unit9隨機誤差的統(tǒng)計學基本分析

1、Unit 9 Basic Statistical Analysis of Random Errors（隨機誤差的統(tǒng)計學基本分析）Random errors are those variables that remain after mistakes are detected and eliminated and all systematic errors have been removed or corrected from the measured values（隨機誤差是在錯誤被察覺【detect和消除【eliminate】后，并且所有系統(tǒng)誤差被從測量值中移除或修正后，保留下的那些變量【v

2、ariable變量、變化n.）They are beyond the control of the observer.（它們是觀測者無法控制的）So the random errors are errors the occurrence of which does not follow a deterministic pattern（因此隨機誤差是不遵循某個確定性【deterministic確定性的模式【pattern】而發(fā)生的誤差）In mathematical statistics, they are considered as stochastic variables, and desp

3、ite their irregular behavior, the study of random errors in any well-conducted measuring process or experiment has indicated that random errors follow the following empirical rules:（在數(shù)理統(tǒng)計【mathematical statistics中，它們被當成隨機變量【stochastic variable，盡管它們的行為無規(guī)律，在任一正 確的【well-conducted原意為品行端正的，這里指測量實驗和活動是無誤的測

4、量活動和實驗中，對的隨機誤差的研究顯示 indicate隨機誤差遵循以下經驗法則【empirical rule丨：）（1）A random error will not exceed a certain amount（隨即誤差不會超過一個確定的值）（2）Positive and negative random errors may occur at the same frequency.（正負誤差出現(xiàn)的頻率相同）（3）Errors that are small in magnitude are more likely to occur than those that are larger in

5、 magnitude（誤差數(shù)值【magnitude量值、大小小的比數(shù)值大的誤差出現(xiàn)可能性大【be likely to可能）（4）The mean of random errors tends to zero as the sample size tends to infinite（當【as樣本大小sample size趨近于無窮【infinite時，隨機誤差的平均值趨近于0）In mathematical statistics, random errors follow statistical behavioral laws such as the laws of probability.（在

6、數(shù)理統(tǒng)計中，隨機誤差遵循統(tǒng)計學的【statistical 行為【behavioral行為的規(guī)律，如概率法則）A characteristic theoretical pattern of error distribution occurs upon analysis of a large number of repeated measurements of a quantity, which conform to normal or Gaussian distribution. 一個誤差分布的典型理論模式出現(xiàn)于對某 一量的大量重復觀測中，這些重復觀測遵從【conform to遵照正態(tài)分布或高斯

7、分布【在對一個量進行大量重復觀測分析后，得到一個誤差分布的理論特征一一正態(tài)或高斯分布The plot of error sizes versus probabilities would approach a smooth curve of the characteristic bell-shape（誤差大小與【versus與、與 的矢系、與相對概率的尖系圖，接近一條光滑的特有的【characteristic特有的鐘形曲線。）This curve is known as the normal error distribution curve（這條曲線被稱為正態(tài)分布曲線）It is also ca

8、lled the probability density function of a normal random variable（也叫做正態(tài)隨機變量normal random variable的概率密度【probability density函數(shù)）It is important to notice that the total area of the vertical bars for each plot equals 1.（需特別注意的是每個圖的條形圖總 面積為1。）This is true no matter the value of n （the number of single co

9、mbined measurements） , and thus the area under the smooth normal error distribution curve is equal to 1.無論【no mattern （【獨立觀測數(shù)）是多少，在光滑的誤差正態(tài)分布曲線下的面積都是1。If an eve nt has a probability of 1, it is certain to occur, and therefore the area under the curve represents the sum of all the probabilities of the

10、 occurrence of errors（如果一件事的概率為1，它一定會發(fā)生，因此曲線下方的面積代表了所有誤差發(fā)生的概率。）A number of properties that relate a random variable and its probability density function are useful in our understanding of its behavior.隨機變量的大量屬性和其概率密度函數(shù)，有助于我們對其行為的理解。Mean and standard deviation are two most popular statistical propert

11、ies of a random variable.均值和標準差就是隨機變量 的兩個最常用的統(tǒng)計屬性Gen erally, a ran dom variable which is no rmally distributed with a mea n and standard deviation can be written in symbol form as N®,2）（一般地，一個通常由平均值和標準偏差描述的隨機變量可以用符號【symbol】表示為N （卩,2）。They can be explained as follows（【它們可以】解釋如下）Mean: The most co

12、mmonly used measure of central tendency is the mean of a set of data （a sample）（平均值：應用最普遍的中心趨向的估計【measure!就是一系列數(shù)據（一個樣本）的平均值）The concept of mean refers to the most probable value of the random variable（平均值的概念【concept】涉及到隨機變量的最或是值）It is also called by any of the several terms expectation, expected val

13、ue, mean or average（還可以由其它幾個術語來稱 呼它一一期望、期望值、平均值或平均值）The mean is defined as（平均值定義為）公式Where xi are the observations, n is the sample size, or total number of observations in the sample, and x is the mean which is also called most probable value （MPV）（xi是觀測值，n是樣本大小，或者叫樣本內觀測值的總數(shù)，x是平均值，經常被稱為最或是值（MPV）The

14、MPV is the closest approximation to the true value that can be easily achieved from a set of data（MPV是最接近真值的近似值【approximation】，可以很容易由一系列數(shù)據得到。）It can be shown that the arithmetic mean of a set of independent observations is an unbiased estimate of the mean of the population（可以看出It can be shown that】一

15、系列獨立independent觀測值的算數(shù)平均值【arithmetic mean】是一個樣 本 population的期望值卩的無偏估計un biased estimate。）Standard deviation is a numerical value indicating the amount of variation about a central value（標準偏差是一個數(shù)值numerical value】，它指示indicate相對于中值的偏離）In order to appreciate the concept upon which indices index 的復數(shù)】 of p

16、recision devolve one must consider a measure that takes into account all the values in a set of data（考慮一系列數(shù)據的所有值精度指標必需顧及一個量，這個量考慮到【takes into account考慮】一組【a set of數(shù)據的所 有值）Such a measure is the deviation from the mean x of each observed value xi i.e. （xi x） , and the mean of the squares of the deviat

17、ions may be used, and this is called the varianee,（這個量是每個觀測值xi相對于平均值x的離差【deviation，也就是，（xix），離差的平方的平均值被采用， 稱之為方差（T2）公式Where 卩 is the mean （expectation） of the population（這里卩是對象總體【樣本】的平均值（期望值）。）The square root of the varianee is called standard deviation © （方差的平方根被稱為標準差©）Theoretically the s

18、tandard deviation, which is the value on the X axis of the probability curve that occurs at the points of i nflecxion【估 i十應為 in flexi ons 拐點】 of the curve, is obtai ned from an in finite number of variables known as the population （理論上來說，標準差是概率曲線拐點的X軸坐標，由無窮多的變量（被稱為樣本）得到）In practice, however, only a

19、sample of variables is available and S is used as an unbiased estimator.然而，實際上，變量的樣本是有限的，且將 S稱為6的無偏估計。Account is taken of the small number of variables in the sample by using （n-1） as the divisor, which is referred to in statistics as the Bessel correction; hence, variance is計算Account is taken of中使用

20、有限【small小的】數(shù)量的變量，且將（n-1 ）作為除數(shù)，這就是統(tǒng)計學中的白賽爾公式，因 此，方差可表述為：公式To obtain an index of precision in the same units as the original data, therefore the square root of the variance is used, and this also called the standard deviation S（為了獲得與源數(shù)據一樣單位的精度指標，因此采用了方差的平方根，又叫做標準差S）The standard deviation is the measur

21、e of the dispersion or spread of the random variable.（標準差是隨機變量分布的量度 標準。）A survey measurement, such as a distance or angle, after mistakes are eliminated and systematic errors corrected, is a random variable（一個測量值，例如距離或角度觀測值，在粗差被去除、系統(tǒng)誤差被修正后，就是一個隨機變量。）If a distance is measured 20 times, it is not unus

22、ual to get values for each of the measurements that differ slightly from its true value that is never known.（如果一段距離被測了 20次，每次的測量值與永遠未知的真值有些微的差值是很正常的）So owing to random variability, an error was defined as the d iff ere nee between a random variable, the measured value （observation） and the constant,

23、 the true value ie. error= measured value因此，由于【owing to】隨機可變性，誤差被定義為隨機變量、測量值和常量、真值之間的差值，也就是，誤差二測量值真值And a correction （residual） , which is the negative of the error in practice, was defined as correction between the MPV and measured value Le. correction=MPV-measured value（改正值，習慣上【in practice】是誤差的負值，

24、定義為MPV和測量值之間的修正值，也就是改正值二MPV 測量值）When the so-called true values are available to compare with calculated values, the mean square error （MSE） is given by （當所謂的真值可以與計算值相比較時，誤差均方差（MSE ）為：）公式In which xi is the measured value, x is the true value and n is the number of measurements（其中xi是測量值，x是真值，n是觀測數(shù)）Pro

25、pagation of errors （or error propagation） : Much data in surveyi ng is obtained in directly from various combinations of observations誤差傳播：測量的許多數(shù)據是間接由各種測量值綜合得到的【combination是名詞，這里翻譯時用到了詞性轉換】For instance the coordinates of a line are a function of its length and bearing（例如，一條直線的坐標就是其長度和方位的函數(shù)）As each me

26、asurement contains an error, it is necessary to consider the combined effect of these errors on the derived quantit y.（由于每項測量值都包含誤差，必需考慮這些源數(shù)據的誤差的聯(lián)合影響）Error propagation is one of the many aspects of analyzing errors（誤差傳播是各種誤差分析方法的其中之一）It is the mathematical processused to estimate the expectedrandom

27、error in a computed or indirectly measured quantity, caused by one or more identified and estimated random errors in one or more identified variables that influe nee the precision of the qua ntity.它（誤差傳播）是一種數(shù)學方法process或者譯為過程】，用來估計estimate在一個計算出的或間接測量的參量 quantity的期望隨機誤差【或偶然誤差】，該參量是由一個或多個獨立【identified變量的偶然誤差引起的，從而影響該量的精度。）The general procedure is to differentiate with respect to each of the observed quantities in turn and sum them to