When measuring any quantity, there is invariably some deviation from the true value, due to the fact that no instrument can give an accurate result. In order to determine permissible deviations the obtained data from the exact value, the representations of relative and unconditional error are used.
You will need
- – measurement results;
- - calculator.
Instructions
1. First of all, take several measurements with an instrument of the same value in order to have a chance of calculating the actual value. The more measurements are taken, the more accurate the result will be. Let's say weigh an apple on an electronic scale. It is possible that you got results of 0.106, 0.111, 0.098 kg.
2. Now calculate the actual value of the quantity (real, because it is impossible to detect the true one). To do this, add up the resulting totals and divide them by the number of measurements, that is, find the arithmetic mean. In the example, the actual value would be (0.106+0.111+0.098)/3=0.105.
3. To calculate the unconditional error of the first measurement, subtract the actual value from the total: 0.106-0.105=0.001. In the same way, calculate the unconditional errors of the remaining measurements. Please note that regardless of whether the result turns out to be a minus or a plus, the sign of the error is invariably positive (that is, you take the absolute value).
4. In order to obtain the relative error of the first measurement, divide the absolute error by the actual value: 0.001/0.105=0.0095. Please note that the relative error is usually measured as a percentage, therefore multiply the resulting number by 100%: 0.0095x100% = 0.95%. In the same way, calculate the relative errors of other measurements.
5. If the true value is already known, immediately begin calculating the errors, eliminating the search for the arithmetic mean of the measurement results. Immediately subtract the resulting total from the true value, and you will discover an unconditional error.
6. After this, divide the absolute error by the true value and multiply by 100% - this will be the relative error. Let's say the number of students is 197, but it was rounded to 200. In this case, calculate the rounding error: 197-200=3, relative error: 3/197x100%=1.5%.
Error is a value that determines the permissible deviations of the obtained data from the exact value. There are concepts of relative and unconditional error. Finding them is one of the tasks of a mathematical review. However, in practice, it is more important to calculate the error in the spread of some measured indicator. Physical devices have their own possible error. But it’s not the only thing that needs to be considered when determining the indicator. To calculate the scatter error σ, it is necessary to carry out several measurements of this quantity.
You will need
- Device for measuring the required value
Instructions
1. Measure the value you need with a device or other measuring device. Repeat measurements several times. The larger the values obtained, the higher the accuracy of determining the scatter error. Traditionally, 6-10 measurements are taken. Write down the resulting set of measured value values.
2. If all the obtained values are equal, therefore, the scatter error is zero. If there are different values in the series, calculate the error of scatter. There is a special formula to determine it.
3. According to the formula, calculate first average value <х>from the obtained values. To do this, add up all the values and divide their sum by the number of measurements taken n.
4. Determine one by one the difference between the entire value obtained and the average value<х>. Write down the results of the differences obtained. After this, square all the differences. Find the sum of the given squares. You will save the final total amount received.
5. Evaluate the expression n(n-1), where n is the number of measurements you take. Divide the total from the previous calculation by the resulting value.
6. Take the square root of the quotient of the division. This will be the error in the spread of σ, the value you measured.
When carrying out measurements, it is impossible to guarantee their accuracy; every device gives a certain error. In order to find out the measurement accuracy or the accuracy class of the device, you need to determine the unconditional and relative error .
You will need
- – several measurement results or another sample;
- - calculator.
Instructions
1. Take measurements at least 3-5 times to be able to calculate the actual value of the parameter. Add up the resulting results and divide them by the number of measurements, you get the real value, which is used in tasks instead of the true one (it is impossible to determine it). Let's say, if the measurements gave a total of 8, 9, 8, 7, 10, then the actual value will be equal to (8+9+8+7+10)/5=8.4.
2. Discover unconditional error of the entire measurement. To do this, subtract the actual value from the measurement result, neglecting the signs. You will receive 5 unconditional errors, one for each measurement. In the example they will be equal to 8-8.4 = 0.4, 9-8.4 = 0.6, 8-8.4 = 0.4, 7-8.4 = 1.4, 10-8.4 =1.6 (total modules taken).
3. To find out the relative error any dimension, divide the unconditional error to the actual (true) value. After this, multiply the resulting total by 100%; traditionally this value is measured as a percentage. In the example, discover the relative error thus: ?1=0.4/8.4=0.048 (or 4.8%), ?2=0.6/8.4=0.071 (or 7.1%), ?3=0.4/ 8.4=0.048 (or 4.8%), ?4=1.4/8.4=0.167 (or 16.7%), ?5=1.6/8.4=0.19 (or 19 %).
4. In practice, to display the error particularly accurately, the standard deviation is used. In order to detect it, square all the unconditional measurement errors and add them together. Then divide this number by (N-1), where N is the number of measurements. By calculating the root of the resulting total, you will obtain the standard deviation, which characterizes error measurements.
5. In order to discover the ultimate unconditional error, find the minimum number that is obviously greater than the unconditional error or equal to it. In the example considered, simply select the largest value - 1.6. It is also occasionally necessary to discover the limiting relative error, in this case, find a number greater than or equal to the relative error, in the example it is 19%.
An inseparable part of any measurement is some error. It represents a good review of the accuracy of the research conducted. According to the form of presentation, it can be unconditional and relative.
You will need
- - calculator.
Instructions
1. Errors in physical measurements are divided into systematic, random and impudent. The former are caused by factors that act identically when measurements are repeated many times. They are continuous or change regularly. They can be caused by incorrect installation of the device or imperfection of the chosen measurement method.
2. The second appear from the power of causes, and causeless disposition. These include incorrect rounding when calculating readings and power environment. If such errors are much smaller than the scale divisions of this measuring device, then it is appropriate to take half the division as the absolute error.
3. Miss or daring error represents the result of tracking, one that is sharply different from all the others.
4. Unconditional error approximate numerical value is the difference between the result obtained during the measurement and the true value of the measured value. The true or actual value especially accurately reflects the physical quantity being studied. This error is the easiest quantitative measure of error. It can be calculated using the following formula: ?Х = Hisl – Hist. It can take on positive and negative meanings. For a better understanding, let's look at an example. The school has 1205 students, when rounded to 1200 the absolute error equals: ? = 1200 – 1205 = 5.
5. There are certain rules for calculating the error of values. Firstly, unconditional error the sum of 2 independent quantities is equal to the sum of their unconditional errors: ?(X+Y) = ?X+?Y. A similar approach is applicable for the difference of 2 errors. You can use the formula: ?(X-Y) = ?X+?Y.
6. The amendment constitutes an unconditional error, taken with the opposite sign: ?п = -?. It is used to eliminate systematic error.
Measurements physical quantities are invariably accompanied by one or another error. It represents the deviation of the measurement results from the true value of the measured value.
You will need
- -measuring device:
- -calculator.
Instructions
1. Errors may arise as a result of the power of various factors. Among them, we can highlight the imperfection of means or methods of measurement, inaccuracies in their manufacture, failure to special conditions when conducting research.
2. There are several systematizations of errors. According to the form of presentation, they can be unconditional, relative and reduced. The first represent the difference between the calculated and actual value of a quantity. They are expressed in units of the phenomenon being measured and are found using the formula:?x = hisl-hist. The latter are determined by the ratio of unconditional errors to the true value of the indicator. The calculation formula has the form:? = ?x/hist. It is measured in percentages or shares.
3. The reduced error of the measuring device is found as the ratio?x to the normalizing value xn. Depending on the type of device, it is taken either equal to the measurement limit or assigned to a certain range.
4. According to the conditions of origin, they distinguish between basic and additional. If the measurements were carried out under typical conditions, then the 1st type appears. Deviations caused by values outside the typical range are additional. To evaluate it, the documentation usually establishes standards within which the value can change if the measurement conditions are violated.
5. Also, errors in physical measurements are divided into systematic, random and daring. The first are caused by factors that act when measurements are repeated many times. The second appear from the power of causes, and causeless disposition. A miss represents the outcome of tracking, the one that is radically different from all the others.
6. Depending on the nature of the measured quantity, they can be used different methods measurement error. The first of them is the Kornfeld method. It is based on calculating the confidence interval ranging from the smallest to the maximum total. The error in this case will be half the difference between these totals: ?x = (xmax-xmin)/2. Another method is the calculation of the mean square error.
Measurements can be carried out with varying degrees accuracy. At the same time, even precision instruments are not absolutely accurate. Unconditional and relative error may be small, but in reality they are virtually unchanged. The difference between the approximate and exact values of a certain quantity is called unconditional error. In this case, the deviation can be either large or small.
You will need
- – measurement data;
- - calculator.
Instructions
1. Before calculating the unconditional error, take several postulates as initial data. Eliminate daring errors. Assume that the necessary corrections have already been calculated and included in the total. Such an amendment could be, say, moving the starting point of measurements.
2. Take as an initial position that random errors are known and taken into account. This implies that they are smaller than the systematic ones, that is, unconditional and relative, characteristic of this particular device.
3. Random errors affect the outcome of even highly accurate measurements. Consequently, every result will be more or less close to the unconditional, but there will invariably be discrepancies. Determine this interval. It can be expressed by the formula (Xism-?X)?Xism? (Hism+?X).
4. Determine the value that is as close as possible to the true value. In real measurements, the arithmetic mean is taken, which can be determined using the formula shown in the figure. Take the total as the true value. In many cases, the reading of the reference instrument is accepted as accurate.
5. Knowing the true measurement value, you can detect an unconditional error that must be considered in all subsequent measurements. Find the value of X1 - the data of a certain measurement. Determine the difference?X by subtracting the smaller number from the larger number. When determining the error, only the modulus of this difference is taken into account.
Note!
As usual, in practice it is impossible to carry out an absolutely accurate measurement. Consequently, the maximum error is taken as the reference value. She represents highest value absolute error module.
Helpful advice
In utilitarian measurements, the value of the unconditional error is usually taken to be half lowest price division. When working with numbers, the absolute error is taken to be half the value of the number, which is subsequently exact numbers discharge. To determine the accuracy class of an instrument, the most important thing is the ratio of the absolute error to the total measurement or to the length of the scale.
Measurement errors are associated with imperfection of instruments, instruments, and methodology. Accuracy also depends on the observation and state of the experimenter. Errors are divided into unconditional, relative and reduced.
Instructions
1. Let a single measurement of a quantity give the result x. The true value is denoted by x0. Then unconditional error?x=|x-x0|. It estimates the unconditional measurement error. Unconditional error consists of 3 components: random errors, systematic errors and misses. Usually, when measuring with an instrument, half the division value is taken as an error. For a millimeter ruler this would be 0.5 mm.
2. The true value of the measured value is in the interval (x-?x; x+?x). In short, this is written as x0=x±?x. The main thing is to measure x and ?x in the same units and write the numbers in the same format, say the whole part and three digits after the decimal point. It turns out unconditional error gives the boundaries of the interval in which, with some probability, the true value is located.
3. Relative error expresses the ratio of the unconditional error to the actual value of the quantity: ?(x)=?x/x0. This is a dimensionless quantity and can also be written as a percentage.
4. Measurements can be direct or indirect. In direct measurements, the desired value is immediately measured with the appropriate device. Let's say the length of a body is measured with a ruler, the voltage with a voltmeter. In indirect measurements, a value is found using the formula for the relationship between it and the measured values.
5. If the result is a connection between 3 easily measured quantities that have errors?x1, ?x2, ?x3, then error indirect measurement?F=?[(?x1 ?F/?x1)?+(?x2 ?F/?x2)?+(?x3 ?F/?x3)?]. Here?F/?x(i) are the partial derivatives of the function with respect to any of the easily measured quantities.
Helpful advice
Errors are daring inaccuracies in measurements that occur due to malfunction of instruments, inattentiveness of the experimenter, or violation of the experimental methodology. In order to reduce the likelihood of such mistakes, when taking measurements, be careful and describe the results obtained in detail.
The result of any measurement is inevitably accompanied by a deviation from the true value. The measurement error can be calculated using several methods depending on its type, for example, statistical methods for determining the confidence interval, standard deviation, etc.
Instructions
1. There are several reasons why errors measurements. These are instrument inaccuracy, imperfect methodology, as well as errors caused by the inattention of the operator taking measurements. In addition, the true value of a parameter is often taken to be its actual value, which in fact is only particularly possible, based on a review of a statistical sample of the results of a series of experiments.
2. Error is a measure of the deviation of a measured parameter from its true value. According to Kornfeld's method, a confidence interval is determined, one that guarantees a certain degree of security. In this case, the so-called confidence limits are found within which the value fluctuates, and the error is calculated as the half-sum of these values:? = (xmax – xmin)/2.
3. This is an interval estimate errors, which makes sense to carry out with a small statistical sample size. A point estimate consists of calculating the mathematical expectation and standard deviation.
4. The mathematical expectation is the integral sum of a series of products of 2 tracking parameters. These are, in fact, the values of the measured quantity and its probability at these points: M = ?xi pi.
5. The classic formula for calculating the standard deviation involves calculating the average value of the analyzed sequence of values of the measured value, and also considers the volume of a series of experiments performed:? = ?(?(xi – xav)?/(n – 1)).
6. According to the method of expression, unconditional, relative and reduced errors are also distinguished. The unconditional error is expressed in the same units as the measured value and is equal to the difference between its calculated and true value:?x = x1 – x0.
7. The relative measurement error is related to the unconditional error, but is more highly effective. It has no dimension and is sometimes expressed as a percentage. Its value is equal to the ratio of the unconditional errors to the true or calculated value of the measured parameter:?x = ?x/x0 or?x = ?x/x1.
8. The reduced error is expressed by the relationship between the unconditional error and some conventionally accepted value x, which is constant for all measurements and is determined by the calibration of the instrument scale. If the scale starts from zero (one-sided), then this normalizing value is equal to its upper limit, and if it’s two-sided, it’s equal to the width of each of its ranges:? = ?x/xn.
Self-monitoring for diabetes is considered an important component of treatment. A glucometer is used to measure blood sugar at home. The possible error of this device is higher than that of laboratory glycemic analyzers.
Measuring blood sugar is necessary to assess the effectiveness of diabetes treatment and to adjust the dose of medications. How many times a month you will need to measure your sugar depends on the prescribed therapy. Occasionally, blood sampling for review is necessary several times during the day, sometimes 1-2 times a week is enough. Self-monitoring is especially necessary for pregnant women and patients with type 1 diabetes.
Permissible error for a glucometer according to international standards
The glucometer is not considered a high-precision device. It is intended only for the approximate determination of blood sugar concentration. The possible error of a glucometer according to world standards is 20% when glycemia is more than 4.2 mmol/l. Let's say, if during self-control a sugar level of 5 mmol/l is recorded, then the real concentration value is in the range from 4 to 6 mmol/l. Possible error of the glucometer in standard conditions measured as a percentage, not mmol/l. The higher the indicators, the larger the error in absolute numbers. Let's say, if blood sugar reaches about 10 mmol/l, then the error does not exceed 2 mmol/l, and if sugar is about 20 mmol/l, then the difference with the result laboratory measurement can be up to 4 mmol/l. In most cases, the glucometer overestimates glycemic levels. The standards allow the stated measurement error to be exceeded in 5% of cases. This means that every twentieth study can significantly distort the results.
Permissible error for glucometers from various companies
Glucometers are subject to mandatory certification. The documents accompanying the device usually indicate figures for the possible measurement error. If this item is not in the instructions, then the error corresponds to 20%. Some glucometer manufacturers place special emphasis on measurement accuracy. There are devices from European companies that have a possible error of less than 20%. The best figure today is 10-15%.
Error in the glucometer during self-monitoring
The permissible measurement error characterizes the operation of the device. Several other factors also affect the accuracy of the survey. Abnormally prepared skin, too small or huge volume of blood drop received, unacceptable temperature regime– all this can lead to errors. Only if all the rules of self-control are followed can one rely on the stated possible research error. You can learn the rules of self-monitoring with the help of a glucometer from your doctor. The accuracy of the glucometer can be checked at service center. Manufacturers' warranties provide free consultations and troubleshooting.
Let some random variable a measured n times under the same conditions. The measurement results gave a set n different numbers
Absolute error- dimensional value. Among n Absolute error values are necessarily both positive and negative.
For the most probable value of the quantity A usually taken average value of measurement results
.
The greater the number of measurements, the closer the average value is to the true value.
Absolute errori
.
Relative errori-th measurement is called quantity
Relative error is a dimensionless quantity. Usually the relative error is expressed as a percentage, for this e i multiply by 100%. The magnitude of the relative error characterizes the accuracy of the measurement.
Average absolute error is defined like this:
.
We emphasize the need to sum the absolute values (modules) of the quantities D and i. Otherwise, the result will be identically zero.
Average relative error is called the quantity
.
For a large number of measurements.
Relative error can be considered as the error value per unit of the measured value.
The accuracy of measurements is judged by comparing the errors of the measurement results. Therefore, measurement errors are expressed in such a form that to assess the accuracy it is enough to compare only the errors of the results, without comparing the sizes of the objects being measured or knowing these sizes very approximately. It is known from practice that the absolute error in measuring an angle does not depend on the value of the angle, and the absolute error in measuring length depends on the value of the length. The greater the length, the this method and measurement conditions, the absolute error will be greater. Consequently, the absolute error of the result can be used to judge the accuracy of the angle measurement, but the accuracy of the length measurement cannot be judged. Expressing the error in relative form makes it possible to compare the accuracy of angular and linear measurements in known cases.
Basic concepts of probability theory. Random error.
Random error called the component of measurement error that changes randomly during repeated measurements of the same quantity.
When repeated measurements of the same constant, unchanging quantity are carried out with the same care and under the same conditions, we obtain measurement results - some of them differ from each other, and some of them coincide. Such discrepancies in measurement results indicate the presence of random error components in them.
Random error arises from the simultaneous influence of many sources, each of which in itself has an imperceptible effect on the measurement result, but the total influence of all sources can be quite strong.
Random errors are an inevitable consequence of any measurements and are caused by:
a) inaccuracy of readings on the scale of instruments and instruments;
b) non-identity of conditions for repeated measurements;
c) random changes external conditions(temperature, pressure, force field etc.) that cannot be controlled;
d) all other influences on measurements, the causes of which are unknown to us. The magnitude of random error can be minimized by repeating the experiment many times and corresponding mathematical processing of the results obtained.
A random error can take on different absolute value values that are impossible to predict for a given measurement act. This error in equally can be either positive or negative. Random errors are always present in an experiment. In the absence of systematic errors, they cause scatter of repeated measurements relative to the true value.
Let us assume that the period of oscillation of a pendulum is measured using a stopwatch, and the measurement is repeated many times. Errors in starting and stopping the stopwatch, an error in the reading value, a slight unevenness in the movement of the pendulum - all this causes scattering of the results of repeated measurements and therefore can be classified as random errors.
If there are no other errors, then some results will be somewhat overestimated, while others will be somewhat underestimated. But if, in addition to this, the clock is also behind, then all the results will be underestimated. This is already a systematic error.
Some factors can cause both systematic and random errors at the same time. So, by turning the stopwatch on and off, we can create a small irregular spread in the starting and stopping times of the clock relative to the movement of the pendulum and thereby introduce a random error. But if, moreover, we are in a hurry to turn on the stopwatch every time and are somewhat late to turn it off, then this will lead to a systematic error.
Random errors are caused by parallax error when counting instrument scale divisions, shaking of the foundation of a building, the influence of slight air movement, etc.
Although it is impossible to exclude random errors in individual measurements, mathematical theory random phenomena allow us to reduce the influence of these errors on the final measurement result. It will be shown below that for this it is necessary to make not one, but several measurements, and the smaller the error value we want to obtain, the more measurements need to be made.
Due to the fact that the occurrence of random errors is inevitable and unavoidable, the main task of any measurement process is to reduce errors to a minimum.
The theory of errors is based on two main assumptions, confirmed by experience:
1. With a large number of measurements, random errors are of the same magnitude, but different sign, that is, errors in the direction of increasing and decreasing the result occur quite often.
2. Errors that are large in absolute value are less common than small ones, thus, the probability of an error occurring decreases as its magnitude increases.
The behavior of random variables is described by statistical patterns, which are the subject of probability theory. Statistical definition probabilities w i events i is the relation
Where n- total number of experiments, n i- the number of experiments in which the event i happened. In this case, the total number of experiments should be very large ( n®¥). With a large number of measurements, random errors obey a normal distribution (Gaussian distribution), the main features of which are the following:
1. The greater the deviation of the measured value from the true value, the less likely it is for such a result.
2. Deviations in both directions from the true value are equally probable.
From the above assumptions it follows that in order to reduce the influence of random errors it is necessary to measure this value several times. Suppose we are measuring some quantity x. Let it be produced n measurements: x 1 , x 2 , ... x n- using the same method and with the same care. It can be expected that the number dn obtained results, which lie in some fairly narrow interval from x before x + dx, must be proportional:
The size of the interval taken dx;
Total number of measurements n.
Probability dw(x) that some value x lies in the range from x before x + dx, is defined as follows :
(with the number of measurements n ®¥).
Function f(X) is called the distribution function or probability density.
As a postulate of the error theory, it is accepted that the results of direct measurements and their random errors, when there are a large number of them, obey the law of normal distribution.
The distribution function of a continuous random variable found by Gauss x has the following form:
, where mis -
distribution parameters .
The parameter m of the normal distribution is equal to the mean value b xñ a random variable, which, for an arbitrary known distribution function, is determined by the integral
.
Thus, the value m is the most probable value measured quantity x, i.e. her best estimate.
The parameter s 2 of the normal distribution is equal to the variance D of the random variable, which in the general case is determined by the following integral
.
Square root from the variance is called the standard deviation of the random variable.
The average deviation (error) of the random variable ásñ is determined using the distribution function as follows
The average measurement error ásñ, calculated from the Gaussian distribution function, is related to the value of the standard deviation s as follows:
< s > = 0.8s.
The parameters s and m are related to each other as follows:
.
This expression allows you to find the standard deviation s if there is a normal distribution curve.
The graph of the Gaussian function is presented in the figures. Function f(x) is symmetrical about the ordinate drawn at the point x = m; passes through a maximum at the point x = m and has an inflection at points m ±s. Thus, variance characterizes the width of the distribution function, or shows how widely the values of a random variable are scattered relative to its true value. The more accurate the measurements, the closer to the true value the results of individual measurements, i.e. the value s is less. Figure A shows the function f(x) for three values of s .
Area of a figure enclosed by a curve f(x) and vertical lines drawn from points x 1 and x 2 (Fig.B) , numerically equal to the probability of the measurement result falling into the interval D x = x 1 - x 2, which is called the confidence probability. Area under the entire curve f(x) is equal to the probability of a random variable falling into the interval from 0 to ¥, i.e.
,
since the probability of a reliable event is equal to one.
Using the normal distribution, error theory poses and solves two main problems. The first is an assessment of the accuracy of the measurements taken. The second is an assessment of the accuracy of the average arithmetic value measurement results.5. Confidence interval. Student's coefficient.
Probability theory allows us to determine the size of the interval in which, with a known probability w the results of individual measurements are found. This probability is called confidence probability, and the corresponding interval (<x>±D x)w called confidence interval. The confidence probability is also equal to the relative proportion of results that fall within the confidence interval.
If the number of measurements n is sufficiently large, then the confidence probability expresses the proportion of the total number n those measurements in which the measured value was within the confidence interval. Each confidence probability w corresponds to its confidence interval. w 2 80%. The wider the confidence interval, the greater the likelihood of getting a result within that interval. In probability theory, a quantitative relationship is established between the value of the confidence interval, confidence probability and the number of measurements.
If we choose as a confidence interval the interval corresponding to the average error, that is, D a =áD Añ, then for a sufficiently large number of measurements it corresponds to the confidence probability w 60%. As the number of measurements decreases, the confidence probability corresponding to such a confidence interval (á Añ ± áD Añ), decreases.
Thus, to estimate the confidence interval of a random variable, one can use the value of the average error áD Añ .
To characterize the magnitude of the random error, it is necessary to specify two numbers, namely, the value of the confidence interval and the value of the confidence probability . Indicating only the magnitude of the error without the corresponding confidence probability is largely meaningless.
If the average measurement error ásñ is known, the confidence interval written as (<x> ± ásñ) w, determined with confidence probability w= 0,57.
If the standard deviation s is known distribution of measurement results, the specified interval has the form (<x>± t w s) w, Where t w- coefficient depending on the confidence probability value and calculated using the Gaussian distribution.
Most commonly used quantities D x are given in table 1.
Terms measurement error And measurement error are used interchangeably.) It is only possible to estimate the magnitude of this deviation, for example, using statistical methods. At the same time, for true meaning the average statistical value obtained by statistical processing of the results of a series of measurements is accepted. This obtained value is not exact, but only the most probable. Therefore, it is necessary to indicate in the measurements what their accuracy is. To do this, the measurement error is indicated along with the result obtained. For example, record T=2.8±0.1 c. means that the true value of the quantity T lies in the range from 2.7 s. before 2.9 s. some specified probability (see confidence interval, confidence probability, standard error).
In 2006, it was adopted at the international level new document, dictating the conditions for carrying out measurements and establishing new rules for comparing state standards. The concept of “error” became obsolete, and the concept of “measurement uncertainty” was introduced instead.
Determination of error
Depending on the characteristics of the measured quantity, various methods are used to determine the measurement error.
- The Kornfeld method consists in choosing a confidence interval ranging from the minimum to the maximum measurement result, and the error as half the difference between the maximum and minimum measurement result:
- Mean square error:
- Root mean square error of the arithmetic mean:
Error classification
According to presentation form
- Absolute error - Δ X is an estimate of the absolute measurement error. The magnitude of this error depends on the method of its calculation, which, in turn, is determined by the distribution of the random variable X meas . In this case the equality:
Δ X = | X true − X meas | ,
Where X true is the true value, and X meas - measured value must be fulfilled with some probability close to 1. If the random variable X meas is distributed according to the normal law, then, usually, its standard deviation is taken as the absolute error. Absolute error is measured in the same units as the quantity itself.
- Relative error- the ratio of the absolute error to the value that is accepted as true:
The relative error is a dimensionless quantity, or measured as a percentage.
- Reduced error- relative error, expressed as the ratio of the absolute error of the measuring instrument to the conventional accepted value a value that is constant over the entire measurement range or part of the range. Calculated by the formula
Where X n- normalizing value, which depends on the type of scale of the measuring device and is determined by its calibration:
If the instrument scale is one-sided, i.e. lower measurement limit equal to zero, That X n determined equal to the upper limit of measurement;
- if the instrument scale is double-sided, then the normalizing value is equal to the width of the instrument’s measurement range.
The given error is a dimensionless quantity (can be measured as a percentage).
Due to the occurrence
- Instrumental/instrumental errors- errors that are determined by the errors of the measuring instruments used and are caused by imperfections in the operating principle, inaccuracy of scale calibration, and lack of visibility of the device.
- Methodological errors- errors due to the imperfection of the method, as well as simplifications underlying the methodology.
- Subjective / operator / personal errors- errors due to the degree of attentiveness, concentration, preparedness and other qualities of the operator.
In technology, instruments are used to measure only with a certain predetermined accuracy - the main error allowed by the normal in normal conditions operation for this device.
If the device operates under conditions other than normal, then an additional error occurs, increasing the overall error of the device. Additional errors include: temperature, caused by a deviation of the ambient temperature from normal, installation, caused by a deviation of the device’s position from the normal operating position, etc. The normal ambient temperature is taken to be 20°C, and the normal Atmosphere pressure 01.325 kPa.
A generalized characteristic of measuring instruments is the accuracy class, determined by the maximum permissible main and additional errors, as well as other parameters affecting the accuracy of measuring instruments; the meaning of the parameters is established by standards for certain types of measuring instruments. The accuracy class of measuring instruments characterizes their precision properties, but is not a direct indicator of the accuracy of measurements performed using these instruments, since the accuracy also depends on the measurement method and the conditions for their implementation. Measuring instruments, the limits of permissible basic error of which are specified in the form of given basic (relative) errors, are assigned accuracy classes selected from a range the following numbers: (1; 1.5; 2.0; 2.5; 3.0; 4.0; 5.0; 6.0)*10n, where n = 1; 0; -1; -2, etc.
By nature of manifestation
- Random error- error that varies (in magnitude and sign) from measurement to measurement. Random errors can be associated with imperfection of instruments (friction in mechanical devices, etc.), shaking in urban conditions, with imperfection of the measurement object (for example, when measuring the diameter of a thin wire, which may not have a completely round cross-section as a result of imperfections in the manufacturing process ), with the characteristics of the measured quantity itself (for example, when measuring the quantity elementary particles passing per minute through a Geiger counter).
- Systematic error- an error that changes over time according to a certain law (a special case is a constant error that does not change over time). Systematic errors may be associated with instrument errors (incorrect scale, calibration, etc.) not taken into account by the experimenter.
- Progressive (drift) error- an unpredictable error that changes slowly over time. It is a non-stationary random process.
- Gross error (miss)- an error resulting from an oversight by the experimenter or a malfunction of the equipment (for example, if the experimenter incorrectly read the number of divisions on the instrument scale, if a short circuit occurred in the electrical circuit).
In practice, usually the numbers on which calculations are performed are approximate values of certain quantities. For brevity, the approximate value of a quantity is called an approximate number. The true value of a quantity is called an exact number. An approximate number has practical value only when we can determine with what degree of accuracy it is given, i.e. estimate its error. Let us recall the basic concepts from general course mathematics.
Let's denote: x- exact number (true value of the quantity), A- approximate number (approximate value of a quantity).
Definition 1. The error (or true error) of an approximate number is the difference between the number x and its approximate value A. Approximate number error A we will denote . So,
Exact number x most often it is unknown, so it is not possible to find the true and absolute error. On the other hand, it may be necessary to estimate the absolute error, i.e. indicate the number that the absolute error cannot exceed. For example, when measuring the length of an object with this tool, we must be sure that the error in the resulting numerical value will not exceed a certain number, for example 0.1 mm. In other words, we must know the absolute error limit. We will call this limit the maximum absolute error.
Definition 3. Maximum absolute error of the approximate number A is a positive number such that , i.e.
Means, X by deficiency, by excess. The following notation is also used:
| . | (2.5) |
It is clear that the maximum absolute error is determined ambiguously: if a certain number is the maximum absolute error, then any larger number There is also a maximum absolute error. In practice, they try to choose the smallest and simplest possible entry (from 1-2 significant figures) number satisfying inequality (2.3).
Example.Determine the true, absolute and maximum absolute error of the number a = 0.17, taken as an approximate value of the number.
True error: 
Absolute error: 
The maximum absolute error can be taken as a number and any larger number. In decimal notation we will have: Replacing this number with a larger and possibly simpler notation, we accept:
Comment. If A is an approximate value of the number X, and the maximum absolute error is equal to h, then they say that A is an approximate value of the number X up to h.
Knowing the absolute error is not enough to characterize the quality of a measurement or calculation. Let, for example, such results be obtained when measuring length. Distance between two cities S 1=500 1 km and the distance between two buildings in the city S 2=10 1 km. Although the absolute errors of both results are the same, what is significant is that in the first case an absolute error of 1 km falls on 500 km, in the second - on 10 km. The measurement quality in the first case is better than in the second. The quality of a measurement or calculation result is characterized by relative error.
Definition 4. Relative error of the approximate value A numbers X is called the ratio of the absolute error of a number A to the absolute value of a number X:
Definition 5. Maximum relative error of the approximate number A is called a positive number such that .
Since , it follows from formula (2.7) that it can be calculated using the formula
| . | (2.8) |
For the sake of brevity, in cases where this does not cause misunderstandings, instead of “maximum relative error” we simply say “relative error”.
The maximum relative error is often expressed as a percentage.
Example 1. . Assuming , we can accept = . By dividing and rounding (necessarily upward), we get =0.0008=0.08%.
Example 2.When weighing the body, the result was obtained: p = 23.4 0.2 g. We have = 0.2. . By dividing and rounding, we get =0.9%.
Formula (2.8) determines the relationship between absolute and relative errors. From formula (2.8) it follows:
| . | (2.9) |
Using formulas (2.8) and (2.9), we can, if the number is known A, using a given absolute error, find the relative error and vice versa.
Note that formulas (2.8) and (2.9) often have to be applied even when we do not yet know the approximate number A with the required accuracy, but we know a rough approximate value A. For example, you need to measure the length of an object with a relative error of no more than 0.1%. The question is: is it possible to measure the length with the required accuracy using a caliper, which allows you to measure the length with an absolute error of up to 0.1 mm? We may not have measured an object with an exact instrument yet, but we know that a rough approximation of the length is about 12 cm. Using formula (1.9) we find the absolute error:
This shows that using a caliper it is possible to perform measurements with the required accuracy.
In the process of computational work, it is often necessary to switch from absolute to relative error, and vice versa, which is done using formulas (1.8) and (1.9).
In the process of measuring something, you need to take into account that the result obtained is not yet final. To more accurately calculate the desired value, it is necessary to take into account the error. Calculating it is quite simple.
How to find the error - calculation
Types of errors:
- relative;
- absolute.
What is needed for the calculation:
- calculator;
- results of several measurements of one quantity.
How to find an error - sequence of actions
- Measure the value 3 – 5 times.
- Add up all the results and divide the resulting number by their number. This number is the real value.
- Calculate the absolute error by subtracting the value obtained in the previous step from the measurement results. Formula: ∆Х = Hisl – Hist. During the calculations, you can get both positive and negative values. In any case, the result module is taken. If it is necessary to find out the absolute error of the sum of two quantities, then calculations are carried out according to the following formula: ∆(X+Y) = ∆X+∆Y. It also works when it is necessary to calculate the error of the difference between two quantities: ∆(X-Y) = ∆X+∆Y.
- Find out the relative error for each measurement. In this case, you need to divide the resulting absolute error by the actual value. Then multiply the quotient by 100%. ε(x)=Δx/x0*100%. The value may not be converted into a percentage.
- To obtain a more accurate error value, it is necessary to find the standard deviation. Finding it is quite simple: calculate the squares of all absolute error values, and then find their sum. The result obtained must be divided by a number (N-1), in which N is the number of all measurements. The last step is to extract the root of the result. After such calculations, the standard deviation will be obtained, which usually characterizes the measurement error.
- To find the maximum absolute error, it is necessary to find the smallest number whose value is equal to or greater than the value of the absolute error.
- The maximum relative error is sought using the same method, only you need to find a number that is greater than or equal to the relative error value.
Measurement errors arise for various reasons and affect the accuracy of the obtained value. Knowing what the error is, you can find out a more accurate value of the measurement taken.