Over the past few weeks, some of our readers have reported unsuccessful attempts to analyze scientific projects. The failure analysis should include the calculation of the difference between results and expectations. This can be done by calculating the percentage of errors observed in the experiment. Then, error analysis should identify sources of error that explain why your results and expectations differ. The sources of the error must be specific.
Experimental errors and
This chapter is primarily a guide to troubleshooting experimental measurement errors. Much of the material has been extensively tested with science students at various levels at the University of Toronto.
Whole books could and have been written on this topic, but here we will focus on the most important. However, our experience shows that an iterative approach to this material is best for beginners. This means that users first look at the material in this chapter. Then try using the material yourself. then review the material; so …
EDA provides functionality to simplify the computations needed to propagate errors. These functions are presented in section 3.3. These error propagation functions are briefly described in Section 3.5.
3.1.1 Purpose of error analysis
Students who only attend lectures and read science textbooks easily get the false impression that science is manipulation Lation with exact and ideal numbers. Courses and guides often include phrases such as:
A particle falling under the action of gravity experiences a constant acceleration of 9.8 m /. Yes …
This specification is not complete for the experimental scientist. Does this mean the acceleration is closer to 9.8 than 9.9 or 9.7? Does this mean that the acceleration is closer to 9.80000 than to 9.80001 or 9.79999? Often the answer depends on the context. If the carpenter says the length is “only 8”, it probably means that the length is closer to 8 0/16 “than to 8 1/16” or 7 15/16 “. When the machinist says the length is” everything. ” 200 millimeters, which probably means that it is closer to 200.00 mm than 200.05 mm or 199.95 mm.
We all know that gravitational acceleration varies from place to place on the Earth’s surface. It also depends on the height above the surface, and gravimeters that can be used to measure the clearance between the ground and the table are readily available. Moreover, any physical measure, such as g, can only be determined experimentally, and since there is no perfect experimental equipment, it is impossible even in principle to ever know g exactly sti. Thus, the g specification above is only useful as a possible student exercise. To make it meaningful, it needs to be changed to something like:
A 5 g ball bearing that fell by gravity on March 13, 1995 in room 126 of the McLennan Physics Laboratory at the University of Toronto at a distance of 1.0 ± 0.1 m above the ground under the action of gravity constant acceleration of 9.81 ± 0.03 m / …
There are two questions about measurement. First, how “accurate” is it, in other words, did the experiment go right and were all the necessary factors taken into account? The answer to this question depends on the ability of the experimenter to identify and eliminate all systematic errors. This is explained in section 3.4.
The second question concerns the “precision” of the experiment. In this case, the accuracy of the result is indicated: the experimenter claims that the accuracy of the result is about 0.03 m / s. The next two sections detail how measurement accuracy is determined. However, the following points are important:
1. The person who made the measurement probably had an “instinct” for accuracy “tied” the error to the result, mainly to convey that feeling to other people. Common sense should always prevail over mathematical manipulation.
2. In complex experiments, failure analysis can identify the underlying failures and thus indicate where additional effort needs to be made to improve the experiment.
3. In experimental physical science, there is hardly a case where the correct analysis of errors involves comparing the result with the number in the book. The correct experiment is being conducted correctly and not according to other measurements.
4. The maximum possible accuracy for a particular experiment is always limited by the device. Measurements of polarization in high energy physics require tens of thousands of man-hours and cost hundreds of thousands of dollars, and a good measurement is twice. Electrodynamic experiments are much cheaper and often give results for 8 or more significant numbers. In any case, the experimenter must struggle with the equipment to obtain the most accurate and accurate measurement.
3.1.2 Different types of oshibok
As mentioned above, there are two types of errors associated with the experimental result: “precision” and “precision”. Familiar text explains the difference as follows:
Accuracy refers to the random distribution of errors associated with a particular experiment or even a particular type of experiment. Accuracy refers to the presence of systematic errors — for example, differences between laboratories. For example, with a high-quality pendulum clock, in which the pendulum is not quite set to the correct length, it would be possible to measure the time very accurately but inaccurately. Pugh E.M., Winslow G.H., p. 6.
The goal of a good test is to minimize both precision errors and precision errors.
Usually in a particular experiment there is one type or another of errors, and the experimenter goes to great lengths to reduce them. For example, when measuring the height of a geranium sample to determine the mean, the random variation in the plant sample is likely to be much larger than any possible inaccuracies in the ruler used. Likewise, in many experiments in the life sciencesnor in the life sciences, the experimenter is most concerned with improving the accuracy of his measurements. Of course, precision errors prevail in some experiments in biology and life sciences.
On the other hand, when titrating a HCl acid sample with NaOH base using a phenolphthalein indicator, the main error in determining the initial acid concentration is probably one of the following: (1) the accuracy of the marking on the burette side; (2) the transition zone of the phenolphthalein indicator; or (3) the experimenter’s ability to decompose the last drop of NaOH. Therefore, the determination accuracy is probably much worse than the accuracy. This happens often with chemical experiments, but certainly not in all.
Question. Most experiments use theoretical formulas, and these formulas are usually approximate. Is the approximation error a precision or correctness error?
There is a wealth of literature on the topics of this chapter. Here are some popular views.
D.K. Byrd, Experiment: An Introduction to Measurement Theory and Experiment Design (Prentice Hall, 1962)
Pugh E.M., Winslow G.H. Analysis of physical quantities (Addison-Wesley, 1966)
J.R. Taylor, An Introduction to Failure Analysis (University Science Books, 1982)
There is also a web document written by the author of EDA that teaches this topic to first-year physics students at the University of Toronto. The following hyperlink refers to this document.
3.2 Determine accuracy
3.2.1 Standard deviation
In the 19th century, Gauss’s assistants carried out astronomical measurements. However, they did not succeed in repeating their results exactly. Eventually Gauss got angry and burst into the laboratory, claiming that he was going to show these people how to take measurements once and for all. The only problem was that Gauss couldn’t exactly repeat his measurements either!
Upon awakening, Gauss drew up a histogram of the results of a particular measurement and discovered the famous Gaussian or bell-shaped curve.
The first acquaintance with this form is the distribution of grades for the course. Here is an example of such a distribution using the EDA EDAHistogram function.
In : =
In : =
In : =
We use the standard Mathematica package to create a PDF of such a “Gaussian” or “normal” distribution. The mean is selected as 78 and the standard deviation as 10. The mean and standard deviation are defined below.
In : =
In : =
Exit  =
Then we normalize the distribution so that the maximum value is close to the maximum number in the histogram and save the result.
In : =
In this chart, the average is greater