9随机误差的统计学基本分析

Unit 9 Basic Statistical Analysis of Random Errors （随机误差的统计学基本分析） 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.（随机误差是在错误被察觉【detect】和消除【eliminate】后，并且所有系统误差被从测量值中移除或修正后，保留下的那些变量【variable变量、变化n.】） They are beyond the control of the observer.（它们是观测者无法控制的） So the random errors are errors the occurrence of which does not follow a deterministic pattern.（因此随机误差是不遵循某个确定性【deterministic确定性的】模式【pattern】而发生的误差） In mathematical statistics, they are considered as stochastic variables, and despite 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:（在数理统计【mathematical statistics】中，它们被当成随机变量【stochastic variable】，尽管它们的行为无规律，在任一正确的【well-conducted原意为品行端正的，这里指测量实验和活动是无误的】测量活动和实验中，对的随机误差的研究显示【indicate】随机误差遵循以下经验法则【empirical rule】：） ⑴A random error will not exceed a certain amount.（随即误差不会超过一个确定的值） ⑵Positive and negative random errors may occur at the same frequency.（正负误差出现的频率相同） ⑶Errors that are small in magnitude are more likely to occur than those that are larger in magnitude.（误差数值【magnitude量值、大小】小的比数值大的误差出现可能性大【be likely to 可能】） ⑷The mean of random errors tends to zero as the sample size tends to infinite.（当【as】样本大小【sample size】趋近于无穷【infinite】时，随机误差的平均值趋近于0） In mathematical statistics, random errors follow statistical behavioral laws such as the laws of probability.（在数理统计中，随机误差遵循统计学的【statistical】行为【behavioral行为的】规律，如概率法则） 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.（发生在一个量的大量重复观测分析【analysisn.】中的误差分布的一个特征理论模式，遵照【conform to遵照】正态或高斯分布）【在对一个量进行大量重复观测分析后，得到一个误差分布的理论特征——正态或高斯分布】 The plot of error sizes versus probabilities would approach a smooth curve of the characteristic bell-shape.（误差大小与【versus与、与……的关系、与……相对】概率的关系图，接近一条光滑的特有的【characteristic特有的】钟形曲线。） This curve is known as the normal error distribution curve.（这条曲线被称为正态分布曲线） It is also called the probability density function of a normal random variable.（也叫做正态随机变量【normal random variable】的概率密度【probability density】函数） It is important to notice that the total area of the vertical bars for each plot equals 1.（需特别注意的是，每个图的条形图总面积为1。） This is true no matter the value of n (the number of single combined measurements), and thus the area under the smooth normal error distribution curve is equal to 1.（无论【no matter】n（单一的联合的测量数目【独立观测数】）是多少，在光滑的误差正态分布曲线下的面积都是1。） If an event 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 occurrence of errors.（如果一件事的概率为1，它一定会发生，因此曲线下方的面积代表了所有误差发生的概率。） A number of properties that relate a random variable and its probability density function are useful in our understanding of its behavior.（有许多工具【property】与随机变量和它的概率密度函数有关，有助于我们理解它的行为） Mean and standard deviation are two most popular statistical properties of a random variable.（平均值和标准偏差就是两个最常用的随机变量的统计工具【property】） Generally, a random variable which is normally distributed with a mean and standard deviation can be written in symbol form as N(μ,σ2).（一般地，一个通常由平均值和标准偏差描述的随机变量可以用符号【symbol】表示为N(μ,σ2)。 They can be explained as follows.（【它们可以】解释如下） Mean: The most commonly used measure of central tendency is the mean of a set of data (a sample).（平均值：最普遍应用的中心趋向的估计【measure】就是一系列数据（一个样本）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】涉及到随机变量的最或是值） It is also called by any of the several terms—expectation, expected value, mean or average. （还可以由其它几个术语来称呼它——期望、预期值、平均值或平均值） The mean is defined as （平均值定义为）

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是观测值，n是样本大小，或者叫样本内观测值的总数，x是平均值，经常被称为最或是值（MPV） The 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数据得到。） It can be shown that the arithmetic mean of a set of independent observations is an unbiased estimate of the meanμof the population.（可以看出【It can be shown that】一系列独立【independent】观测值的算数平均值【arithmetic mean】是一个样本【population】的期望值μ的无偏估计【unbiased estimate】。）

Standard deviation is a numerical value indicating the amount of variation about a central value.（标准偏差是一个数值【numerical value】，指示【indicate】相对于中值的偏离） In order to appreciate the concept upon which indices【index的复数】 of precision devolve one must consider a measure that takes into account all the values in a set of data.（考虑一系列数据的所有值精度指标 必需顾及一个量，这个量考虑到【takes into account考虑】一组【a set of】数据的所有值） 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 deviations may be used, and this is called the varianceσ2,（这个量是每个观测值xi相对于平均值x 的离差【deviation】，也就是，(xi- x)，离差的平方的平均值被采用，称之为方差σ2）