1.1.5 Error
1)Errors
The deviations between measured values(experimental result)x and true values a are called errors,which are represented by“ε”and can be positive or negative.
2)Causes of errors
①Theoretical approximation and hypothesis.
②Accuracy of instruments.
③Experimental apparatus.
④Changes of experimental condition.
⑤Observers.
3)Classification of errors
According to the nature of their influences,errors can be divided into systematic errors and random errors.
(1)Systematic error:when the same physical quantity is measured many times under the same condition,the value and symbol of the error always remain unchanged or change depending on certain rules.The causes are as follows:
①Instrumental error:it is caused by manufacturing defects,improper use or non-calibration of instruments.For example,inaccurate mass of weights,unequal arms of the balance,inaccurate zero point of the meter,etc.
②Theoretical and methodological error:inadequate principle or method of measuring the theoretical formula approximation,experimental conditions and methods failing to meet the requirements perfectly are all causes of error.In experiments,this sort of errors is likely to derive from mistakenly ignored factors as follows:frictional resistance,wire and contact resistance and heat dissipation.
③Error of device:errors caused by improper arrangement,installation or adjustment of the measuring devices,both circuital and optical paths.
④Artificial error:error caused by habits,psychological or physiological characteristics of the measurer like pressing the stopwatch too fast or too slowly.
There are some methods to reduce systematic error,such as improve the accuracy of experimental instruments,revise the theoretical formula,modify the experimental conditions and improve our experimental skills and techniques.
(2)Random error(accidental error):when repeating the measurement of a certain physical quantity under the same conditions,the measurement results“jump”around a certain value rather than staying the same.And the value and symbol of the error change indeterminably.
The law of accidental error and its probability characteristics:
According to the statistical analysis of a large number of experimental data,the accidental error obeys certain laws of probability distribution when multiple observations are carried out under the same condition.The most common ones are Gaussian distribution(normal distribution)and t distribution(please refer to some books about error theory).The Gaussian distribution curve is shown in Figure 1.1.1.The horizontal axis x is the accidental error of measurement,and the vertical axis f(x)is the probability density of the error.The cell area of f(x)dx under the curve,or the shadow part,is the probability of occurrence of errors from x to x+dx.The function f(x)is derived by Carl Friedrich Gauss(1777-1855).(https://www.daowen.com)
Where σ is the standard error and e is the base of the natural logarithm.If the precision index is set at
,then
Figure 1.1.1 Gauss distribution curve
When x=0,
.Therefore,the smaller the σ is,the bigger the f(0)is.The steeper the error distribution curve is,and the more concentrated the data is(that is the greater the precision index h,the higher the measurement precision).As shown in Figure 1.1.2.
Figure 1.1.2 Gauss distribution curves with different σ
From the distribution curve of accidental error,the error possesses these probabilistic characteristics:
①Unimodality:the smaller the absolute value of error is,the greater the probability of error occurrence becomes.Conversely,the greater the absolute value is,the smaller the probability of error is.
②Symmetry:when the absolute values of errors are equal,the probabilities of occurrence of errors are equal.
③Boundedness:the probability of error with a large absolute value is almost equal to zero,that is to say,it seldom occurs.Meaning the accidental error will not exceed a certain limit under given conditions.
④Causes:many unexpected accidental factors cause these errors.
There are some reduction methods,such as increase the times of measurements and take the arithmetic mean value of multiple equal precision measurement columns as the result.
According to different calculation methods,they can be divided into absolute errors and relative errors.
(1)Absolute error:referring to the difference between the measured value xi and the true value a.The arithmetic mean of multiple equal precision measurement columns is taken as the true value.
(2)Relative error:representing the percentage ratio of the absolute value of the absolute error to the true value:
