></P></TD>
<TD  vAlign=top width=120>
<P ><FONT face="Times New Roman">36</FONT></P>
<P ><FONT face="Times New Roman">75</FONT></P>
<P ><FONT face="Times New Roman">96</FONT></P>
<P ><FONT face="Times New Roman">72</FONT></P>
<P ><FONT face="Times New Roman">52</FONT></P>
<P ><FONT face="Times New Roman">18</FONT></P>
<P ><FONT face="Times New Roman">0</FONT></P>
<P ><FONT face="Times New Roman">14</FONT></P>
<P ><FONT face="Times New Roman">56</FONT></P>
<P ><FONT face="Times New Roman">81</FONT></P>
<P ><FONT face="Times New Roman">80</FONT></P>
<P ><FONT face="Times New Roman">100</FONT></P>
<P ><FONT face="Times New Roman">72</FONT></P>
<P ><FONT face="Times New Roman">49</FONT></P></TD></TR>
<TR>
<TD  vAlign=top width=91>
<P >T<FONT face="Times New Roman">otals</FONT></P></TD>
<TD  vAlign=top width=108>
<P ><FONT face="Times New Roman"> <p></p></FONT></P></TD>
<TD  vAlign=top width=96>
<P ><FONT face="Times New Roman">117</FONT></P></TD>
<TD  vAlign=top width=96>
<P >+<FONT face="Times New Roman">15</FONT></P></TD>
<TD  vAlign=top width=120>
<P ><FONT face="Times New Roman">801</FONT></P></TD></TR></TABLE>
<P ><FONT size=3><FONT face="Times New Roman"> <p></p></FONT></FONT></P>
<P ><FONT size=3><FONT face="Times New Roman">  We shall assume that the original measurements were made as accurately as possible,but that they are given here only to the mearest 0.02 m (i.e.2 cm).Thus the group labeled “1.66” contains all those men whose true measurements were between 1065 and 1067 m.One si biable to run into trouble if the exact methods of recording the measurements and grouping them are not specified exactly.In the example just given the mid-point of the interval labeled”1.66” m.But suppose that the original readings were made only to the nearest 0.01 m (i.e. 1 cm )and then “rounded up “to the nearest multiple of 0.02 m.We should then have “1.65”, which covers the range 1.645 to 1.655,included with “1.66”.The interval “1.66”would then contain all measurements lying between 1.645 m and 1.665 m .for which the mid-point is 1.655 m. The difference of 5 mm from the supposed value of 1.66 m could lead to serious inaccuracy in certain types of investigation.</FONT></FONT></P>
<P ><v:group><lock aspectratio="t" v:ext="edit" position="t" rotation="t"></lock><v:shape><FONT size=3><FONT face="Times New Roman"><v:fill detectmouseclick="t"></v:fill><v:path connecttype="none" extrusionok="t"></v:path><lock v:ext="edit" text="t"></lock></FONT></FONT></v:shape><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:shape><v:path arrowok="t"><FONT face="Times New Roman" size=3></FONT></v:path></v:shape><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><FONT face="Times New Roman" size=3></FONT></v:line><v:line><v:stroke endarrow="block"><FONT face="Times New Roman" size=3></FONT></v:stroke></v:line><w:anchorlock></w:anchorlock></v:group><v:shape><FONT size=3><FONT face="Times New Roman"><v:imagedata cropbottom="65520f" croptop="-65520f"></v:imagedata><lock v:ext="edit" position="t" rotation="t"></lock></FONT></FONT></v:shape></P>  A convenient visual way of presenting such data is shown in fig. 1, in which the area of each rectangle is ,on the scale used, equal to the observed proportion or percentage of individuals whose height falls in the corresponding group.The total area covered by all the rectangles therefore adds up to unity or 100per cent .This diagram is called a histogram.It is easily constructed when ,sa here ,all the groups are of the same width.It is also easily adapted to the case when the intervals are uneqal, provided we remember that the areas of the rectangles must be proportional to the numbers of units concerned.If, for example, we wished to group togcther the entries for the three groups 1.80,1.82 and 1.84 m,totaling 7 individuals or 6 per cent of the total,then we should need a rectangle whose base covered 3working groups on the horizontal scale but whose height was only 2 units on the vertical scale shown in the diagram.In this way we can make allowance for unequal grouping intervals ,but it is usually less troublesome if we can manageto keep them all the same width.In some books histograms are drawn so that the area of each rectangle is equal to the actual number (instend of the proportion) of individuals in the corresponding group.It is better, however, to use proportions, sa different histograms can then be compared directly.
 楼主| 发表于 2004-5-6 09:43:51 | 显示全部楼层
< 0cm 0cm 0pt"><FONT face="Times New Roman">The general appearance of the rectangles in Fig.1 is quite striking ,especially the tall hump in the centre and the rapidly falling tails on each side.There are certain minor irregularities in the pattern, and these would, in general ,be more ronounced if the size of the sample were smaller. Conversely, weth larger samples we usually find that the set of rectangles presents a more regular appearance. This suggests that if we had a very large number of measurements ,the ultimate shape of the picture for a suitably small width of rectangle would be something very like a smooth curve,Such a curve could be regarded as representing the true ,theoretical or ideal distribution of heights in a very (or ,better,infinitely)large population of individuals.</FONT></P>< 0cm 0cm 0pt"><FONT face="Times New Roman">  What sort of ideal curve can we expect ? There are seveala theoretical reasons for expecting the so-called Gaussiao or “normal “curve to turn up in practice;and it is an empirical fact that such a curve lften describes with sufficient accuracy the shape of histograms based on large numbers of obscrvations. Moreover,the normal curve is one of the easiest to handle theoretically,and it leads to types of statistical analysis that can be carried out with a minimum amount of computation. Hence the central importance of this distribution in statistical work .</FONT></P>< 0cm 0cm 0pt"><FONT face="Times New Roman">  The actual mathematical equation of the normal curve is where u is the mean or average value and </FONT><v:shapetype><v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></v:shapetype><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman"> is the standard deviation, which is a measure of the concentration of frequency about the mean. More will be said about </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">and </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman"> later .The ideal variable x may take any value from </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman"> to </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><FONT face="Times New Roman">.However ,some real measurements,like stature, may be essentially positive. But if small values are very rare ,the ideal normal curve may be a sufficiently close approximation. Those readers who are anxious to avoid as much algebraic manipulation as possible can be reassured by the promise that no derect use will be made in this book of the equation shown. Most of the practical numerical calculations to which it leads are fairly simple.</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">  Fig. 1 shows a normal curve, with its typocal symmetrical bell shape , fitted by suitable methods to the data embodied in the rectangles. This is not to say that the fitted curve is actually the true, ideal one to which the histogram approxime.tes; it is merely the best approximation we can find.</FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman">  The mormal curve used above is the curve we have chosen to represent the frequency distribution of stature for thr ideal or infinitely large population. This ideal poplation should be contrasted with the limited sample of obsrever. Values that turns up on any occasion when we make actual measurements in the real world. In the survey mentioned above we had a sample of 117 men .If the community were sufficiently large for us to collect several samples of this size, we should find that few if any of the corresponding histograms were exactly the same ,although they might all be taken as illustrating the underlying frequency distribution. The differences between such histograms constitute what we call sampling variation, and this becomes more prominent at the size of sample decreases.</FONT></P>
 楼主| 发表于 2004-5-6 09:44:08 | 显示全部楼层
< 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><B><FONT face="Times New Roman">Vocabulary <p></p></FONT></B></P>< 0cm 0cm 0pt"><FONT face="Times New Roman">Socio-medical survey </FONT>社会医疗调查表<FONT face="Times New Roman">    visual   </FONT>可见的。</P>< 0cm 0cm 0pt"><FONT face="Times New Roman">distribute  </FONT>分布(动词)<FONT face="Times New Roman">               percentage   </FONT>百分比</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">distribution   </FONT>分布(名词)<FONT face="Times New Roman">            individual    </FONT>个人,个别</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">histogram  </FONT>直方图,矩形图<FONT face="Times New Roman">            mean        </FONT>平均值,中数</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">hump      </FONT>驼背,使隆起<FONT face="Times New Roman">             standard deviation  </FONT>标准差</P><P 0cm 0cm 0pt"><FONT face="Times New Roman">normal distribution </FONT>正态分布<FONT face="Times New Roman">           sample varianice   </FONT>样本方差</P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P><P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P>
您需要登录后才可以回帖 登录 | 注-册-帐-号

本版积分规则

小黑屋|手机版|Archiver|数学建模网 ( 湘ICP备11011602号 )

GMT+8, 2024-3-29 20:49 , Processed in 0.057133 second(s), 18 queries .

Powered by Discuz! X3.4

Copyright © 2001-2021, Tencent Cloud.

快速回复 返回顶部 返回列表