Errors in indirect measurements are the addition rule for them. Calculation of errors of indirect measurements
The result of direct measurement is not the true value X measured quantity, and a series of n values 
. Let it now
Summing up the last equality, we get
(7)
Where 
arithmetic mean of measured values 
. Thus,
(8)
Very important consequences follow from this simple result. Indeed, when 
And 
.
This means that for an infinitely large number of dimensions 
and, therefore, at finite n the greater the number of measurements, the closer the result is to the arithmetic mean. It also follows that when assessing ∆
X as 
it is advisable to take 
.
On practice n of course and 
. The task of the mathematical theory of random error includes estimating the interval
which contains the true value of the measured quantity. Interval (9) is called confidence interval, and the value 
–absolute error of the result of a series of measurements. Evaluation theory ∆
X is quite complex, so only its main results will be considered here. First of all, it should be noted that, since X– random variable, error ∆
X can only be determined to one degree or another reliability
α
, which is also called confidence probability. Confidence probability is the probability that the true value of the measured quantity X falls within the confidence interval (9). If you put α
=1 (100%), then this will correspond to a reliable event, i.e. the likelihood that X takes on some value in the interval ( 
). Wherein 
. Obviously, this choice of reliability α
inappropriate. At small α
confidence interval ∆
X determined with low reliability. In what follows we will assume α
=0.90 or 0.95. Confidence interval and reliability are interrelated. To estimate the boundaries of the confidence interval, the English mathematician W. Gosset (who published his works under the pseudonym Student) introduced the coefficient in 1908:
(10)
equal to the error ratio ∆ X to mean square error*
(11)
Coefficient 
depends on reliability α
, as well as on the number of measurements n and is called Student's coefficient. This coefficient is tabulated (see Appendix 1), therefore, calculating 
and setting the confidence probability α
, it's not hard to find a random error:
(12)
Calculation of the error of indirect measurements.
In indirect measurements, the measured quantity f is found from the functional dependency:
Where x, y, z– results of direct measurements. Formula for ∆ f can be obtained by replacing the differentials in (2) with errors and taking all terms modulo
(13)
Relation (13) is recommended for estimating the error ∆ f, caused by instrument errors of the value x, y, z,... To estimate the error associated with random errors in direct measurements, the following ratio is recommended:
(14)
It should really be noted that formulas (13) and (14) lead to almost identical results. The derivatives in (13) and (14) are taken at averages, i.e. at the measured values of the arguments.
Very often function f represented by a power dependence on the arguments
(15)
where c, n, m and p are constants. A special case of formula (15) are the relations 
,
and etc.
Exercise. Show that for a function of the form (15), formulas (13) and (14) take the form:
(13)
(14)
From relations (13) and (14) it follows that for power functions the calculation of errors is significantly simplified, and it is advisable to first find the relative error, which is expressed through the relative error of direct measurements, and then find the absolute error
(16)
Under 
is understood as a function of the average (measured) values of the arguments
.
Algorithm for calculating errors
- For direct measurements
1. Calculate the arithmetic mean of the results 
series of n measurements:
Comment: when calculating 
It’s more convenient to start from the formula:
Where 
- any convenient value close to 
.
2. Find deviations of individual measurements from the average value
Comment. At
can be put
and count 
according to the formula
5. If
,
then the random error can not be counted.
6. Otherwise, set the confidence probability 
and find the Student coefficient from the table
.
Note 1. If the instrument error 
has the same order of magnitude as 
, then the absolute error of the result of a series of measurements is found by the formula:
Where 
Almost in quality 
you can take the table value 
corresponding to the largest of the values given in it P(For example, n=500
)
.
Note 2. For a large number of measurements 
can be put
Where 
.
8. Present the measurement result in the form:
- For indirect measurements
Error 
indirect measurement can be calculated using one of the formulas (13), (14), (13*), (14*). The last two formulas are valid for power-law dependences, and relations (13) and (14) are of a general nature.
Summary of relationships for calculating indirect measurement error 
for some simple functional dependencies is presented in the table.
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Formulas for calculating errors |
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Example. Let the Joule heat Q be calculated by the formula
Since this is a power-law dependence, it is advisable to use formula (13*)
Rules for presenting measurement results and their errors
Uncertainties can only be estimated, so it is usually sufficient to indicate the uncertainty to one significant figure. For example, Δm=0.2 g. 
d. Recording T
= 3.0 g means that the measurement was made with an accuracy of tenths of a gram. However, in intermediate calculations it is advisable to leave more significant figures.
The rules for rounding numbers (measurement results) are illustrated in the table (pay attention to the features of rounding the number 5).
Table Rounding to tenths of significant figures
It is customary to round the measurement result so that the numerical value ends with a digit of the same digit as the error value. For example, record
cm.
unacceptable, because The very value of the error Δl = 0.1 cm indicates that the 018 numbers of the result cannot be guaranteed. You need to write it like this:
cm.
In most cases, the final goal of laboratory work is to calculate the desired quantity using some formula that includes directly measured quantities. Such measurements are called indirect. As an example, we give the formula for the density of a cylindrical solid body
where r is the density of the body, m- body mass, d– cylinder diameter, h- his high.
Dependence (A.5) in general can be represented as follows:
Where Y– indirectly measured quantity, in formula (A.5) this is density r; X 1 , X 2 ,... ,X n– directly measured quantities, in formula (A.5) these are m, d, And h.
The result of an indirect measurement cannot be accurate, since the results of direct measurements of quantities X 1 , X 2, ... ,X n always contain an error. Therefore, with indirect measurements, as with direct ones, it is necessary to estimate the confidence interval (absolute error) of the obtained value DY and relative error e.
When calculating errors in the case of indirect measurements, it is convenient to follow the following sequence of actions:
1) obtain the average values of each directly measured quantity b X 1ñ, á X 2ñ, …, á X nñ;
2) obtain the average value of the indirectly measured quantity b Yñ by substituting the average values of directly measured quantities into formula (A.6);
3) estimate the absolute errors of directly measured quantities DX 1 , DX 2 , ..., DXn, using formulas (A.2) and (A.3);
4) based on the explicit form of the function (A.6), obtain a formula for calculating the absolute error of an indirectly measured value DY and calculate it;
6) write down the measurement result taking into account the error.
Below, without derivation, is a formula that allows one to obtain formulas for calculating the absolute error if the explicit form of the function (A.6) is known:
where ¶Y¤¶ X 1 etc. – partial derivatives of Y with respect to all directly measurable quantities X 1 , X 2 , …, X n (when the partial derivative is taken, for example with respect to X 1, then all other quantities X i in the formula are considered constant), D X i– absolute errors of directly measured quantities, calculated according to (A.3).
Having calculated DY, find the relative error.
However, if function (A.6) is a monomial, then it is much easier to first calculate the relative error, and then the absolute one.
Indeed, dividing both sides of equality (A.7) into Y, we get
But since , we can write
Now, knowing the relative error, determine the absolute one.
As an example, we obtain a formula for calculating the error in the density of a substance, determined by formula (A.5). Since (A.5) is a monomial, then, as stated above, it is easier to first calculate the relative measurement error using (A.8). In (A.8) under the root we have the sum of squared partial derivatives of logarithm measured quantity, so first we find the natural logarithm of r:
ln r = ln 4 + ln m– ln p –2 ln d–ln h,
and then we will use formula (A.8) and obtain that
As can be seen, in (A.9) the average values of directly measured quantities and their absolute errors, calculated by the method of direct measurements according to (A.3), are used. The error introduced by the number p is not taken into account, since its value can always be taken with an accuracy exceeding the accuracy of measurement of all other quantities. Having calculated e, we find .
If indirect measurements are independent (the conditions of each subsequent experiment differ from the conditions of the previous one), then the values of the quantity Y are calculated for each individual experiment. Having produced n experiences, get n values Y i. Next, taking each of the values Y i(Where i– experiment number) for the result of direct measurement, calculate á Yñ and D Y according to formulas (A.1) and (A.2), respectively.
The final result of both direct and indirect measurements should look like this:
Where m– exponent, u– units of measurement of quantity Y.
If the desired physical quantity cannot be measured directly by the device, but is expressed through the measured quantities using a formula, then such measurements are called indirect.
As with direct measurements, you can calculate the mean absolute (arithmetic mean) error or the mean square error of indirect measurements.
General rules for calculating errors for both cases are derived using differential calculus.
Let the physical quantity j( x, y, z, ...) is a function of a number of independent arguments x, y, z, ..., each of which can be determined experimentally. By direct measurements, quantities are determined and their average absolute errors or root mean square errors are estimated.
The average absolute error of indirect measurements of the physical quantity j is calculated using the formula
where are the partial derivatives of φ with respect to x, y, z, calculated for the average values of the corresponding arguments.
Since the formula uses the absolute values of all terms of the sum, the expression for estimates the maximum error in measuring the function for given maximum errors of the independent variables.
Mean square error of indirect measurements of physical quantity j
Relative maximum error of indirect measurements of physical quantity j
where, etc.
Similarly, we can write the relative root mean square error of indirect measurements j
If the formula represents an expression convenient for logarithmization (that is, a product, fraction, power), then it is more convenient to first calculate the relative error. To do this (in the case of average absolute error), you need to do the following.
1. Take the logarithm of the expression for the indirect measurement of a physical quantity.
2. Differentiate it.
3. Combine all terms with the same differential and put it out of brackets.
4. Take the expression in front of various modulo differentials.
5. Formally replace the differential symbols with absolute error symbols D.
Then, knowing e, you can calculate the absolute error Dj using the formula
Example 1. Derivation of a formula for calculating the maximum relative error of indirect measurements of cylinder volume.
Expression for indirect measurement of a physical quantity (original formula)
Diameter size D and cylinder height h measured directly by instruments with direct measurement errors, respectivelyD D and D h.
Let's take the logarithm of the original formula and get
Let us differentiate the resulting equation
Replacing the differential symbols with the absolute error symbols D, we finally obtain a formula for calculating the maximum relative error of indirect measurements of cylinder volume
Let us first consider the case when the quantity at depends on only one variable X, which is found by direct measurement,
Average<y> can be found by substituting in (8) X average<X>.
.
The absolute error can be considered as the increment of function (8) with the increment of argument ∆ X(total error of the measured value X). For small values of ∆ X it is approximately equal to the differential of the function
, (9)
where is the derivative of the function calculated at . The relative error will be equal to
.
Let the quantity being determined at is a function of several variables x i,
. (10)
It is assumed that the errors of all quantities in the working formula are random, independent and calculated with the same confidence probability (for example R= 0.95). The error of the desired value will have the same confidence probability. In this case, the most probable value of the quantity<at> determined by formula (10), using the most probable values of quantities for calculation X i, i.e. their average values:
<at> = f(<x 1 >, <x 2 >, …,<x i >, …,<x m >).
In this case, the absolute error of the final result Δ at determined by the formula
, (11)
where ∂ at/∂X i – partial derivatives of the function at by argument X i , calculated for the most probable values of quantities X i. The partial derivative is the derivative that is calculated from the function at by argument X i provided that all other arguments are considered constant.
Relative error of value at we get by dividing ∆ at on<y>
. (12)
Taking into account that (1/ at) dy/dx represents the derivative with respect to X from natural logarithm at the relative error can be written as follows
. (13)
Formula (12) is more convenient to use in cases where, depending on (10), the measured quantities x i are included mainly in the form of terms, and formula (13) is convenient for calculations when (10) is a product of quantities X i. In the latter case, preliminary logarithm of expression (10) significantly simplifies the form of partial derivatives. Measured quantity at is a dimensional quantity and it is impossible to logarithm a dimensional quantity. To eliminate this incorrectness, you need to separate at to a constant having a given dimension. After logarithmization, you get an additional term that does not depend on the quantities X i and therefore will disappear when taking partial derivatives, since the derivative of a constant value is equal to zero. Therefore, when taking logarithms, the presence of such a term is simply assumed.
Considering the simple relationship between absolute and relative errors ε y = Δ at/<at>, easily based on the known value Δ at calculate ε y and vice versa.
The functional relationship between the errors of direct measurements and the error of indirect measurements for some simple cases is given in Table. 3.
Let us consider some special cases that arise when calculating measurement errors. The above formulas for calculating errors in indirect measurements are valid only when all X i are independent quantities and are measured by various instruments and methods. In practice, this condition is not always met. For example, if any physical quantities in dependence (10) are measured by the same device, then the instrument errors Δ X i pr of these quantities will no longer be independent, and the instrumental error of the indirectly measured quantity Δ at pr in this case it will be slightly larger than with “quadratic summation”. For example, if the area of the plate is length l and width b measured with one caliper, then the relative instrument error of indirect measurement will be
(ΔS/S) pr = (Δ l/l) pr + ( Δb/b) etc,
those. errors are summed up arithmetically (errors Δ l at Δb of the same sign and their values are the same), instead of the relative instrumental error
with independent errors.
Table 3
Functional connection between errors of direct and indirect measurements
| Working formula | Formula for calculating error |
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When carrying out measurements, there may be cases when the values X i have different values that are specially changed or specified during the experiment, for example, the viscosity of a liquid using the Poiseuille method is determined for different heights of the liquid column above the capillary, or the acceleration of gravity g is determined using a mathematical pendulum for different lengths). In such cases, the value of the indirectly measured quantity should be calculated at in each of the n experiments separately, and take the average value as the most probable value, i.e. 
. Random error Δ at sl calculated as the error in direct measurement. Calculation of instrument error Δ at pr is produced through partial derivatives using formula (11), and the final total error of the indirectly measured value is calculated using the formula
Now it is necessary to consider the question of how to find the error of a physical quantity U, which is determined by indirect measurements. General view of the measurement equation
Y=f(X 1 , X 2 , … , Xn), (1.4)
Where X j– various physical quantities that are obtained by the experimenter through direct measurements, or physical constants known with a given accuracy. In a formula, they are function arguments.
In measurement practice, two methods of calculating the error of indirect measurements are widely used. Both methods give almost the same result.
Method 1. The absolute D is found first, and then the relative one d errors. This method is recommended for measurement equations that contain sums and differences of arguments.
General formula for calculating the absolute error in indirect measurements of a physical quantity Y for any type f functions has the form:
where are the partial derivatives of the function Y=f(X 1 , X 2 , … , Xn) by argument X j,
General error of direct measurements of quantity X j.
To find the relative error, you must first find the average value of the quantity Y. To do this, it is necessary to substitute the arithmetic average values of the quantities into the measurement equation (1.4) X j.
That is, the average value Y equals: . Now it is easy to find the relative error: .
Example: find the error in volume measurement V cylinder. Height h and diameter D cylinder we consider determined by direct measurements, and let the number of measurements n= 10.
The formula for calculating the volume of a cylinder, that is, the measurement equation has the form:
Let at P= 0,68;
At P= 0,68.
Then, substituting the average values into formula (1.5), we find:
Error D V in this example it depends, as can be seen, mainly on the error in measuring the diameter.
The average volume is equal to: , relative error d V is equal to:
Or d V = 19%.
V=(47±9) mm 3 , d V = 19%, P= 0,68.
Method 2. This method of determining the error of indirect measurements differs from the first method in that it has fewer mathematical difficulties, which is why it is used more often.
First, find the relative error d, and only then absolute D. This method is especially convenient if the measurement equation contains only products and ratios of arguments.
The procedure can be considered using the same specific example - determining the error when measuring the volume of a cylinder
Let us keep all the numerical values of the quantities included in the formula the same as in the calculations using method 1.
Let mm, ; at P= 0,68;
; at P=0.68.
Number rounding error p(see Fig. 1.1)
Using method 2 you should do this:
1) take the logarithm of the measurement equation (take the natural logarithm)
find the differentials of the left and right sides, considering independent variables,
2) replace the differential of each value with the absolute error of the same value, and the “minus” signs, if they are in front of the errors, with “plus”:
3) it would seem that using this formula it is already possible to give an estimate for the relative error, but this is not so. It is required to estimate the error in such a way that the confidence probability of this estimate coincides with the confidence probabilities of estimating the errors of those terms that appear on the right side of the formula. To do this, for this condition to be met, you need to square all the terms of the last formula, and then take the square root of both sides of the equation:
Or in other notations, the relative volume error is equal to:
Moreover, the probability of this estimate of the volume error will coincide with the probability of estimating the errors of the terms included in the radical expression:
Having made the calculations, we will make sure that the result coincides with the estimate according to method 1:
Now, knowing the relative error, we find the absolute one:
D V=0.19 47=9.4 mm 3 , P=0,68.
Final result after rounding:
V= (47 ± 9) mm 3, d V = 19%, P=0,68.
Control questions
1. What is the task of physical measurements?
2. What types of measurements are distinguished?
3. How are measurement errors classified?
4. What are absolute and relative errors?
5. What are misses, systematic and random errors?
6. How to evaluate systematic error?
7. What is the arithmetic mean of a measured value?
8. How to estimate the magnitude of the random error, how is it related to the standard deviation?
9. What is the probability of detecting the true value of the measured value in the range from X av - s before X av + s?
10. If we choose the value as an estimate for the random error 2s or 3s, then with what probability will the true value fall within the intervals defined by these estimates?
11. How to summarize errors and when should this be done?
12. How to round the absolute error and the average value of the measurement result?
13. What methods exist for assessing errors in indirect measurements? How to proceed with this?
14. What should be recorded as the measurement result? What values should I indicate?