An assessment of the accuracy of the experimental results is mandatory, since the obtained values may lie within the possible experimental error, and the derived patterns may turn out to be unclear and even incorrect. Accuracy is the degree of correspondence of measurement results to the actual value of the measured quantity. Concept of accuracy associated with the concept of error: the higher the accuracy, the smaller the measurement error, and vice versa. The most accurate instruments cannot show the actual value of a value; their readings contain an error.
The difference between the actual value of the measured quantity and the measured one is called absolute error measurements. Almost within absolute error 
understand the difference between the measurement result using more accurate methods or instruments of higher accuracy (exemplary) and the value of this value obtained by the device used in the study:
Absolute error cannot, however, serve as a measure of accuracy, since, for example, 
at = 100 mm it is quite small, but at = 1 mm it is very large. Therefore, to assess the accuracy of measurements, the concept is introduced relative error 
, equal to the ratio of the absolute error of the measurement result to the measured value
. (1.8)
For measure accuracy measured quantity is understood to be the reciprocal
. Hence, the smaller the relative error 
, the higher the measurement accuracy. For example, if the relative measurement error is obtained equal to 2%, then they say that the measurements were made with an error of no more than 2%, or with an accuracy of at least 0.5%, or with an accuracy of at least 1/0.02 = 50. The term should not be used "accuracy" instead of the terms "absolute error" and "relative error". For example, it is incorrect to say “the mass was measured with an accuracy of 0.1 mg,” since 0.1 mg is not accuracy, but the absolute error in measuring mass.
There are systematic, random and gross measurement errors.
Systematic errors are associated mainly with the errors of measuring instruments and remain constant with repeated measurements.
Random errors caused by uncontrollable circumstances, such as friction in devices. Random measurement errors can be expressed in several concepts.
Under ultimate(maximum) absolute error understand its value at which the probability of the error falling within the interval 
so great that the event can be considered almost certain. In this case, only in some cases the error can go beyond the specified interval. A measurement with such an error is called a rough measurement (or miss) and is excluded from consideration when processing the results.
The value of the measured quantity can be represented by the formula
which should be read as follows: the true value of the measured quantity is within the range from 
before 
.
The method of processing experimental data depends on the nature measurements, which can be direct and indirect, single and multiple. Measurements of quantities are made once when it is impossible or difficult to repeat the measurement conditions. This usually occurs during measurements in industrial and sometimes laboratory conditions.
The value of the measured quantity during a single measurement by the device may differ from the true values by no more than the value of the maximum error allowed by the accuracy class of the device ,
. (1.9)
As follows from relation (1.9), instrument accuracy class expresses the largest permissible error 
as a percentage of the nominal value 
(limit) scale of the device. All devices are divided into eight accuracy classes: 0.05; 0.1; 0.2; 0.5; 1.0; 1.5; 2.5 and 4.0.
It must be remembered that the accuracy class of a device does not yet characterize the accuracy of measurements obtained when using this device, since relative error measurements in the initial part of the scale more(less accuracy), than at the end of the scale with an almost constant absolute error. It is the presence of this property of indicating instruments that explains the desire to choose the measurement limit of the device in such a way that during operation of the device the scale was counted in the area between the middle of the scale and its end mark or, in other words, in the second half of the scale.
Example. Let the wattmeter be rated at 250 W (= 250 W) with an accuracy class = 0.5 measured power = 50 W. It is required to determine the maximum absolute error and relative measurement error. For this device, an absolute error of 0.5% of the upper measurement limit is allowed in any part of the scale, i.e. from 250 W, which is
Limit relative error at measured power 50 W
.
From this example it is clear that the accuracy class of the device ( = 0.5) and the maximum relative measurement error at an arbitrary point on the instrument scale (in the example, 2.5% for 50 W) are not equal in the general case (they are equal only for the nominal value of the instrument scale).
Indirect measurements are used when direct measurements of the desired quantity are impracticable or difficult. Indirect measurements are reduced to measuring independent quantities A, B, C..., associated with the desired value by functional dependence 
.
Maximum relative error indirect measurements of a quantity is equal to the differential of its natural logarithm, and one should take sum of absolute values all members of such an expression (take with a plus sign):
In thermotechnical experiments, indirect measurements are used to determine the thermal conductivity of a material, heat transfer and heat transfer coefficients. As an example, consider the calculation of the maximum relative error for indirect measurement of thermal conductivity.
The thermal conductivity of a material using the cylindrical layer method is expressed by the equation
.
The logarithm of this function has the form
and the differential taking into account the rule of signs (everything is taken with a plus)
Then the relative error in measuring the thermal conductivity of the material, considering 
And 
, will be determined by the expression
The absolute error in measuring the length and diameter of a pipe is taken to be equal to half the value of the smallest scale division of a ruler or caliper, temperature and heat flow - according to the readings of the corresponding instruments, taking into account their accuracy class.
When determining the values of random errors, in addition to the maximum error, the statistical error of repeated (several) measurements is calculated. This error is established after measurements using methods of mathematical statistics and error theory.
Error theory recommends using the arithmetic mean as an approximate value of the measured value:
, (1.12)
where is the number of measurements of the quantity .
To assess the reliability of measurement results taken equal to the average value, it is used standard deviation of the result of several measurements(arithmetic mean)
Measurement error is the deviation of the measurement result from the true value of the measured value. The smaller the error, the higher the accuracy. Types of errors are presented in Fig. eleven.
Systematic error– component of the measurement error that remains constant or changes naturally with repeated measurements of the same quantity. Systematic errors include, for example, errors from the discrepancy between the actual value of the measure with which the measurements were made and its nominal value (errors in instrument readings due to incorrect scale calibration).
Systematic errors can be studied experimentally and eliminated from measurement results by introducing appropriate corrections.
Amendment– the value of a quantity of the same name as the one being measured, added to the value obtained during measurements in order to eliminate systematic error.
Random error is a component of the measurement error that changes randomly with repeated measurements of the same quantity. For example, errors due to variations in the readings of the measuring device, errors in rounding or counting of the readings of the device, temperature fluctuations during the measurement process, etc. They cannot be established in advance, but their influence can be reduced by repeated repeated measurements of one value and processing of experimental data based on probability theory and mathematical statistics.
To gross errors(misses) refer to random errors that significantly exceed the errors expected under given measurement conditions. For example, incorrect reading on the instrument scale, incorrect installation of the part being measured during the measurement process, etc. Gross errors are not taken into account and are excluded from the measurement results, because are the result of a miscalculation.
Fig. 11. Error classification
Absolute error– measurement error, expressed in units of the measured value. Absolute error determined by the formula.
= meas. – , (1.5)
Where change- measured value; - true (actual) value of the measured quantity.
Relative measurement error– the ratio of the absolute error to the true value of a physical quantity (PV):
= or 100% (1.6)
In practice, instead of the true value of the PV, the actual value of the PV is used, by which we mean a value that differs from the true one so little that for this specific purpose this difference can be neglected.
Reduced error– is defined as the ratio of the absolute error to the normalizing value of the measured physical quantity, that is:
, (1.7)
Where X N – normalizing value of the measured quantity.
Standard value X N selected depending on the type and nature of the instrument scale. This value is taken equal to:
The final value of the working part of the scale. X N = X K, if the zero mark is on the edge or outside the working part of the scale (uniform scale Fig. 12, A - X N = 50; rice. 12, b - X N = 55; power scale - X N = 4 in Fig. 12, e);
The sum of the final values of the scale (without taking into account the sign), if the zero mark is inside the scale (Fig. 12, V - X N= 20 + 20 = 40; Fig. 12, G - X N = 20 + 40 = 60);
The length of the scale, if it is significantly uneven (Fig. 12, d). In this case, since the length is expressed in millimeters, the absolute error is also expressed in millimeters.
Rice. 12. Types of scales
Measurement error is the result of the superposition of elementary errors caused by various reasons. Let us consider the individual components of the total measurement error.
Methodological error is caused by the imperfection of the measurement method, for example, an incorrectly selected basing (installation) scheme for the product, an incorrectly selected sequence of measurements, etc. Examples of methodological error are:
- Reading error– occurs due to insufficiently accurate reading of the instrument and depends on the individual abilities of the observer.
- Interpolation error when counting- occurs from an insufficiently accurate eye assessment of the fraction of the scale division corresponding to the position of the pointer.
- Parallax error arises as a result of sighting (observation) of an arrow located at a certain distance from the scale surface in a direction not perpendicular to the scale surface (Fig. 13).
- Error due to measuring force arise due to contact deformations of surfaces at the point of contact between the surfaces of the measuring instrument and the product; thin-walled parts; elastic deformations of installation equipment, such as brackets, stands or tripods.
 |
Fig. 13. Diagram of the occurrence of errors due to parallax.
Parallax error n directly proportional to distance h pointer 1 from scale 2 and the tangent of the angle φ of the observer’s line of sight to the scale surface n = h× tg φ(Fig. 13).
Instrumental error– is determined by the error of the measuring instruments used, i.e. the quality of their manufacture. An example of instrumental error is skew error.
Skew error occurs in devices whose design does not comply with the Abbe principle, which consists in the fact that the measurement line should be a continuation of the scale line, for example, the skew of the caliper frame changes the distance between jaws 1 and 2 (Fig. 14).
Error in determining the measured size due to skew lane = l× cosφ. When fulfilling Abbe's principle l× cosφ= 0 accordingly lane . = 0.
Subjective errors are related to the individual characteristics of the operator. As a rule, this error occurs due to errors in readings and operator inexperience.
The types of instrumental, methodological and subjective errors discussed above cause the appearance of systematic and random errors, which make up the total measurement error. They can also lead to gross measurement errors. The total measurement error may include errors due to the influence of measurement conditions. These include basic And additional errors.
Fig. 14. Measurement error due to skew of caliper jaws.
Basic error is the error of the measuring instrument under normal operating conditions. As a rule, normal operating conditions are: temperature 293 ± 5 K or 20 ± 5 ° C, relative humidity 65 ± 15% at 20 ° C, power supply voltage 220 V ± 10% with a frequency of 50 Hz ± 1%, atmospheric pressure from 97.4 to 104 kPa, absence of electric and magnetic fields.
In operating conditions, which often differ from normal ones due to a wider range of influencing quantities, additional error measuring instruments.
Additional error arises as a result of instability of the object’s operating mode, electromagnetic interference, fluctuations in power supply parameters, the presence of moisture, shock and vibration, temperature, etc.
For example, a temperature deviation from the normal value of +20°C leads to a change in the length of parts of measuring instruments and products. If it is impossible to meet the requirements for normal conditions, then a temperature correction D should be introduced into the result of linear measurements X t, determined by the formula:
D X t = X MEASURE .. [α 1 (t 1 -20)- α 2 (t 2 -20)](1.8)
Where X MEASURE. - measured size; α 1 And α 2- coefficients of linear expansion of materials of the measuring instrument and product; t 1 And t 2- temperatures of measuring instruments and products.
The additional error is normalized in the form of a coefficient indicating “by how much” or “how much” the error changes when the nominal value deviates. For example, stating that a voltmeter has a temperature error of ±1% per 10°C means that for every 10°C change in environment an additional 1% error is added.
Thus, increasing the accuracy of dimensional measurement is achieved by reducing the influence of individual errors on the measurement result. For example, you need to select the most accurate instruments, set them to zero (size) using high-grade length gauges, entrust measurements to experienced specialists, etc.
Static errors are constant, not changing during the measurement process, for example, incorrect setting of the reference point, incorrect setting of the SI.
Dynamic errors are variables in the measurement process; they can monotonically decrease, increase or change periodically.
For each measuring instrument, the error is given only in one form.
If the SI error under constant external conditions is constant over the entire measurement range (given by one number), then
D = ± a. (1.9)
If the error varies within the specified range (set by a linear dependence), then
D = ± (a + bx) (1.10)
At D = ± a the error is called additive, and when D =± (a+bx) – multiplicative.
If the error is expressed as a function D = f(x), then it is called nonlinear.
Physical quantities are characterized by the concept of “error accuracy”. There is a saying that by taking measurements you can come to knowledge. This way you can find out the height of the house or the length of the street, like many others.
Introduction
Let us understand the meaning of the concept of “measure a quantity”. The measurement process is to compare it with homogeneous quantities, which are taken as a unit.
Liters are used to determine volume, grams are used to calculate mass. To make calculations more convenient, the SI system of international classification of units was introduced.
For measuring the length of the stick in meters, mass - kilograms, volume - cubic liters, time - seconds, speed - meters per second.
When calculating physical quantities, it is not always necessary to use the traditional method; it is enough to use the calculation using a formula. For example, to calculate indicators such as average speed, you need to divide the distance traveled by the time spent on the road. This is how the average speed is calculated.
When using units of measurement that are ten, one hundred, thousand times higher than the accepted measurement units, they are called multiples.
The name of each prefix corresponds to its multiplier number:
- Deca.
- Hecto.
- Kilo.
- Mega.
- Giga.
- Tera.
In physical science, powers of 10 are used to write such factors. For example, a million is written as 10 6 .
In a simple ruler, length has a unit of measurement - centimeters. It is 100 times less than a meter. A 15 cm ruler is 0.15 m long.
A ruler is the simplest type of measuring instrument for measuring lengths. More complex devices are represented by a thermometer - to a hygrometer - to determine humidity, an ammeter - to measure the level of force with which electric current propagates.
How accurate will the measurements be?
Take a ruler and a simple pencil. Our task is to measure the length of this stationery.
First you need to determine what the division price indicated on the scale of the measuring device is. On the two divisions, which are the closest strokes of the scale, numbers are written, for example, “1” and “2”.
It is necessary to count how many divisions are between these numbers. If counted correctly it will be "10". Let us subtract from the number that is larger the number that will be smaller and divide by the number that is the division between the digits:
(2-1)/10 = 0.1 (cm)
So we determine that the price that determines the division of stationery is the number 0.1 cm or 1 mm. It is clearly shown how the price indicator for division is determined using any measuring device.
When measuring a pencil with a length that is slightly less than 10 cm, we will use the knowledge gained. If there were no fine divisions on the ruler, it would be concluded that the object has a length of 10 cm. This approximate value is called the measurement error. It indicates the level of inaccuracy that can be tolerated when making measurements.
By determining the parameters of the length of a pencil with a higher level of accuracy, with a larger division price, greater measurement accuracy is achieved, which ensures a smaller error.
In this case, absolutely accurate measurements cannot be taken. And the indicators should not exceed the size of the division price.
It has been established that the measurement error is ½ of the price, which is indicated on the graduations of the device used to determine the dimensions.
After taking measurements of a pencil of 9.7 cm, we will determine its error indicators. This is the interval 9.65 - 9.85 cm.
The formula that measures this error is the calculation:
A = a ± D (a)
A - in the form of a quantity for measuring processes;
a is the value of the measurement result;
D - designation of absolute error.
When subtracting or adding values with an error, the result will be equal to the sum of the error indicators, which is each individual value.
Introduction to the concept
If we consider depending on the method of its expression, we can distinguish the following varieties:
- Absolute.
- Relative.
- Given.
The absolute measurement error is indicated by the letter “Delta” in capital. This concept is defined as the difference between the measured and actual values of the physical quantity that is being measured.
The expression of absolute measurement error is the units of the quantity that needs to be measured.
When measuring mass, it will be expressed, for example, in kilograms. This is not a measurement accuracy standard.
How to calculate the error of direct measurements?
There are ways to depict measurement errors and calculate them. To do this, it is important to be able to determine a physical quantity with the required accuracy, to know what the absolute measurement error is, that no one will ever be able to find it. Only its boundary value can be calculated.
Even if this term is used conventionally, it indicates precisely the boundary data. Absolute and relative measurement errors are indicated by the same letters, the difference is in their spelling.
When measuring length, the absolute error will be measured in the units in which the length is calculated. And the relative error is calculated without dimensions, since it is the ratio of the absolute error to the measurement result. This value is often expressed as a percentage or fraction.
Absolute and relative measurement errors have several different methods of calculation, depending on what physical quantity.
Concept of direct measurement
The absolute and relative errors of direct measurements depend on the accuracy class of the device and the ability to determine the weighing error.
Before we talk about how the error is calculated, it is necessary to clarify the definitions. Direct measurement is a measurement in which the result is directly read from the instrument scale.
When we use a thermometer, ruler, voltmeter or ammeter, we always carry out direct measurements, since we directly use a device with a scale.
There are two factors that influence the effectiveness of the readings:
- Instrument error.
- The error of the reference system.
The absolute error limit for direct measurements will be equal to the sum of the error that the device shows and the error that occurs during the counting process.
D = D (flat) + D (zero)
Example with a medical thermometer
The error indicators are indicated on the device itself. A medical thermometer has an error of 0.1 degrees Celsius. The counting error is half the division value.
D ots. = C/2
If the division value is 0.1 degrees, then for a medical thermometer you can make the following calculations:
D = 0.1 o C + 0.1 o C / 2 = 0.15 o C
On the back of the scale of another thermometer there is a specification and it is indicated that for correct measurements it is necessary to immerse the entire back of the thermometer. not specified. All that remains is the counting error.
If the scale division value of this thermometer is 2 o C, then it is possible to measure temperature with an accuracy of 1 o C. These are the limits of the permissible absolute measurement error and the calculation of the absolute measurement error.
A special system for calculating accuracy is used in electrical measuring instruments.
Accuracy of electrical measuring instruments
To specify the accuracy of such devices, a value called accuracy class is used. The letter “Gamma” is used to designate it. To accurately determine the absolute and relative measurement error, you need to know the accuracy class of the device, which is indicated on the scale.
Let's take an ammeter for example. Its scale indicates the accuracy class, which shows the number 0.5. It is suitable for measurements on direct and alternating current and belongs to electromagnetic system devices.
This is a fairly accurate device. If you compare it with a school voltmeter, you can see that it has an accuracy class of 4. You must know this value for further calculations.
Application of knowledge
Thus, D c = c (max) X γ /100
We will use this formula for specific examples. Let's use a voltmeter and find the error in measuring the voltage provided by the battery.
Let's connect the battery directly to the voltmeter, first checking whether the needle is at zero. When connecting the device, the needle deviated by 4.2 divisions. This state can be characterized as follows:
- It can be seen that the maximum U value for this item is 6.
- Accuracy class -(γ) = 4.
- U(o) = 4.2 V.
- C=0.2 V
Using these formula data, the absolute and relative measurement error is calculated as follows:
D U = DU (ex.) + C/2
D U (ex.) = U (max) X γ /100
D U (ex.) = 6 V X 4/100 = 0.24 V
This is the error of the device.
The calculation of the absolute measurement error in this case will be performed as follows:
D U = 0.24 V + 0.1 V = 0.34 V
Using the formula discussed above, you can easily find out how to calculate the absolute measurement error.
There is a rule for rounding errors. It allows you to find the average between the absolute and relative error limits.
Learning to determine weighing error
This is one example of direct measurements. Weighing has a special place. After all, lever scales do not have a scale. Let's learn how to determine the error of such a process. Accuracy is affected by the accuracy of the weights and the perfection of the scales themselves.
We use lever scales with a set of weights that must be placed on the right pan of the scale. To weigh, take a ruler.
Before starting the experiment, you need to balance the scales. Place the ruler on the left bowl.
The mass will be equal to the sum of the installed weights. Let us determine the error in measuring this quantity.
D m = D m (scales) + D m (weights)
The error in mass measurement consists of two terms associated with scales and weights. To find out each of these values, factories producing scales and weights provide products with special documents that allow the accuracy to be calculated.
Using tables
Let's use a standard table. The error of the scale depends on what mass is put on the scale. The larger it is, the correspondingly larger the error.
Even if you put a very light body, there will be an error. This is due to the friction process occurring in the axes.
The second table is for a set of weights. It indicates that each of them has its own mass error. The 10 gram has an error of 1 mg, the same as the 20 gram. Let's calculate the sum of the errors of each of these weights taken from the table.
It is convenient to write the mass and mass error in two lines, which are located one below the other. The smaller the weights, the more accurate the measurement.
Results
In the course of the material reviewed, it was established that it is impossible to determine the absolute error. You can only set its boundary indicators. To do this, use the formulas described above in the calculations. This material is proposed for study at school for students in grades 8-9. Based on the knowledge gained, you can solve problems to determine the absolute and relative errors.
The result of a measurement is the value of a quantity found by measuring it. The result obtained always contains some error.
Thus, the measurement task includes not only finding the value itself, but also estimating the error allowed during the measurement.
The absolute measurement error D refers to the deviation of the measurement result of a given value A from its true meaning A x
D= A – Ax. (IN 1)
In practice, instead of the true value which is unknown, the actual value is usually used.
The error calculated using formula (B.1) is called the absolute error and is expressed in units of the measured value.
The quality of measurement results is usually conveniently characterized not by the absolute error D, but by its ratio to the measured value, which is called the relative error and is usually expressed as a percentage:
ε = (D / A) 100 %. (AT 2)
The relative error ε is the ratio of the absolute error to the measured value.
The relative error ε is directly related to the measurement accuracy.
Measurement accuracy is the quality of a measurement, reflecting the closeness of its results to the true value of the measured value. Measurement accuracy is the reciprocal of its relative error. High measurement accuracy corresponds to small relative errors.
The magnitude and sign of the error D depends on the quality of the measuring instruments, the nature and conditions of the measurements, and the experience of the observer.
All errors, depending on the reasons for their occurrence, are divided into three types: A) systematic; b) random; V) misses.
Systematic errors are errors whose magnitude is the same in all measurements carried out by the same method using the same measuring instruments.
Systematic errors can be divided into three groups.
1. Errors, the nature of which is known and the magnitude can be determined quite accurately. Such errors are called corrections. For example, A) when determining the length, the elongation of the measured body and the measuring ruler due to temperature changes; b) when determining weight - an error caused by “weight loss” in the air, the magnitude of which depends on temperature, humidity and atmospheric air pressure, etc.
The sources of such errors are carefully analyzed, the magnitude of the corrections is determined and taken into account in the final result.
2. Errors of measuring instruments δ cl t. For the convenience of comparing devices with each other, the concept of reduced error d pr (%) has been introduced
Where A k– some normalized value, for example, the final value of the scale, the sum of the values of a two-sided scale, etc.
The accuracy class of a device d class t is a physical quantity that is numerically equal to the greatest permissible reduced error, expressed
as a percentage, i.e.
d cl p = d pr max
Electrical measuring instruments are usually characterized by an accuracy class ranging from 0.05 to 4.
If an accuracy class of 0.5 is indicated on the device, this means that the device readings have an error of up to 0.5% of the entire operating scale of the device. Errors in measuring instruments cannot be excluded, but their largest value D max can be determined.
The value of the maximum absolute error of a given device is calculated according to its accuracy class
(AT 4)
When measuring with a device whose accuracy class is not specified, the absolute measurement error is usually equal to half the value of the smallest scale division.
3. The third type includes errors whose existence is not suspected. For example: it is necessary to measure the density of some metal; for this, the volume and mass of the sample are measured.
If the sample being measured contains voids inside, for example, air bubbles trapped during casting, then the density measurement is carried out with systematic errors, the magnitude of which is unknown.
Random errors are those errors whose nature and magnitude are unknown.
Random measurement errors arise due to the simultaneous influence on the measurement object of several independent quantities, the changes of which are of a fluctuation nature. It is impossible to exclude random errors from measurement results. It is only possible, on the basis of the theory of random errors, to indicate the limits between which the true value of the measured quantity lies, the probability of the true value being within these limits, and its most probable value.
Misses are observational errors. The source of errors is the lack of attention of the experimenter.
You should understand and remember:
1) if the systematic error is decisive, that is, its value is significantly greater than the random error inherent in this method, then it is enough to perform the measurement once;
2) if random error is decisive, then the measurement should be performed several times;
3) if the systematic Dsi and random Dcl errors are comparable, then the total D total measurement error is calculated based on the law of adding errors, as their geometric sum
It is almost impossible to determine the true value of a physical quantity absolutely accurately, because any measurement operation is associated with a number of errors or, in other words, inaccuracies. The reasons for errors can be very different. Their occurrence may be associated with inaccuracies in the manufacture and adjustment of the measuring device, due to the physical characteristics of the object under study (for example, when measuring the diameter of a wire of non-uniform thickness, the result randomly depends on the choice of the measurement site), random reasons, etc.
The experimenter’s task is to reduce their influence on the result, and also to indicate how close the result obtained is to the true one.
There are concepts of absolute and relative error.
Under absolute error measurements will understand the difference between the measurement result and the true value of the measured quantity:
∆x i =x i -x and (2)
where ∆x i is the absolute error of the i-th measurement, x i _ is the result of the i-th measurement, x and is the true value of the measured value.
The result of any physical measurement is usually written in the form:
where is the arithmetic mean value of the measured value, closest to the true value (the validity of x and ≈ will be shown below), is the absolute measurement error.
Equality (3) should be understood in such a way that the true value of the measured quantity lies in the interval [ - , + ].
Absolute error is a dimensional quantity; it has the same dimension as the measured quantity.
The absolute error does not fully characterize the accuracy of the measurements taken. In fact, if we measure segments 1 m and 5 mm long with the same absolute error ± 1 mm, the accuracy of the measurements will be incomparable. Therefore, along with the absolute measurement error, the relative error is calculated.
Relative error measurements is the ratio of the absolute error to the measured value itself:
Relative error is a dimensionless quantity. It is expressed as a percentage:
In the example above, the relative errors are 0.1% and 20%. They differ markedly from each other, although the absolute values are the same. Relative error gives information about accuracy
Measurement errors
According to the nature of the manifestation and the reasons for the occurrence of errors, they can be divided into the following classes: instrumental, systematic, random, and misses (gross errors).
Errors are caused either by a malfunction of the device, or a violation of the methodology or experimental conditions, or are of a subjective nature. In practice, they are defined as results that differ sharply from others. To eliminate their occurrence, it is necessary to be careful and thorough when working with devices. Results containing errors must be excluded from consideration (discarded).
Instrument errors. If the measuring device is in good working order and adjusted, then measurements can be made on it with limited accuracy determined by the type of device. It is customary to consider the instrument error of a pointer instrument to be equal to half the smallest division of its scale. In instruments with digital readout, the instrument error is equated to the value of one smallest digit of the instrument scale.
Systematic errors are errors whose magnitude and sign are constant for the entire series of measurements carried out by the same method and using the same measuring instruments.
When carrying out measurements, it is important not only to take into account systematic errors, but it is also necessary to ensure their elimination.
Systematic errors are conventionally divided into four groups:
1) errors, the nature of which is known and their magnitude can be determined quite accurately. Such an error is, for example, a change in the measured mass in the air, which depends on temperature, humidity, air pressure, etc.;
2) errors, the nature of which is known, but the magnitude of the error itself is unknown. Such errors include errors caused by the measuring device: a malfunction of the device itself, a scale that does not correspond to the zero value, or the accuracy class of the device;
3) errors, the existence of which may not be suspected, but their magnitude can often be significant. Such errors occur most often in complex measurements. A simple example of such an error is the measurement of the density of some sample containing a cavity inside;
4) errors caused by the characteristics of the measurement object itself. For example, when measuring the electrical conductivity of a metal, a piece of wire is taken from the latter. Errors can occur if there is any defect in the material - a crack, thickening of the wire or inhomogeneity that changes its resistance.
Random errors are errors that change randomly in sign and magnitude under identical conditions of repeated measurements of the same quantity.
Related information.
