unit 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.

一个误差分布的典型理论模式出现于对某一量的大量重复观测中，这些重复观测遵从【conform to遵照】正态分布或高斯分布【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（【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。）