Being an exact science, mathematics does not tolerate bringing situations to the general without taking into account the features of a particular example. In particular, it is impossible to make a correct measurement literally “by eye” in mathematics and physics without taking into account the resulting error.
What is it about?
Scientists have found different types of error, so today we can safely say that not a single decimal point is left without attention. Of course, it is impossible without rounding, otherwise all people on the planet would only be engaged in counting, going deep into thousandths and ten thousandths. As you know, many numbers cannot be divided by each other without a remainder, and the measurements obtained during the experiments are an attempt to divide the continuous into separate parts in order to measure them.
In practice, the accuracy of measurements and calculations is really very important, as it is one of the main parameters that allow us to speak about the correctness of the data. The types of errors reflect how close the obtained figures are to reality. As for the quantitative expression: the measurement error is what shows how true the result is. The accuracy is better ifthe error turned out to be smaller.
Laws of science
According to the regularities found in the currently existing theory of errors, in a situation where the accuracy of the result should be twice as high as the current one, the number of experiments will have to be quadrupled. In the case when the accuracy is increased three times, there should be more experiments by 9 times. The systematic error is excluded.
Metrology considers the measurement of errors to be one of the most important steps to guarantee the uniformity of measurements. You have to consider: accuracy is affected by a wide range of factors. This led to the development of a very complex system of classification, which operates only with the proviso that it is conditional. In real conditions, the results strongly depend not only on the inherent error of the process, but also on the features of the very process of obtaining information for analysis.
Classification system
Types of error identified by modern scientists:
- absolute;
- relative;
- reduced.
This category can be divided into other groups, based on what are the reasons for the inaccuracies of the calculations and experiments. They say they have appeared:
- systematic error;
- accident.
The first value is constant, depends on the features of the measuring process and remains unchanged if the conditions are preserved with each subsequent manipulation
But the random error can change if the tester repeats similar studies using the same apparatus and being in conditions identical to the first period.
Systematic, random error appear simultaneously and occur in any test. The value of a random variable is not known in advance, since it is provoked by unpredictable factors. Despite the impossibility of elimination, algorithms have been developed to reduce this value. They are used at the stage of processing the data obtained during the research.
Systematic, in comparison with random, is distinguished by the clarity of the sources that provoke it. It is detected in advance and can be considered by scientists, taking into account the relationship with its causes.
And if you understand in more detail?
To have a complete understanding of the concept, you need to know not only the types of error, but also what are the components of this phenomenon. Mathematicians distinguish the following components:
- related to methodology;
- tool-conditioned;
- subjective.
When calculating the error, the operator depends on specific, only inherent, individual characteristics. It is they that form the subjective component of the error that violates the accuracy of information analysis. Perhaps the reason will be a lack of experience, sometimes - in errors associated with the start of the countdown.
Mostly, the calculation of the error takes into account two other points, that is, instrumental and methodical.
Important Ingredients
Accuracy and error are concepts without which neither physics, nor mathematics, nor a number of other natural and exact sciences based on them is possible.
At the same time, it must be remembered that all methods known to mankind for obtaining data in the course of experiments are imperfect. This is what provoked a methodological error, which is absolutely impossible to avoid. It is also affected by the accepted system of calculation and inaccuracies inherent in the calculation formulas. Of course, the need to round results also has an impact.
They highlight gross blunders, i.e. errors caused by incorrect behavior of the operator during the experiment, as well as breakdown, incorrect functioning of devices or the occurrence of an unforeseen situation.
You can detect gross error in values by analyzing the received data and identifying incorrect values when comparing data with special criteria.
What do math and physics talk about today? The error can be prevented by preventive measures. Several rational ways of reducing this concept have been invented. To do this, one or another factor leading to the inaccuracy of the result is eliminated.
Category and classification
There are errors:
- absolute;
- methodical;
- random;
- relative;
- reduced;
- instrumental;
- main;
- additional;
- systematic;
- personal;
- static;
- dynamic.
The error formula for different types is different, since in each case it takes into account a number of factors that influenced the formation of data inaccuracy.
If we talk about mathematics, then with such an expression, only relative and absolute errors are distinguished. But when the interaction of changes occurs in a given time period, we can talk about the presence of dynamic, static components.
The error formula, which takes into account the interaction of the target object with external conditions, contains an additional, main figure. The dependence of the readings on the input data for a particular experiment will indicate a multiplicative error or an additive one.
Absolute
This term is commonly understood as data calculated by highlighting the difference between the indicators taken during the experiment and the real ones. The following formula was invented:
A Qn=Qn - A Q0
And Qn are the data you are looking for, Qn are the ones identified in the experiment, and zero are the base numbers with which the comparison is made.
Reduced
This term is commonly understood as a value that expresses the ratio between the absolute error and the norm.
When calculating this type of error, not only the shortcomings associated with the operation of the instruments involved in the experiment are important, but also the methodological component, as well as the approximate reading error. The last value is provokedthe shortcomings of the division scale present on the measuring device.
Instrumental error is closely related to this concept. It occurs when the device was produced incorrectly, erroneously, incorrectly, which is why the readings given by it become insufficiently accurate. However, now our society is at such a level of technological progress, when the creation of devices that do not have instrumental error at all is still unattainable. What can we say about outdated samples used in school and student experiments. Therefore, when calculating control, laboratory work, it is unacceptable to neglect the instrumental error.
Methodical
This variety is provoked by one of two reasons or by a complex:
- the mathematical model used in the research turned out to be insufficiently accurate;
- incorrect measurement methods selected.
Subjective
The term is applied to a situation where, when obtaining information in the course of calculations or experiments, errors were made due to insufficient qualifications of the person performing the operation.
It cannot be said that it occurs only when an uneducated or stupid person took part in the project. In particular, the error is provoked by the imperfection of the human visual system. Therefore, the reasons may not depend directly on the participant of the experiment, however, they are classified as a human factor.
Static anddynamics for error theory
A certain error is always related to how the input and output value interact. In particular, the process of interconnection in a given time interval is analyzed. It is customary to talk about:
- The error that appears when calculating a certain value that is constant in a given time period. This is called static.
- Dynamic, associated with the appearance of a difference, detected by measuring non-constant data, the type described in the paragraph above.
What is primary and what is secondary?
Of course, the margin of error is provoked by the main quantities that affect a specific task, however, the influence is not uniform, which allowed the researchers to subdivide the group into two categories of data:
- Calculated under normal operating conditions with standard numerical expressions of all affecting figures. These are called the main ones.
- Additional, formed under the influence of atypical factors that do not correspond to normal values. The same type is also spoken of in the case when the main value goes beyond the limits of the norm.
What's happening around?
The term “norm” has been mentioned more than once above, but no explanation has been given of what kind of conditions in science are usually called normal, as well as a mention of what other types of conditions distinguish.
So, normal conditions are those conditions when all the quantities affecting the workflow are within the normal values identified for them.
But the workers -term applicable to the conditions under which changes in quantities occur. In comparison with the normal ones, the frames here are much wider, however, the influencing quantities must fit into the work area specified for them.
The working norm of the influencing quantity assumes such an interval of the value axis when normalization is possible due to the introduction of an additional error.
What does the input value affect?
When calculating the error, you have to remember that the input value affects what types of error occur in a particular situation. At the same time, they talk about:
- additive, which is characterized by an error calculated as the sum of different values taken modulo. At the same time, the indicator is not affected by how large the measured value is;
- multiplicative that will change when the measured value is affected.
It should be remembered that the absolute additive is an error that has no connection with the value, which is the purpose of the experiment to measure. In any part of the range of values, the indicator remains constant, it is not affected by the parameters of the measuring instrument, including sensitivity.
Additive error indicates how small the value obtained by applying the selected measurement tool can be.
But the multiplicative one will change not randomly, but proportionally, as it is related to the parameters of the measured value. How large the error is is calculated by examining the sensitivity of the device, since the value will be proportional to it. This subtype of error arises precisely because the input value acts on the measuring tool and changes its parameters.
How to remove the error?
In some cases, the error can be excluded, although this is not true for every species. For example, if we are talking about the above, the error class in this case depends on the parameters of the device and the value can be changed by choosing a more accurate, modern tool. At the same time, measurement flaws due to the technical features of the machines used cannot be completely ruled out, since there will always be factors that reduce the reliability of the data.
Classic there are four methods to eliminate or minimize the error:
- Remove the cause, the source before the start of the experiment.
- Elimination of error in the course of data acquisition activities. For this, substitution methods are used, they try to compensate by sign and oppose observations to each other, and also resort to symmetrical observations.
- Correction of the results obtained in the course of making edits, that is, a computational way to eliminate the error.
- Determination of what are the limits of systematic error, taking them into account in the case when it cannot be eliminated.
The best option is to eliminate the causes, sources of error duringexperimental data acquisition. Despite the fact that the method is considered to be the most optimal, it does not complicate the workflow, on the contrary, it even makes it easier. This is due to the fact that the operator does not need to eliminate the error already in the course of directly obtaining data. You don't have to edit the finished result, adjusting it to the standards.
But when it was decided to eliminate errors already in the course of measurements, they resorted to one of the popular technologies.
Known exceptions
The most widely used is the introduction of edits. To use them, you need to know exactly what the systematic error inherent in a particular experiment.
In addition, the substitution option is in demand. Resorting to it, specialists instead of the value they are interested in use a substituted value placed in a similar environment. This is common when electrical quantities need to be measured.
Opposition - a method that requires experiments to be carried out twice, while the source in the second stage affects the result in the opposite way compared to the first. The logic of work is close to this method of a variant called "compensation by sign", when the value in one experiment should be positive, in the other - negative, and a specific value is calculated by comparing the results of two measurements.
