东南大学误差处理与数据处理复习课

Correctly arrange the experiments, design or select instruments and apply measuring methods, thus construct the optimal system to obtain the best result. 正确组织实 验，合理设计或选择仪器和测量方法，以便在合理优化的 系统下得到理想结果。
arithmetic mean is given as follows: 设 l1 , l2 , , ln 为n次测量所得的值，则算术平均值按如下计算：
l1 , l2 ,, ln be the results of n measurements, then the
l1 l 2 l n 1 n x li n n i 1
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Significant Digit
35.6 Vs 0.0356 0.0027 ≠0.00270
2400 ???
2.4x103 or 2.40x103 or 2.400x103
Standard expression: ax10n; 1≤a<10 L=20.531±0.01 L=20.53±0.01=(2.053±0.001)x101 For important measurements: L=15.214±0.042
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Examples of rounding-off operation 数字舍入运算示例 Original data: 3.14159 2.71729 4.51050 3.21550 6.378501 7.691499 5.43460 Rounded off data: 3.142 2.717 4.510 3.216 6.379 7.691 5.435
n

lim
n i 1
i
0
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Arithmetic Mean 算数平均值
When measuring the same object with equal precision, the results are different due to random errors, and the arithmetic mean should be used as the final measurement result. 对某一量进行多次等精度测量，由于随机误差的存在，测量结果 各不相同，应以全部测量值的算术平均值作为最后测量结果。 the significance of arithmetic mean 算术平均值的意义: let
Theory of Error & Data Processing
误差理论与数据处理
Review Lesson (复习课)
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期末考试
时间：2015年1月6日周三 14:00~15:40 (100分钟) 地点： 闭卷，请携带计算器
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The purposes of error studying
Error
Absolute Error
绝对误差
Relative Error
相对误差
System Error
系统误差
Random Abnormal Error Error
随机误差 粗大误差
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Correctness, Precision & Accuracy
High correctness 高准确度 Low precision 低精密度
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Properties of Normal Distribution 正态分布的特性:



The distribution is symmetric to Y-axis, i.e. for error w/ the same absolute value, positive error and negative error appear w/ same probability 分布曲线关于Y-轴对称，就是说，绝对值 相等的正负误差出现的概率相同;（对称性） Error w/ smaller absolute value appears w/ higher probability 绝对值小的误差出现的概率大;（单峰性） Under certain condition, the absolute value of random errors are bounded within a certain range 在一定的条件下，随机误差 的绝对值不会查过一定界限;（有界性） The average of random errors goes to 0 as the number of measurement increases 随着测量次数增加，随机误 差的平均值趋于0; （抵偿性）
Low correctness 低准确度 High precision 高精密度
Low correctness 低准确度 Low precision 低精密度
High correctness 高准确度 High precision 高精密度
Correctness
Precision Random Error System Error Accuracy System & Ramdom Error
4 Average error平均误差: | | f ( )d 5


2 Probable error或然误差: 0.6745 3
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Normal Distribution 正态分布
The figure shows normal distribution curve and related parameters. 正态分布密度曲线和相应的各种参数如图所示 σ is the X value of inflexion point A; θ is the X value of point B , the gravity center of the right half of the curve; line parallel to Y-axis at x=ρ evenly splits the area of the right half of the curve. σ是曲线拐点 A 的横坐标; θ是 B 点的横坐标, 也就是右半部曲线下方 面积的重心; 在 x=ρ处与Y轴平行的直线等分曲线右半部分下方面 积.
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Definitions
Absolute Error 绝对误差= 测得值－真值 绝对误差 Correction 修正值 修正值= 真值－测得值 L＝L－L0 C = L－ L0 = －L
Relative Error 相对误差 绝对误差 绝对误差 r L L 真值 测得值 L0 L 相对误差 Fiducial Error 最大示值误差 仪器(最大)引用误差 仪器引用误差 测量范围上限
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Residual Error 残余误差
vi li x
n i 1
vi li nx
i 1
n
vi 0
i 1
n
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单次测量的标准差
Major sources of error 误差的主要来源
Device （设备）
Environment （环境）
Method （方法）
People （人员）
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Classifications
Classifications of errors 误差分类:
In terms of expression 按表示方式 分类 In terms of property 按性质分类
xm rm xm
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The sources of error (误差来源)
The sources of error are everywhere, and almost every step of the measuring procedure brings in error. 误差无处不在，几乎测量的每一个步骤都会导入误差。
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Rules of data computation数据运算规则
For summation and subtraction, the computing data with the fewest number of decimal digits should be the reference, and other data may keep an extra decimal digit, but the final result should have the fewest number of decimal digits as the reference; 在加减运算时，各运算数据以小数位数最少的数据位数为准，其余各 数据可多保留一位小数，但最后结果应保留最少的小数位数； For multiplication and division, the computing data with the fewest number of significant digits should be the reference, and other data may keep an extra significant digit, but the final result should have the fewest number of significant digits as the reference; 在乘除运算时，各运算数据以有效位数最少的数据为准，其余各数据 可以多保留一位有效位数，但最后的结果应象参考数据一样保留最 少的有效位数； For computation of square and square root, the computing numbers are treated the same way as in multiplication and division; 在平方和开方运算时，按照乘除运算处理；