1.4.4 Method of successive differences
The method of successive differences is a common method of data processing in physical experiments.It is generally used to process the data obtained from a linear equally spaced measurement.According to the theory of errors,multi-measurement is usually employed in order to decrease the errors in measurement.But in a linear equally spaced measurement,only the first as well as the last data are valid if we choose the general averaging method,yet the data in the middle shall be cancelled out.This does not mean the characteristics that multiple measurements can decrease the experimental error.
Now we take the simplest linear function y=bx+a(both a and b are constants)as an example to illustrate the steps of the method of successive difference.
Measuring frequency:
Independent variable:
Dependent variable:
According to the definition of average value,it can be obtained that:
It can be seen that although 10 groups of data have been measured,only the first and the tenth group of data are valid for the result,while the ones in the middle are all cancelled out.Thus,it is necessary to make some changes to the processing of data,so as to maintain the advantage of decreasing measurement error in multiple measurements.
Usually,the above 10 data yi and xi are divided into the front half and rear half,i.e.
And then the corresponding items are subtracted,i.e.(https://www.daowen.com)
To acquire their averages,respectively:
If the symbol“k”denotes the number in the two synthesized data sets and k=n/2,then the formula for calculating the average increment can be obtained as follows:
Then we may assume that the method of successive difference can not only make full use of all experimental data and estimate the best value and measurement error,but also determine the value of a and b to obtain the analytical expressions for y and x.
Example 1.4.2
In the experiment of measuring the stiffness coefficient of springs,the measured natural length of spring l0=10.00 cm.The spring lengths l1=10.81 cm,l2=11.60 cm,l3=12.43 cm,l4=13.22 cm,l5=14.01 cm,l6=14.83 cm,l7=15.62 cm respectively with successive 10 g increment of the weights.Divide the data of length into two groups:the front half l0,l1,l2,l3 and the rear half:l4,l5,l6,l7.
The data of the front group is subtracted from that of the rear group:
Then the proportional coefficient:
