Definition of measurement error

Common methods for representing errors include absolute error, relative error, and citation error.

1. Absolute error:

Definition: The difference between the measured value x * and the measured true value x is called the absolute error of the approximate value x *, abbreviated as ε.

Calculation formula: Absolute error=measured value - true value;

2. Relative error:

Definition: The numerical value obtained by multiplying the absolute error caused by measurement by the ratio of the measured (agreed) true value by 100%, expressed as a percentage.

Calculation formula: Relative error=(measured value - true value)/true value x 100% (i.e. the percentage of absolute error to true value);

3. Citation error

Definition: The ratio of the absolute error of a measurement to the full range value of an instrument is called the reference error of the instrument, which is often expressed as a percentage.

Calculation formula: Reference error=(maximum absolute error/instrument range) x 100%

The smaller the citation error, the higher the accuracy of the instrument, and the citation error is related to the range of the instrument. Therefore, when using instruments with the same accuracy, it is often necessary to compress the range range to reduce measurement errors

01 instance

Using a multimeter to measure voltage of 1.005V, assuming the true voltage value is 1V, the multimeter range is 10V, and the accuracy (citation error) is 0.1% F.S. Is the multimeter testing error within the allowable range?

The analysis process is as follows:

Absolute error: E=1.005V -1V=+0.005V;

Relative error: δ=0.005V/1V x 100%=0.5%;

Reference error of multimeter: 10V x 0.1% F.S=0.1V;

Due to an absolute error of 0.005V<0.1V, a multimeter with a reference error of 0.1% F.S was used for the 10V range. The relative error of measuring 1V was 0.5%, which is still within the allowable range of error.

2. Generation of measurement errors

Absolute error objectively exists but cannot be determined by people, and absolute error is inevitable. Relative error can be minimized as much as possible.

The components of error can be divided into random error and systematic error, namely: error=measurement result - true value=random error+systematic error

Therefore, any error can be decomposed into the algebraic sum of systematic error and random error, and systematic error:

1. System error

Definition: The difference between the average value of an infinite number of measurements taken on the same subject under repeatability conditions and the true value of the subject being measured.

Cause: Measurement errors caused by inherent errors in measuring tools (or instruments), theoretical deficiencies in measuring principles or methods, experimental operations, and limitations in the psychological and physiological conditions of experimenters.

Characteristic: Under the same measurement conditions, the measurement results obtained from repeated measurements are always biased towards larger or smaller values, and the error value is constant or varies according to a certain pattern.

Optimization method: The method can usually change the measurement tool or method, and can also consider correction values for the measurement results.

2. Random error

Definition: Random error, also known as accidental error, refers to the difference between the measurement result and the average result of a large number of repeated measurements to be measured.

Cause: Even in the ideal situation of completely eliminating system errors, repeated measurements of the same measurement object will still result in measurement errors due to various accidental and unpredictable uncertainties.

Characteristics: It refers to multiple repeated measurements of the same measurement object, and the error of the measurement result shows irregular fluctuations, which may be positive or negative deviation, and the absolute value of the error fluctuates irregularly.

However, the distribution of errors follows statistical laws and exhibits the following three characteristics:

Unimodal, meaning that there are more small errors than large errors;

Symmetry, which means that the probability of positive and negative errors is equal;

Boundedness, which means that the probability of a large error is almost zero.

Optimization method: According to the distribution law of random errors, increasing the number of measurements and processing the measurement results according to statistical theory can reduce random errors.

3 Precision, precision, and accuracy

Precision and error can be said to be twin brothers, because the existence of error gives rise to the concept of precision. In short, instrument accuracy refers to the degree to which the measured value of an instrument is close to the true value, usually expressed as relative percentage error (also known as relative equivalent error).

1. The magnitude of measurement error reflects the precision of measurement

Using the same measuring tool and method to measure multiple times under the same conditions, if the random error of the measurement value is small, that is, the fluctuation of each measurement result is small, it indicates good measurement repeatability, which is called good measurement precision or good stability.

2. The magnitude of system error reflects the level of accuracy that measurement may achieve

According to error theory, when the number of measurements increases infinitely, the random error can approach zero, and the degree of deviation between the obtained measurement results and the true value - measurement accuracy - will fundamentally depend on the size of the system error.

3. Accuracy is the general term for the accuracy and precision of a measurement

In actual measurement, the main factors that affect accuracy may be systematic errors or random errors, and of course, both may have an impact on measurement accuracy that cannot be ignored. In some measuring instruments, the commonly used concept of accuracy actually includes two aspects: systematic error and random error. For example, commonly used instruments are often classified based on accuracy.

4 Instrument accuracy level and range

Accuracy is an important quality indicator for instruments, and precision levels are commonly used to standardize and represent them. The accuracy level is the maximum relative percentage error after removing the sign and%. According to national unified regulations, there are levels such as 0.05, 0.02, 0.1, 0.2, 1.5, etc. The smaller the number, the higher the accuracy of the instrument.

The accuracy of an instrument is not only related to absolute error, but also to the measurement range of the instrument. If two instruments with the same absolute error have different measurement ranges, the instrument with a larger measurement range will have a smaller relative percentage error and higher instrument accuracy; On the contrary, two instruments with the same accuracy level have a larger absolute error for instruments with a larger range.

5. Selection of application accuracy

In practical applications, the range and accuracy of the instrument should be selected based on the actual measurement situation. Instruments with low accuracy levels may not necessarily have the best measurement effect.