2021年1月23日发(作者：th)
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< br>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
】
：
）

⑴
A random error will not exceed a certain amount.
（随即误差不会超过
一个确定的值）

⑵
Positive and negative random errors may occur at the same frequency.
（正负误差出现的频率相同）

⑶
Errors 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】
）

⑷
The
mean
of
random
errors
tends
to
zero
as
the
sample
size
tends
to
infinite.
（当【
as
】样本大小【
sample siz e
】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< br>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【
analysis n.
】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
【< br>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 be havior.
（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
,
σ
2
).
（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< br>【
symbol
】表示为
N(
μ
,
σ
2)
。

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、< br>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.
（
MP V
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 mean
μ
of the population.
（可
以看出【
It can be shown that< br>】一系列独立【
independent
】观测值的算
数平均值【
ar ithmetic
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
】< br>，
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 variance
σ
2
,
（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
σ
2
）



Where
μ

is the mean (expectation) of the population.
（这里
μ
是对 象总
体【样本】的平均值（期望值）
。
）