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In data recording and processing, experimental data is often tabulated. Tabulation is an effec tive method to record experimental data in sequence and also a clear way to show various relationships between data as well, which is beneficial to find out and analyze problems just to avoiding mistakes.
Requirements of tabulation are listed as follows:
(1) Tabulation must be simple and clear for finding out the relationship between related quantities and processing data. In addition, the title must be shown.
(2) In the tabulation, the meaning of symbol standing for physical quantity must be marked clearly as well as its unit. The unit and the order of magnitude of each quantity should be given in the title bar once and for all.
(3) The data must correctly reflect the accuracy of the instrument, requiring the significant figures to be precise. When scientific notation is required, the involution of “10” should be put in the title bar, and the decimal points have to be aligned.
(4) Necessary instructions should be concise and accurate.
For example, in Table 1.4.1, a group of measured data of cylinder diameter D using a vernier scale caliper (0.05 mm) and the corresponding processed data are listed. It can be seen from the table that the sum of deviation is zero (or closed to zero), indicating to the correct calculation.
Table 1.4.1 Table of measured and processed data of cylinder diameter D
Graphic interpretation is used to expressing the relationship between physical quantities and al so to get a simple analytical function formula by drawing geometric lines with a series of corresponding measured values on graph paper.
1) Benefits of plotting
Plotting is often used in experimental data processing. The benefits are as follows:
(1) The function relationship can be displayed intuitively, and certain experimental results such as slope and intercept can be found directly in the line chart.
(2) Corresponding values that are not measured in the experiment can be obtained from the line chart with interpolation and extrapolation.
(3) Some data with flaws can be found in the graph.
(4) A standard data curve of the instrument can help not only correct systematic errors but also speculate other measurement values.
2) Requirements of plotting
(1) Select suitable coordinate paper such as rectangular coordinate paper, logarithmic coordinate paper and polar coordinate paper.
(2) The size of coordinate paper and the scale of coordinate axis should depend on the significant figures of measured data and requirements for precision of results. In principle, the minimum gird of the coordinate paper is limited by the last significant digit of experimental data, which gives proper enlargement a shot to become possible. Though the origin of coordinates being outside the graph makes sense, it would be better to have the graph in the middle of the coordinate paper.
(3) The physical quantities, units and division values should be marked on coordinate axis.
(4) Symbols such as + , X, △, and ☉ are used to mark data points on coordinate paper,while identical symbols are used on the same curve.
(5) The curves are supposed to be smooth during connection. Particularly, the calibration curve should involve all calibration points. It is not strictly required that all points need to be on the curve. But the deviation of the points from the curve should be more evenly distributed on both sides. Figure 1.4.1 is an example of graphic interpretation for the relationship between current I and voltage U .
Figure 1.4.1 The relationship between current I and voltage U
The least square method is the most accurate one in a series of approximate calculations, by which the best value can be determined from a group of measured values with the same accuracy.Take the linear equation as an example: the slope m and intercept b can be obtained once the line equation is determined. So linear regression (linear fitting) is the process of determining the size of m and b from the experimental data set ( x i , y i ).
According to the least square method principle, if the best fitted linear equation is Y = f ( x ),the sums of squares of the deviation values between measured values y i and corresponding values on the fitted line is the minimum, i. e.
In order to simplify the question, it is supposed that every measured value x i is of equal accuracy, and only y i is accompanied by obviously measured accidental error. Substitute the linear equation into Equation (1.4.1), we may get the following formula:
So m and b should be solutions to the following equations:
i. e.
Solve Equation (1.4.4) to obtain:
Where
In order to examine the linear fitting, the correlation coefficient is defined as follows:
It can be proved that the absolute value of r always lays between 0 and 1. The closer the abso lute value approaches 1, the better the linear relationship between x and y is.
Example 1.4.1
Data is shown in Table 1.4.2.
Table 1.4.2 Measuring results for surface tension coefficient of water α acquired at different temperatures
Suppose the primary function form α = mT + b , where T stands for the thermodynamic tempera ture. Evaluate m and b by ordinary least squares. Firstly, work the following quantities out:
Then, m and b can be evaluated through Equation (1.4.5) and Equation (1.4.6), i. e.
Finally, the analytic equation for relationship between surface tension coefficient α and temper ature T can be concluded as α = -(0.16 T + 119.6)× 10 3 N/ m.
For exponential function, logarithmic function and power function, we can obtain the linear relationship by the variable substitution, which is suitable for least square fitting.
We can also use a computer to perform the linear regression, directly solving the experimental equation and obtaining the relevant parameters and their errors. This approach is now viable due to the popularization of computers.
The method of successive differences is a common method of data processing in physical experi ments. 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 y i and x i are divided into the front half and rear half, i. e.
And then the corresponding items are subtracted, i. e.
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 l 0 = 10.00 cm. The spring lengths l 1 = 10.81 cm, l 2 = 11.60 cm, l 3 = 12.43 cm, l 4 =13.22 cm, l 5 = 14.01 cm, l 6 = 14.83 cm, l 7 = 15.62 cm respectively with successive 10 g increment of the weights. Divide the data of length into two groups: the front half l 0 , l 1 , l 2 , l 3 and the rear half: l 4 , l 5 , l 6 , l 7 .
The data of the front group is subtracted from that of the rear group:
Then the proportional coefficient:
