Approximation error. Relative error of the approximate number

measurement result

The error of the measurement result allows you to determine those figures of the result that are reliable. When calculating the error value, especially with the help of calculators, the error value is obtained from a large number signs. This gives the impression of a high measurement accuracy, which is not true, since the initial data for the calculation are most often the normalized error values ​​of the SI used, which are indicated with only one or two significant figures. As a result, the final value of the calculated error should not contain more than two significant figures. Metrology has the following rules:

1. The error of the measurement result is indicated by two significant digits if the first of them is 3 or less, and one - if the first digit is 4 or more.

Significant digits of a number are considered to be all digits from the first digit on the left, not equal to zero, to the last digit on the right, while zeros written as a factor of 10 n are not taken into account.

2. The measurement result is rounded to the same decimal place that ends the rounded absolute error value. (For example, the result is 85.6342, the error is 0.01. The result is rounded to 85.63. The same result with an error within 0.012 should be rounded to 85.634).

3. Rounding is done only in the final answer, and all preliminary calculations are carried out with one or two extra signs.

4. Rounding should be performed immediately to the desired number of significant digits, gradual rounding leads to errors.

When rounding the numerical values ​​of the error and the measurement result, the following general rounding rules should be followed.

Extra digits in integers are replaced by zeros, and in decimal fractions they are discarded. (For example, the number 165245 is rounded to 165200 while maintaining four significant digits, and the number 165.245 is rounded to 165.2).

If the decimal ends in zeros, they are discarded only up to the digit that corresponds to the digit of the error. (For example, the measurement result is 235.200, the error is 0.05. The result is rounded to 235.20. The same result with an error within 0.015 should be rounded to 235.200).

If the first (counting from left to right) of the zero-replaced or discarded digits is less than 5, the remaining digits do not change .

If the first of these digits is 5 and is followed by no digits or zeros, then if the last digit in the rounded number is even or zero, it remains unchanged , if odd - increases by one . (For example, the number 1234.50 is rounded up to 1234, and the number 8765.50 is rounded up to 8766).

If the first of the zero-replaced or discarded digits is greater than or equal to 5, but followed by a significant digit, then the last remaining digit is increased by one . (For example, the number 6783.6, while maintaining four significant figures, is rounded up to 6784, and the number 12.34520 is rounded up to 12.35).

Particular attention should be paid to recording the measurement result without specifying the error, since the results of 2.4 10 3 V and 2400V are not identical . The first entry means that the numbers of thousands and hundreds of volts are correct and the true value can be in the range from 2.351kV to 2.449kV. The entry 2400 means that the units of volts are also correct, therefore the true voltage value can be in the range from 2399.51V to 2400.49V.

Therefore, recording the result without specifying the error highly undesirable .

Finally, the rules for recording the measurement result can be formulated as follows.

1) In intermediate calculations, the error values ​​​​save three to four significant digits.

2) The final error value and the result value are rounded according to the above rules.

3) With single technical measurements, when only the main error of the MI is taken into account (MI are used under normal operating conditions), the result is written as:

(For example, voltage measurement result
B, error
B. The result can be written as:

4) For single technical measurements under operating conditions, when the main and additional errors are taken into account according to the standard data on the SI and the resulting error is determined by formula (1.35), the result is written as:

5) In statistical measurements, when only the magnitude of the random error of normally distributed data is determined in the form of a confidence interval, the result is written in accordance with (1.31):

If the boundaries of the confidence interval are not symmetrical, then they are indicated separately.

For example,

6) In statistical measurements, when the bounds of non-excluded systematic errors of the result (NSP) and the confidence interval of the random error of normally distributed data are estimated, but the result is used as an intermediate to find other values ​​(for example, with statistical indirect measurements) or it is supposed to be compared with other results of a similar measurement experiment, the result is written in accordance with (1.39):

if
, then this is indicated additionally, as in paragraph 5.

If the boundaries of the NSP or the boundaries of the confidence interval are not symmetrical, then they are indicated separately:

7) If during the measurement, estimates of the error are obtained under the conditions specified in paragraph 6, but the result is final and it is not expected to be further analyzed and compared with other results, then it is written in accordance with (1.41):

where
is determined by formula (1.40),

if
, this is indicated additionally, as in paragraph 5.

8) During statistical measurements, when the boundaries of the NSP and the confidence interval of random error are estimated, but when processing the results, a distribution law other than normal is identified, estimates of the value of the measurement result and the confidence interval of random error are found according to the corresponding formulas, the result is presented in the form similar to the presentation of the result in 6, but additional information is provided on the form of the experimental data distribution law.

9) If, as in paragraph 8, the results of static measurements are processed and it is known in advance that the law of distribution of experimental data differs from the normal one, but no actions are taken to identify the type of real law for any reason, then the result can be presented in a form similar to the representation result in paragraph 6, but the confidence interval of the random error is determined in accordance with the recommendations of GOST 11.001-73 as
with a confidence level
.

The result entry might look like this:

(at
);
;
;
.

Confidence probability at which the total NSP is determined -
, in which case it may differ from
.

When calculating the values ​​of systematic, random and total errors, especially when using an electronic calculator, a value with a large number of digits is obtained. However, the input data for these calculations is always given with one or two significant figures. Indeed, the accuracy class of the device on its scale is indicated with no more than two significant figures, and it makes no sense to write the standard deviation with more than two significant figures, since the accuracy of this assessment with 10 measurements is not higher than 30%. As a result, only the first one or two significant figures should be left in the final value of the calculated error. In doing so, the following must be taken into account. If the resulting number starts with the number 1 or 2, then discarding the second character leads to a very large error (up to 30–50%), this is unacceptable. If the resulting number starts, for example, with the number 9, then the preservation of the second sign, that is, the indication of the error, for example, 0.94 instead of 0.9, is misinformation, since the original data does not provide such accuracy.

As a result, one can formulate rounding rules the calculated value of the error and the obtained experimental measurement result:

1. The absolute error of the measurement result is indicated by two significant figures if the first of them is equal to 1 or 2, and one, if the first is 3 or more.

2. The mean value of the measured value is rounded to the same decimal place as the rounded value of the absolute error ends.

3. The relative error, expressed as a percentage, is enough to write down in two significant figures.

4. Rounding is performed only in the final answer, and all preliminary calculations are carried out with one extra sign.

Example:
On a voltmeter of accuracy class 2,5 with measurement limit 300 V several repeated measurements of the same voltage were made. It turned out that all measurements gave the same result. 267.5 V.

The absence of differences between the signs indicates that the random error is negligible, so the total error coincides with the systematic one (see Fig. 1a).

First we find the absolute, and then the relative error. The absolute error of the calibration of the device is equal to:

Since the first significant figure of the absolute error is greater than three, this value should be rounded up to 8 V. Relative error:

Two significant digits must be stored in the relative error value: 2,8 %.

Thus, the final response should report “Measured voltage U=(268+8) V with relative error dU=2,8 % ”.

When performing calculations, it often becomes necessary to round numbers, i.e. in replacing them with numbers with fewer significant digits.

There are three ways to round numbers:

Rounding down to k th significant digit is to discard all digits starting with (k+1) th.

Rounding up differs from rounding down in that the last digit stored is increased by one.

Rounding with the smallest error differs from rounding with an excess in that the increase by one of the last digit stored is performed only if the first of the discarded digits is greater than 4.

Exception: if the rounding with the smallest error is to discard only one digit 5, then the last digit stored is not changed if it is even, and increased by 1 if it is odd.

From the above rules for rounding approximate numbers, it follows that the error caused by rounding with the smallest error does not exceed half the unit of the last stored digit, and when rounding with a deficiency or excess, the error may be more than half the unit of the last stored digit, but not more than a whole unit of this discharge.

Let's look at this in the following examples.

1. The error of the amount. Let x a, at-- some approximation of the value b. Let X and at-- absolute errors of the corresponding approximations X and at. Let's find the absolute error limit h a+b amounts x+y, which is an approximation of the sum a+b.

a = x + x,

b = y + y.

Adding these two equalities, we get

a + b = x + y + x + y.

Obviously, the error of the sum of approximations x and at is equal to the sum of the errors of the terms, i.e.

(x + y) = x + y

It is known that the modulus of the sum is less than or equal to the sum of the moduli of the terms. That's why

(x + y) = x + y x + y

This implies that the absolute error of the sum of approximations does not exceed the sum of the absolute errors of the terms. Therefore, the sum of the limits of the absolute errors of the terms can be taken as the limit of the absolute error of the sum.

Denoting the boundary of the absolute error of the quantity a through h a, and b through h b will have

h a+b = h a + h b

2. Difference error. Let x and y be the errors in the approximations x and y of a and b, respectively.

a = x + x,

b = y + y.

Subtract the second from the first equality, we get

a - b = (x - y) + (x - y)

Obviously, the error of the difference of approximations is equal to the difference between the errors of the minuend and the subtrahend, i.e.

(x - y) = x - y),

(x - y) = x + (-y)

And then, arguing in the same way as in the case of addition, we have

(x - y) = x + (-y) x + y

It follows that the absolute error of the difference does not exceed the sum of the absolute errors of the minuend and the subtrahend.

For the boundary of the absolute error of the difference, you can take the sum of the boundaries of the absolute errors of the reduced and the subtrahend. In this way.

h a-b = h a +h b (9)

From formula (9) it follows that the limit of the absolute error of the difference cannot be less than the limit of the absolute error of each approximation. This implies the rule for subtracting approximations, which is sometimes used in calculations.

When subtracting numbers that are approximations of some quantities, as a result, you should leave as many digits after the decimal point as the approximation with the smallest number digits after the decimal point.

3. Product error. Consider the product of numbers X and at, which are approximations of the quantities a and b. Denote by x approximation error X, and through at-- approximation error at,

a = x + x,

b = y + y.

Multiplying these two equalities, we get

Absolute product error hu is equal to

And therefore

Dividing both sides of the resulting inequality by hu, we get

Taking into account that the modulus of the product is equal to the product of the moduli of the factors, we will have

Here the left side of the inequality is the relative error of the product hu, -- relative approximation error X, and is the relative approximation error at. Therefore, discarding the small value here, we obtain the inequality

Thus, the relative error of the product of approximations does not exceed the sum of the relative errors of the factors. It follows that the sum of the limits of the relative errors of the factors is the limit of the relative error of the product, i.e.

E ab = E a + E b (10)

From formula (10) it follows that the limit of the relative error of the product cannot be less than the limit of the relative error of the least accurate of the factors. Therefore, here, as in the previous steps, it makes no sense to keep an excessive number of significant digits in the factors.

It is sometimes useful to use the following rule in calculations to reduce the amount of work: When multiplying approximations with different numbers of significant digits, the result should retain as many significant digits as the approximation with the smallest number of significant digits has.

4. The error of the quotient. If x is an approximation of a with error x, and y is an approximation of b with error y, then

Let us first calculate the absolute error of the quotient:

and then the relative error:

Taking into account that y little compared to y, the absolute value of the fraction can be considered equal to one. Then

it follows from the last formula that the relative error of the quotient does not exceed the sum of the relative errors of the dividend and divisor. Therefore, we can assume that the limit of the relative error of the quotient is equal to the sum of the limits of the relative errors of the dividend and divisor, i.e.

5. Error of degree and root. 1) Let u = a n, where n is a natural number, and let a x. Then if E a-- boundary of the relative approximation error x quantities a, then

and therefore

Thus, the limit of the relative error of the degree is equal to the product of the limit of the relative error of the base by the exponent, i.e.

E u = nE a (11)

2) Let where n is a natural number, and let Oh.

By formula (11)

and hence

error deductible calculation

Thus, the boundary of the relative error of the root n th degree in n times less than the limit of the relative error of the root number.

6. Inverse problem of approximate calculations. In the direct problem, it is required to find the approximate value of the function u=f(x, y, ..., n) according to the given approximate values ​​of the arguments

and margin of error h a, which is expressed in terms of the errors of the arguments of some function

h u = (h x , h y , …, h z ) (12)

In practice, it is often necessary to solve an inverse problem, in which it is required to find out with what accuracy the values ​​of the arguments should be given x, y, …, z to compute the corresponding function values u = f(x, y, …, z) with predetermined accuracy h u .

Thus, when solving the inverse problem, the sought are the error limits of the arguments associated with the given error limit of the function h u equation (12), and the solution of the inverse problem is reduced to compiling and solving the equation h u = (h x , h y , …, h z ) relatively h x , h y , …, h z. Such an equation either has an infinite number of solutions, or has no solutions at all. The problem is considered solved if at least one solution of such an equation is found.

To solve the inverse problem, which is often indeterminate, one has to introduce additional conditions on the ratios of the unknown errors, for example, consider them equal and thereby reduce the problem to an equation with one unknown.

Dealing in calculations with infinite decimals, it is necessary, for convenience, to approximate these numbers, i.e., round them. Approximate numbers are also obtained from various measurements.

It can be useful to know how much the approximate value of a number differs from its exact value. It is clear that the smaller this difference, the better, the more accurately the measurement or calculation is performed.

To determine the accuracy of measurements (calculations), such a concept is introduced as approximation error. They call it differently absolute error. The approximation error is the modulo difference between the exact value of a number and its approximate value.

If a is the exact value of a number, and b is its approximate value, then the approximation error is determined by the formula |a – b|.

Let's assume that as a result of measurements the number 1.5 was obtained. However, as a result of the calculation by the formula, the exact value of this number is 1.552. In this case, the approximation error will be equal to |1.552 – 1.5| = 0.052.

In the case of infinite fractions, the approximation error is determined by the same formula. In place of the exact number, the infinite fraction itself is written. For example, |π – 3.14| = |3.14159... – 3.14| = 0.00159... . Here it turns out that the approximation error is expressed by an irrational number.

As is known, the approximation can be performed both in terms of deficiency and excess. The same number π, when approaching the deficiency with an accuracy of 0.01, is 3.14, and when approaching the excess with an accuracy of 0.01, it is 3.15. The reason for using its deficiency approximation in the calculations is to apply rounding rules. According to these rules, if the first digit to be discarded is five or greater than five, then an excess approximation is performed. If less than five, then by deficiency. Since the third digit after the decimal point of the number π is 1, therefore, when approaching with an accuracy of 0.01, it is performed by deficiency.

Indeed, if we calculate the approximation errors up to 0.01 of the number π in terms of deficiency and excess, we get:

|3,14159... – 3,14| = 0,00159...
|3,14159... – 3,15| = 0,0084...

Since 0.00159...

Speaking about the approximation error, as well as in the case of the approximation itself (by excess or deficiency), indicate its accuracy. So in the above example with the number π, it should be said that it is equal to the number 3.14 with an accuracy of 0.01. After all, the modulus of the difference between the number itself and its approximate value does not exceed 0.01 (0.00159... ≤ 0.01).

Similarly, π is equal to 3.15 up to 0.01, because 0.0084... ≤ 0.01. However, if we talk about greater accuracy, for example, up to 0.005, then we can say that π is equal to 3.14 with an accuracy of 0.005 (since 0.00159 ... ≤ 0.005). We cannot say this in relation to the approximation of 3.15 (since 0.0084 ... > 0.005).

Absolute and relative error of the number.

As characteristics of the accuracy of approximate quantities of any origin, the concepts of absolute and relative errors of these quantities are introduced.

Denote by a the approximation to the exact number A.

Define. The value is called the error of the approximate numbera.

Definition. Absolute error approximate number a is called the value
.

In practice, the exact number A is usually unknown, but we can always indicate the limits in which the absolute error changes.

Definition. Limit absolute error approximate number a is the smallest of the upper bounds for the quantity , which can be found with this method of obtaining the number a.

In practice, as choose one of the upper bounds for , close enough to the smallest.

Because the
, then
. Sometimes they write:
.

Absolute error is the difference between the measurement result

and true (real) value measured value.

The absolute error and the limiting absolute error are not sufficient to characterize the accuracy of a measurement or calculation. The magnitude of the relative error is qualitatively more significant.

Definition. Relative error approximate number a let's call the value:

Definition. Limiting relative error approximate number a we call the value

Because
.

Thus, the relative error actually determines the magnitude of the absolute error per unit of the measured or calculated approximate number a.

Example. Rounding the exact numbers A to three significant figures, determine

absolute D and relative δ errors of the obtained approximate

Given:

Find:

∆-absolute error

δ - relative error

Solution:

=|-13.327-(-13.3)|=0.027

,a 0

*100%=0.203%

Answer:=0.027; δ=0.203%

2. Decimal notation of an approximate number. Significant digit. True signs of a number (definition of true and significant figures, examples; theory about the relationship between relative error and the number of correct signs).

Correct signs of the number.

Definition. A significant digit of an approximate number a is any digit other than zero, and zero if it is between significant digits or is a representative of a stored decimal place.

For example, in the number 0.00507 =
we have 3 significant digits, and in the number 0.005070=
significant digits, i.e. zero on the right, keeping the decimal place, is significant.

Let us agree henceforth to write zeros on the right, if only they are significant. Then, in other words,

all digits of the number a are significant, except for zeros on the left.

In the decimal number system, any number a can be represented as a finite or infinite sum (decimal fraction):

where
,
- the first significant digit, m - an integer, called the most significant decimal place of the number a.

For example, 518.3 =, m=2.

Using the notation, we introduce the concept of correct decimal places (in significant figures) approximately

th number.

Definition. They say that in an approximate number a of the form n - the first significant digits ,

where i= m, m-1,..., m-n+1 are true if the absolute error of this number does not exceed half the unit of the digit expressed by the n-th significant digit:

Otherwise, the last digit
called doubtful.

When writing an approximate number without indicating its error, it is required that all recorded numbers

were true. This requirement is met in all mathematical tables.

The term “n correct signs” characterizes only the degree of accuracy of the approximate number and should not be understood in such a way that the n first significant digits of the approximate number a coincide with the corresponding digits of the exact number A. For example, for the numbers A = 10, a = 9.997, all significant digits are different , but the number a has 3 valid significant digits. Indeed, here m=0 and n=3 (find by selection).